29687

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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 )}


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ultimately show ?thesis by blast


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qed


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lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0))"


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proof


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let ?S = "{n. f$n \<noteq> 0}"


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{assume "\<exists>n. f$n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0)" and "f = 0"


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then have False by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}


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moreover


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{assume f0: "f \<noteq> 0"


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from f0 fps_nonzero_nth have ex: "\<exists>n. f$n \<noteq> 0" by blast


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hence Se: "?S\<noteq> {}" by blast


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from ex obtain n where n: "f$n \<noteq> 0" by blast


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from n have nS: "n \<in> ?S" by blast


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let ?U = "?S \<inter> {0..n}"


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have fU: "finite ?U" by auto


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from n have Ue: "?U \<noteq> {}" by auto


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let ?m = "Min ?U"


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have mU: "?m \<in> ?U" using Min_in[OF fU Ue] .


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hence mn: "?m \<le> n" by simp


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from mU have mf: "f $ ?m \<noteq> 0" by blast


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{fix m assume m: "m < ?m" and f: "f $m \<noteq> 0"


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from m mn have mn': "m < n" by arith


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with f have mU': "m \<in> ?U" by simp


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from Min_le[OF fU mU'] m have False by arith}


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hence "\<forall>m <?m. f$m = 0" by blast


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with mf have "\<exists> n. f $n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0)" by blast}


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ultimately show ?thesis by blast


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qed


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


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by (auto simp add: fps_nth_def Rep_fps_eq[unfolded expand_fun_eq])


351 


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lemma fps_setsum_nth: "(setsum f S) $ n = setsum (\<lambda>k. (f k) $ n) S"


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proof


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


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moreover


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{assume fS: "finite S"


357 
have ?thesis by(induct rule: finite_induct[OF fS]) auto}


358 
ultimately show ?thesis by blast


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qed


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section{* Injection of the basic ring elements and multiplication by scalars *}


362 


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


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lemma fps_const_0_eq_0[simp]: "fps_const 0 = 0" by (simp add: fps_const_def fps_eq_iff)


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lemma fps_const_1_eq_1[simp]: "fps_const 1 = 1" by (simp add: fps_const_def fps_eq_iff)


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lemma fps_const_neg[simp]: " (fps_const (c::'a::ring)) = fps_const ( c)"


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by (simp add: fps_uminus_def fps_const_def fps_eq_iff)


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lemma fps_const_add[simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)"


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by (simp add: fps_plus_def fps_const_def fps_eq_iff)


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lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)"


371 
by (auto simp add: fps_times_def fps_const_def fps_eq_iff intro: setsum_0')


372 


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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)"


374 
unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)


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)"


376 
unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)


377 


378 
lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)"


379 
unfolding fps_eq_iff fps_mult_nth


380 
by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)


381 
lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f$n * c)"


382 
unfolding fps_eq_iff fps_mult_nth


383 
by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)


384 


385 
lemma fps_const_nth[simp]: "(fps_const c) $n = (if n = 0 then c else 0)"


386 
by (simp add: fps_const_def)


387 


388 
lemma fps_mult_left_const_nth[simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n"


389 
by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)


390 


391 
lemma fps_mult_right_const_nth[simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c"


392 
by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)


393 


394 
section {* Formal power series form an integral domain*}


395 


396 
instantiation fps :: (ring_1) ring_1


397 
begin


398 


399 
instance by (intro_classes, auto simp add: diff_minus left_distrib)


400 
end


401 


402 
instantiation fps :: (comm_ring_1) comm_ring_1


403 
begin


404 


405 
instance by (intro_classes, auto simp add: diff_minus left_distrib)


406 
end


407 
instantiation fps :: ("{ring_no_zero_divisors, comm_ring_1}") ring_no_zero_divisors


408 
begin


409 


410 
instance


411 
proof


412 
fix a b :: "'a fps"


413 
assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"


414 
then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0"


415 
and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0" unfolding fps_nonzero_nth_minimal


416 
by blast+


417 
have eq: "({0..i+j} {i}) \<union> {i} = {0..i+j}" by auto


418 
have d: "({0..i+j} {i}) \<inter> {i} = {}" by auto


419 
have f: "finite ({0..i+j} {i})" "finite {i}" by auto


420 
have th0: "setsum (\<lambda>k. a$k * b$(i+j  k)) ({0..i+j} {i}) = 0"


421 
apply (rule setsum_0')


422 
apply auto


423 
apply (case_tac "aa < i")


424 
using i


425 
apply auto


426 
apply (subgoal_tac "b $ (i+j  aa) = 0")


427 
apply blast


428 
apply (rule j(2)[rule_format])


429 
by arith


430 
have "(a*b) $ (i+j) = setsum (\<lambda>k. a$k * b$(i+j  k)) {0..i+j}"


431 
by (rule fps_mult_nth)


432 
hence "(a*b) $ (i+j) = a$i * b$j"


433 
unfolding setsum_Un_disjoint[OF f d, unfolded eq] th0 by simp


434 
with i j have "(a*b) $ (i+j) \<noteq> 0" by simp


435 
then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast


436 
qed


437 
end


438 


439 
instantiation fps :: (idom) idom


440 
begin


441 


442 
instance ..


443 
end


444 


445 
section{* Inverses of formal power series *}


446 


447 
declare setsum_cong[fundef_cong]


448 


449 


450 
instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") inverse


451 
begin


452 


453 
fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where


454 
"natfun_inverse f 0 = inverse (f$0)"


455 
 "natfun_inverse f n =  inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n}"


456 


457 
definition fps_inverse_def:


458 
"inverse f = (if f$0 = 0 then 0 else Abs_fps (natfun_inverse f))"


459 
definition fps_divide_def: "divide \<equiv> (\<lambda>(f::'a fps) g. f * inverse g)"


460 
instance ..


461 
end


462 


463 
lemma fps_inverse_zero[simp]:


464 
"inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0"


465 
by (simp add: fps_zero_def fps_inverse_def)


466 


467 
lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"


468 
apply (auto simp add: fps_one_def fps_inverse_def expand_fun_eq)


469 
by (case_tac x, auto)


470 


471 
instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") division_by_zero


472 
begin


473 
instance


474 
apply (intro_classes)


475 
by (rule fps_inverse_zero)


476 
end


477 


478 
lemma inverse_mult_eq_1[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"


479 
shows "inverse f * f = 1"


480 
proof


481 
have c: "inverse f * f = f * inverse f" by (simp add: mult_commute)


482 
from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n"


483 
by (simp add: fps_inverse_def)


484 
from f0 have th0: "(inverse f * f) $ 0 = 1"


485 
by (simp add: fps_inverse_def fps_one_def fps_mult_nth)


486 
{fix n::nat assume np: "n >0 "


487 
from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto


488 
have d: "{0} \<inter> {1 .. n} = {}" by auto


489 
have f: "finite {0::nat}" "finite {1..n}" by auto


490 
from f0 np have th0: " (inverse f$n) =


491 
(setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n}) / (f$0)"


492 
by (cases n, simp_all add: divide_inverse fps_inverse_def fps_nth_def ring_simps)


493 
from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]


494 
have th1: "setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n} =


495 
 (f$0) * (inverse f)$n"


496 
by (simp add: ring_simps)


497 
have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n  i))"


498 
unfolding fps_mult_nth ifn ..


499 
also have "\<dots> = f$0 * natfun_inverse f n


500 
+ (\<Sum>i = 1..n. f$i * natfun_inverse f (ni))"


501 
unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]]


502 
by simp


503 
also have "\<dots> = 0" unfolding th1 ifn by simp


504 
finally have "(inverse f * f)$n = 0" unfolding c . }


505 
with th0 show ?thesis by (simp add: fps_eq_iff)


506 
qed


507 


508 
lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) \<longleftrightarrow> f$0 = 0"


509 
apply (simp add: fps_inverse_def)


510 
by (metis fps_nth_def fps_nth_def inverse_zero_imp_zero)


511 


512 
lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $0 = 0"


513 
proof


514 
{assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)}


515 
moreover


516 
{assume h: "inverse f = 0" and c: "f $0 \<noteq> 0"


517 
from inverse_mult_eq_1[OF c] h have False by simp}


518 
ultimately show ?thesis by blast


519 
qed


520 


521 
lemma fps_inverse_idempotent[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"


522 
shows "inverse (inverse f) = f"


523 
proof


524 
from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp


525 
from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]


526 
have th0: "inverse f * f = inverse f * inverse (inverse f)" by (simp add: mult_ac)


527 
then show ?thesis using f0 unfolding mult_cancel_left by simp


528 
qed


529 


530 
lemma fps_inverse_unique: assumes f0: "f$0 \<noteq> (0::'a::field)" and fg: "f*g = 1"


531 
shows "inverse f = g"


532 
proof


533 
from inverse_mult_eq_1[OF f0] fg


534 
have th0: "inverse f * f = g * f" by (simp add: mult_ac)


535 
then show ?thesis using f0 unfolding mult_cancel_right


536 
unfolding Rep_fps_eq[of f 0, symmetric]


537 
by (auto simp add: fps_zero_def expand_fun_eq fps_nth_def)


538 
qed


539 


540 
lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field)))


541 
= Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then  1 else 0)"


542 
apply (rule fps_inverse_unique)


543 
apply simp


544 
apply (simp add: fps_eq_iff fps_nth_def fps_times_def fps_one_def)


545 
proof(clarsimp)


546 
fix n::nat assume n: "n > 0"


547 
let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n  i = 1 then  1 else 0"


548 
let ?g = "\<lambda>i. if i = n then 1 else if i=n  1 then  1 else 0"


549 
let ?h = "\<lambda>i. if i=n  1 then  1 else 0"


550 
have th1: "setsum ?f {0..n} = setsum ?g {0..n}"


551 
by (rule setsum_cong2) auto


552 
have th2: "setsum ?g {0..n  1} = setsum ?h {0..n  1}"


553 
using n apply  by (rule setsum_cong2) auto


554 
have eq: "{0 .. n} = {0.. n  1} \<union> {n}" by auto


555 
from n have d: "{0.. n  1} \<inter> {n} = {}" by auto


556 
have f: "finite {0.. n  1}" "finite {n}" by auto


557 
show "setsum ?f {0..n} = 0"


558 
unfolding th1


559 
apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)


560 
unfolding th2


561 
by(simp add: setsum_delta)


562 
qed


563 


564 
section{* Formal Derivatives, and the McLauren theorem around 0*}


565 


566 
definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"


567 


568 
lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n +1) * f $ (n+1)" by (simp add: fps_deriv_def)


569 


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"


571 
unfolding fps_eq_iff fps_add_nth fps_const_mult_left fps_deriv_nth by (simp add: ring_simps)


572 


573 
lemma fps_deriv_mult[simp]:


574 
fixes f :: "('a :: comm_ring_1) fps"


575 
shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"


576 
proof


577 
let ?D = "fps_deriv"


578 
{fix n::nat


579 
let ?Zn = "{0 ..n}"


580 
let ?Zn1 = "{0 .. n + 1}"


581 
let ?f = "\<lambda>i. i + 1"


582 
have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)


583 
have eq: "{1.. n+1} = ?f ` {0..n}" by auto


584 
let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n  i) +


585 
of_nat (i+1)* f $ (i+1) * g $ (n  i)"


586 
let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1)  i) +


587 
of_nat i* f $ i * g $ ((n + 1)  i)"


588 
{fix k assume k: "k \<in> {0..n}"


589 
have "?h (k + 1) = ?g k" using k by auto}


590 
note th0 = this


591 
have eq': "{0..n +1} {1 .. n+1} = {0}" by auto


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"


593 
apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1  i"])


594 
apply (simp add: inj_on_def Ball_def)


595 
apply presburger


596 
apply (rule set_ext)


597 
apply (presburger add: image_iff)


598 
by simp


599 
have s1: "setsum (\<lambda>i. f $ i * g $ (n + 1  i)) ?Zn1 = setsum (\<lambda>i. f $ (n + 1  i) * g $ i) ?Zn1"


600 
apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1  i"])


601 
apply (simp add: inj_on_def Ball_def)


602 
apply presburger


603 
apply (rule set_ext)


604 
apply (presburger add: image_iff)


605 
by simp


606 
have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute)


607 
also have "\<dots> = (\<Sum>i = 0..n. ?g i)"


608 
by (simp add: fps_mult_nth setsum_addf[symmetric])


609 
also have "\<dots> = setsum ?h {1..n+1}"


610 
using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto


611 
also have "\<dots> = setsum ?h {0..n+1}"


612 
apply (rule setsum_mono_zero_left)


613 
apply simp


614 
apply (simp add: subset_eq)


615 
unfolding eq'


616 
by simp


617 
also have "\<dots> = (fps_deriv (f * g)) $ n"


618 
apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)


619 
unfolding s0 s1


620 
unfolding setsum_addf[symmetric] setsum_right_distrib


621 
apply (rule setsum_cong2)


622 
by (auto simp add: of_nat_diff ring_simps)


623 
finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .}


624 
then show ?thesis unfolding fps_eq_iff by auto


625 
qed


626 


627 
lemma fps_deriv_neg[simp]: "fps_deriv ( (f:: ('a:: comm_ring_1) fps)) =  (fps_deriv f)"


628 
by (simp add: fps_uminus_def fps_eq_iff fps_deriv_def fps_nth_def)


629 
lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g"


630 
using fps_deriv_linear[of 1 f 1 g] by simp


631 


632 
lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps)  g) = fps_deriv f  fps_deriv g"


633 
unfolding diff_minus by simp


634 


635 
lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"


636 
by (simp add: fps_deriv_def fps_const_def fps_zero_def)


637 


638 
lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"


639 
by simp


640 


641 
lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"


642 
by (simp add: fps_deriv_def fps_eq_iff)


643 


644 
lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"


645 
by (simp add: fps_deriv_def fps_eq_iff )


646 


647 
lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"


648 
by simp


649 


650 
lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S"


651 
proof


652 
{assume "\<not> finite S" hence ?thesis by simp}


653 
moreover


654 
{assume fS: "finite S"


655 
have ?thesis by (induct rule: finite_induct[OF fS], simp_all)}


656 
ultimately show ?thesis by blast


657 
qed


658 


659 
lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))"


660 
proof


661 
{assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp


662 
hence "fps_deriv f = 0" by simp }


663 
moreover


664 
{assume z: "fps_deriv f = 0"


665 
hence "\<forall>n. (fps_deriv f)$n = 0" by simp


666 
hence "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)


667 
hence "f = fps_const (f$0)"


668 
apply (clarsimp simp add: fps_eq_iff fps_const_def)


669 
apply (erule_tac x="n  1" in allE)


670 
by simp}


671 
ultimately show ?thesis by blast


672 
qed


673 


674 
lemma fps_deriv_eq_iff:


675 
fixes f:: "('a::{idom,semiring_char_0}) fps"


676 
shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0  g$0) + g)"


677 
proof


678 
have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f  g) = 0" by simp


679 
also have "\<dots> \<longleftrightarrow> f  g = fps_const ((fg)$0)" unfolding fps_deriv_eq_0_iff ..


680 
finally show ?thesis by (simp add: ring_simps)


681 
qed


682 


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)"


684 
apply auto unfolding fps_deriv_eq_iff by blast


685 


686 


687 
fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where


688 
"fps_nth_deriv 0 f = f"


689 
 "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"


690 


691 
lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"


692 
by (induct n arbitrary: f, auto)


693 


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"


695 
by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute)


696 


697 
lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n ( (f:: ('a:: comm_ring_1) fps)) =  (fps_nth_deriv n f)"


698 
by (induct n arbitrary: f, simp_all)


699 


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"


701 
using fps_nth_deriv_linear[of n 1 f 1 g] by simp


702 


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"


704 
unfolding diff_minus fps_nth_deriv_add by simp


705 


706 
lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"


707 
by (induct n, simp_all )


708 


709 
lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"


710 
by (induct n, simp_all )


711 


712 
lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"


713 
by (cases n, simp_all)


714 


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"


716 
using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp


717 


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"


719 
using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute)


720 


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"


722 
proof


723 
{assume "\<not> finite S" hence ?thesis by simp}


724 
moreover


725 
{assume fS: "finite S"


726 
have ?thesis by (induct rule: finite_induct[OF fS], simp_all)}


727 
ultimately show ?thesis by blast


728 
qed


729 


730 
lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)"


731 
by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult)


732 


733 
section {* Powers*}


734 


735 
instantiation fps :: (semiring_1) power


736 
begin


737 


738 
fun fps_pow :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where


739 
"fps_pow 0 f = 1"


740 
 "fps_pow (Suc n) f = f * fps_pow n f"


741 


742 
definition fps_power_def: "power (f::'a fps) n = fps_pow n f"


743 
instance ..


744 
end


745 


746 
instantiation fps :: (comm_ring_1) recpower


747 
begin


748 
instance


749 
apply (intro_classes)


750 
by (simp_all add: fps_power_def)


751 
end


752 


753 
lemma eq_neg_iff_add_eq_0: "(a::'a::ring) = b \<longleftrightarrow> a + b = 0"


754 
proof


755 
{assume "a = b" hence "b + a = b + b" by simp


756 
hence "a + b = 0" by (simp add: ring_simps)}


757 
moreover


758 
{assume "a + b = 0" hence "a + b  b = b" by simp


759 
hence "a = b" by simp}


760 
ultimately show ?thesis by blast


761 
qed


762 


763 
lemma fps_sqrare_eq_iff: "(a:: 'a::idom fps)^ 2 = b^2 \<longleftrightarrow> (a = b \<or> a = b)"


764 
proof


765 
{assume "a = b \<or> a = b" hence "a^2 = b^2" by auto}


766 
moreover


767 
{assume "a^2 = b^2 "


768 
hence "a^2  b^2 = 0" by simp


769 
hence "(ab) * (a+b) = 0" by (simp add: power2_eq_square ring_simps)


770 
hence "a = b \<or> a = b" by (simp add: eq_neg_iff_add_eq_0)}


771 
ultimately show ?thesis by blast


772 
qed


773 


774 
lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)"


775 
by (induct n, auto simp add: fps_power_def fps_times_def fps_nth_def fps_one_def)


776 


777 
lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"


778 
proof(induct n)


779 
case 0 thus ?case by (simp add: fps_power_def)


780 
next


781 
case (Suc n)


782 
note h = Suc.hyps[OF `a$0 = 1`]


783 
show ?case unfolding power_Suc fps_mult_nth


784 
using h `a$0 = 1` fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: ring_simps)


785 
qed


786 


787 
lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"


788 
by (induct n, auto simp add: fps_power_def fps_mult_nth)


789 


790 
lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"


791 
by (induct n, auto simp add: fps_power_def fps_mult_nth)


792 


793 
lemma startsby_power:"a $0 = (v::'a::{comm_ring_1, recpower}) \<Longrightarrow> a^n $0 = v^n"


794 
by (induct n, auto simp add: fps_power_def fps_mult_nth power_Suc)


795 


796 
lemma startsby_zero_power_iff[simp]:


797 
"a^n $0 = (0::'a::{idom, recpower}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)"


798 
apply (rule iffI)


799 
apply (induct n, auto simp add: power_Suc fps_mult_nth)


800 
by (rule startsby_zero_power, simp_all)


801 


802 
lemma startsby_zero_power_prefix:


803 
assumes a0: "a $0 = (0::'a::idom)"


804 
shows "\<forall>n < k. a ^ k $ n = 0"


805 
using a0


806 
proof(induct k rule: nat_less_induct)


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)"


808 
let ?ths = "\<forall>m<k. a ^ k $ m = 0"


809 
{assume "k = 0" then have ?ths by simp}


810 
moreover


811 
{fix l assume k: "k = Suc l"


812 
{fix m assume mk: "m < k"


813 
{assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0


814 
by simp}


815 
moreover


816 
{assume m0: "m \<noteq> 0"


817 
have "a ^k $ m = (a^l * a) $m" by (simp add: k power_Suc mult_commute)


818 
also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m  i))" by (simp add: fps_mult_nth)


819 
also have "\<dots> = 0" apply (rule setsum_0')


820 
apply auto


821 
apply (case_tac "aa = m")


822 
using a0


823 
apply simp


824 
apply (rule H[rule_format])


825 
using a0 k mk by auto


826 
finally have "a^k $ m = 0" .}


827 
ultimately have "a^k $ m = 0" by blast}


828 
hence ?ths by blast}


829 
ultimately show ?ths by (cases k, auto)


830 
qed


831 


832 
lemma startsby_zero_setsum_depends:


833 
assumes a0: "a $0 = (0::'a::idom)" and kn: "n \<ge> k"


834 
shows "setsum (\<lambda>i. (a ^ i)$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)$k) {0 .. k}"


835 
apply (rule setsum_mono_zero_right)


836 
using kn apply auto


837 
apply (rule startsby_zero_power_prefix[rule_format, OF a0])


838 
by arith


839 


840 
lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{recpower, idom})"


841 
shows "a^n $ n = (a$1) ^ n"


842 
proof(induct n)


843 
case 0 thus ?case by (simp add: power_0)


844 
next


845 
case (Suc n)


846 
have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: ring_simps power_Suc)


847 
also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n  i)) {0.. Suc n}" by (simp add: fps_mult_nth)


848 
also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n  i)) {n .. Suc n}"


849 
apply (rule setsum_mono_zero_right)


850 
apply simp


851 
apply clarsimp


852 
apply clarsimp


853 
apply (rule startsby_zero_power_prefix[rule_format, OF a0])


854 
apply arith


855 
done


856 
also have "\<dots> = a^n $ n * a$1" using a0 by simp


857 
finally show ?case using Suc.hyps by (simp add: power_Suc)


858 
qed


859 


860 
lemma fps_inverse_power:


861 
fixes a :: "('a::{field, recpower}) fps"


862 
shows "inverse (a^n) = inverse a ^ n"


863 
proof


864 
{assume a0: "a$0 = 0"


865 
hence eq: "inverse a = 0" by (simp add: fps_inverse_def)


866 
{assume "n = 0" hence ?thesis by simp}


867 
moreover


868 
{assume n: "n > 0"


869 
from startsby_zero_power[OF a0 n] eq a0 n have ?thesis


870 
by (simp add: fps_inverse_def)}


871 
ultimately have ?thesis by blast}


872 
moreover


873 
{assume a0: "a$0 \<noteq> 0"


874 
have ?thesis


875 
apply (rule fps_inverse_unique)


876 
apply (simp add: a0)


877 
unfolding power_mult_distrib[symmetric]


878 
apply (rule ssubst[where t = "a * inverse a" and s= 1])


879 
apply simp_all


880 
apply (subst mult_commute)


881 
by (rule inverse_mult_eq_1[OF a0])}


882 
ultimately show ?thesis by blast


883 
qed


884 


885 
lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n  1)"


886 
apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add)


887 
by (case_tac n, auto simp add: power_Suc ring_simps)


888 


889 
lemma fps_inverse_deriv:


890 
fixes a:: "('a :: field) fps"


891 
assumes a0: "a$0 \<noteq> 0"


892 
shows "fps_deriv (inverse a) =  fps_deriv a * inverse a ^ 2"


893 
proof


894 
from inverse_mult_eq_1[OF a0]


895 
have "fps_deriv (inverse a * a) = 0" by simp


896 
hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp


897 
hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0" by simp


898 
with inverse_mult_eq_1[OF a0]


899 
have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0"


900 
unfolding power2_eq_square


901 
apply (simp add: ring_simps)


902 
by (simp add: mult_assoc[symmetric])


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"


904 
by simp


905 
then show "fps_deriv (inverse a) =  fps_deriv a * inverse a ^ 2" by (simp add: ring_simps)


906 
qed


907 


908 
lemma fps_inverse_mult:


909 
fixes a::"('a :: field) fps"


910 
shows "inverse (a * b) = inverse a * inverse b"


911 
proof


912 
{assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)


913 
from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all


914 
have ?thesis unfolding th by simp}


915 
moreover


916 
{assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)


917 
from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all


918 
have ?thesis unfolding th by simp}


919 
moreover


920 
{assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0"


921 
from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp add: fps_mult_nth)


922 
from inverse_mult_eq_1[OF ab0]


923 
have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp


924 
then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"


925 
by (simp add: ring_simps)


926 
then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp}


927 
ultimately show ?thesis by blast


928 
qed


929 


930 
lemma fps_inverse_deriv':


931 
fixes a:: "('a :: field) fps"


932 
assumes a0: "a$0 \<noteq> 0"


933 
shows "fps_deriv (inverse a) =  fps_deriv a / a ^ 2"


934 
using fps_inverse_deriv[OF a0]


935 
unfolding power2_eq_square fps_divide_def


936 
fps_inverse_mult by simp


937 


938 
lemma inverse_mult_eq_1': assumes f0: "f$0 \<noteq> (0::'a::field)"


939 
shows "f * inverse f= 1"


940 
by (metis mult_commute inverse_mult_eq_1 f0)


941 


942 
lemma fps_divide_deriv: fixes a:: "('a :: field) fps"


943 
assumes a0: "b$0 \<noteq> 0"


944 
shows "fps_deriv (a / b) = (fps_deriv a * b  a * fps_deriv b) / b ^ 2"


945 
using fps_inverse_deriv[OF a0]


946 
by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])


947 


948 
section{* The eXtractor series X*}


949 


950 
lemma minus_one_power_iff: "( (1::'a :: {recpower, comm_ring_1})) ^ n = (if even n then 1 else  1)"


951 
by (induct n, auto)


952 


953 
definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"


954 


955 
lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field)))


956 
= 1  X"


957 
by (simp add: fps_inverse_gp fps_eq_iff X_def fps_minus_def fps_one_def)


958 


959 
lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) $n = (if n = 0 then 0 else f $ (n  1))"


960 
proof


961 
{assume n: "n \<noteq> 0"


962 
have fN: "finite {0 .. n}" by simp


963 
have "(X * f) $n = (\<Sum>i = 0..n. X $ i * f $ (n  i))" by (simp add: fps_mult_nth)


964 
also have "\<dots> = f $ (n  1)"


965 
using n by (simp add: X_def cond_value_iff cond_application_beta setsum_delta[OF fN]


966 
del: One_nat_def cong del: if_weak_cong)


967 
finally have ?thesis using n by simp }


968 
moreover


969 
{assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)}


970 
ultimately show ?thesis by blast


971 
qed


972 


973 
lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n  1))"


974 
by (metis X_mult_nth mult_commute)


975 


976 
lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"


977 
proof(induct k)


978 
case 0 thus ?case by (simp add: X_def fps_power_def fps_eq_iff)


979 
next


980 
case (Suc k)


981 
{fix m


982 
have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m  1))"


983 
by (simp add: power_Suc del: One_nat_def)


984 
then have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)"


985 
using Suc.hyps by (auto cong del: if_weak_cong)}


986 
then show ?case by (simp add: fps_eq_iff)


987 
qed


988 


989 
lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n  k))"


990 
apply (induct k arbitrary: n)


991 
apply (simp)


992 
unfolding power_Suc mult_assoc


993 
by (case_tac n, auto)


994 


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))"


996 
by (metis X_power_mult_nth mult_commute)


997 
lemma fps_deriv_X[simp]: "fps_deriv X = 1"


998 
by (simp add: fps_deriv_def X_def fps_eq_iff)


999 


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)"


1001 
by (cases "n", simp_all)


1002 


1003 
lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def)


1004 
lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))"


1005 
by (simp add: X_power_iff)


1006 


1007 
lemma fps_inverse_X_plus1:


1008 
"inverse (1 + X) = Abs_fps (\<lambda>n. ( (1::'a::{recpower, field})) ^ n)" (is "_ = ?r")


1009 
proof


1010 
have eq: "(1 + X) * ?r = 1"


1011 
unfolding minus_one_power_iff


1012 
apply (auto simp add: ring_simps fps_eq_iff)


1013 
by presburger+


1014 
show ?thesis by (auto simp add: eq intro: fps_inverse_unique)


1015 
qed


1016 


1017 


1018 
section{* Integration *}


1019 
definition "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n  1) / of_nat n))"


1020 


1021 
lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a"


1022 
by (simp add: fps_integral_def fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)


1023 


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")


1025 
proof


1026 
have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)


1027 
moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def)


1028 
ultimately show ?thesis


1029 
unfolding fps_deriv_eq_iff by auto


1030 
qed


1031 


1032 
section {* Composition of FPSs *}


1033 
definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where


1034 
fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})"


1035 


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)


1037 


1038 
lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"


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)


1040 


1041 
lemma fps_const_compose[simp]:


1042 
"fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)"


1043 
apply (auto simp add: fps_eq_iff fps_compose_nth fps_mult_nth


1044 
cond_application_beta cond_value_iff cong del: if_weak_cong)


1045 
by (simp add: setsum_delta )


1046 


1047 
lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"


1048 
apply (auto simp add: fps_compose_def fps_eq_iff cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)


1049 
apply (simp add: power_Suc)


1050 
apply (subgoal_tac "n = 0")


1051 
by simp_all


1052 


1053 


1054 
section {* Rules from Herbert Wilf's Generatingfunctionology*}


1055 


1056 
subsection {* Rule 1 *}


1057 
(* {a_{n+k}}_0^infty Corresponds to (f  setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)


1058 


1059 
lemma fps_power_mult_eq_shift:


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")


1061 
proof


1062 
{fix n:: nat


1063 
have "?lhs $ n = (if n < Suc k then 0 else a n)"


1064 
unfolding X_power_mult_nth by auto


1065 
also have "\<dots> = ?rhs $ n"


1066 
proof(induct k)


1067 
case 0 thus ?case by (simp add: fps_setsum_nth power_Suc)


1068 
next


1069 
case (Suc k)


1070 
note th = Suc.hyps[symmetric]


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)


1072 
also have "\<dots> = (if n < Suc k then 0 else a n)  (fps_const (a (Suc k)) * X^ Suc k)$n"


1073 
using th


1074 
unfolding fps_sub_nth by simp


1075 
also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"


1076 
unfolding X_power_mult_right_nth


1077 
apply (auto simp add: not_less fps_const_def)


1078 
apply (rule cong[of a a, OF refl])


1079 
by arith


1080 
finally show ?case by simp


1081 
qed


1082 
finally have "?lhs $ n = ?rhs $ n" .}


1083 
then show ?thesis by (simp add: fps_eq_iff)


1084 
qed


1085 


1086 
subsection{* Rule 2*}


1087 


1088 
(* We can not reach the form of Wilf, but still near to it using rewrite rules*)


1089 
(* If f reprents {a_n} and P is a polynomial, then


1090 
P(xD) f represents {P(n) a_n}*)


1091 


1092 
definition "XD = op * X o fps_deriv"


1093 


1094 
lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"


1095 
by (simp add: XD_def ring_simps)


1096 


1097 
lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"


1098 
by (simp add: XD_def ring_simps)


1099 


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)"


1101 
by simp


1102 


1103 


1104 
fun funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where


1105 
"funpow 0 f = id"


1106 
 "funpow (Suc n) f = f o funpow n f"


1107 


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)"


1109 
by (induct n, simp_all)


1110 


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)


1112 


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)"


1114 
by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)


1115 


1116 
subsection{* Rule 3 is trivial and is given by fps_times_def*}


1117 
subsection{* Rule 5  summation and "division" by (1  X)*}


1118 


1119 
lemma fps_divide_X_minus1_setsum_lemma:


1120 
"a = ((1::('a::comm_ring_1) fps)  X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"


1121 
proof


1122 
let ?X = "X::('a::comm_ring_1) fps"


1123 
let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"


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


1125 
{fix n:: nat


1126 
{assume "n=0" hence "a$n = ((1  ?X) * ?sa) $ n"


1127 
by (simp add: fps_mult_nth)}


1128 
moreover


1129 
{assume n0: "n \<noteq> 0"


1130 
then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}"


1131 
"{0..n  1}\<union>{n} = {0..n}"


1132 
apply (simp_all add: expand_set_eq) by presburger+


1133 
have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}"


1134 
"{0..n  1}\<inter>{n} ={}" using n0


1135 
by (simp_all add: expand_set_eq, presburger+)


1136 
have f: "finite {0}" "finite {1}" "finite {2 .. n}"


1137 
"finite {0 .. n  1}" "finite {n}" by simp_all


1138 
have "((1  ?X) * ?sa) $ n = setsum (\<lambda>i. (1  ?X)$ i * ?sa $ (n  i)) {0 .. n}"


1139 
by (simp add: fps_mult_nth)


1140 
also have "\<dots> = a$n" unfolding th0


1141 
unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]


1142 
unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]


1143 
apply (simp)


1144 
unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]


1145 
by simp


1146 
finally have "a$n = ((1  ?X) * ?sa) $ n" by simp}


1147 
ultimately have "a$n = ((1  ?X) * ?sa) $ n" by blast}


1148 
then show ?thesis


1149 
unfolding fps_eq_iff by blast


1150 
qed


1151 


1152 
lemma fps_divide_X_minus1_setsum:


1153 
"a /((1::('a::field) fps)  X) = Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"


1154 
proof


1155 
let ?X = "1  (X::('a::field) fps)"


1156 
have th0: "?X $ 0 \<noteq> 0" by simp


1157 
have "a /?X = ?X * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) * inverse ?X"


1158 
using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0


1159 
by (simp add: fps_divide_def mult_assoc)


1160 
also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) "


1161 
by (simp add: mult_ac)


1162 
finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])


1163 
qed


1164 


1165 
subsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary


1166 
finite product of FPS, also the relvant instance of powers of a FPS*}


1167 


1168 
definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}"


1169 


1170 
lemma natlist_trivial_1: "natpermute n 1 = {[n]}"


1171 
apply (auto simp add: natpermute_def)


1172 
apply (case_tac x, auto)


1173 
done


1174 


1175 
lemma foldl_add_start0:


1176 
"foldl op + x xs = x + foldl op + (0::nat) xs"


1177 
apply (induct xs arbitrary: x)


1178 
apply simp


1179 
unfolding foldl.simps


1180 
apply atomize


1181 
apply (subgoal_tac "\<forall>x\<Colon>nat. foldl op + x xs = x + foldl op + (0\<Colon>nat) xs")


1182 
apply (erule_tac x="x + a" in allE)


1183 
apply (erule_tac x="a" in allE)


1184 
apply simp


1185 
apply assumption


1186 
done


1187 


1188 
lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys"


1189 
apply (induct ys arbitrary: x xs)


1190 
apply auto


1191 
apply (subst (2) foldl_add_start0)


1192 
apply simp


1193 
apply (subst (2) foldl_add_start0)


1194 
by simp


1195 


1196 
lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}"


1197 
proof(induct xs arbitrary: x)


1198 
case Nil thus ?case by simp


1199 
next


1200 
case (Cons a as x)


1201 
have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}"


1202 
apply (rule setsum_reindex_cong [where f=Suc])


1203 
by (simp_all add: inj_on_def)


1204 
have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all


1205 
have d: "{0} \<inter> {1..<length (a#as)} = {}" by simp


1206 
have seq: "{0} \<union> {1..<length(a#as)} = {0 ..<length (a#as)}" by auto


1207 
have "foldl op + x (a#as) = x + foldl op + a as "


1208 
apply (subst foldl_add_start0) by simp


1209 
also have "\<dots> = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp


1210 
also have "\<dots> = x + setsum (op ! (a#as)) {0..<length (a#as)}"


1211 
unfolding eq[symmetric]


1212 
unfolding setsum_Un_disjoint[OF f d, unfolded seq]


1213 
by simp


1214 
finally show ?case .


1215 
qed


1216 


1217 


1218 
lemma append_natpermute_less_eq:


1219 
assumes h: "xs@ys \<in> natpermute n k" shows "foldl op + 0 xs \<le> n" and "foldl op + 0 ys \<le> n"


1220 
proof


1221 
{from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def)


1222 
hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .}


1223 
note th = this


1224 
{from th show "foldl op + 0 xs \<le> n" by simp}


1225 
{from th show "foldl op + 0 ys \<le> n" by simp}


1226 
qed


1227 


1228 
lemma natpermute_split:


1229 
assumes mn: "h \<le> k"


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)")


1231 
proof


1232 
{fix l assume l: "l \<in> ?R"


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


1234 
from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def)


1235 
from ys have ys': "foldl op + 0 ys = n  m" by (simp add: natpermute_def)


1236 
have "l \<in> ?L" using leq xs ys h


1237 
apply simp


1238 
apply (clarsimp simp add: natpermute_def simp del: foldl_append)


1239 
apply (simp add: foldl_add_append[unfolded foldl_append])


1240 
unfolding xs' ys'


1241 
using mn xs ys


1242 
unfolding natpermute_def by simp}


1243 
moreover


1244 
{fix l assume l: "l \<in> natpermute n k"


1245 
let ?xs = "take h l"


1246 
let ?ys = "drop h l"


1247 
let ?m = "foldl op + 0 ?xs"


1248 
from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def)


1249 
have xs: "?xs \<in> natpermute ?m h" using l mn by (simp add: natpermute_def)


1250 
have ys: "?ys \<in> natpermute (n  ?m) (k  h)" using l mn ls[unfolded foldl_add_append]


1251 
by (simp add: natpermute_def)


1252 
from ls have m: "?m \<in> {0..n}" unfolding foldl_add_append by simp


1253 
from xs ys ls have "l \<in> ?R"


1254 
apply auto


1255 
apply (rule bexI[where x = "?m"])


1256 
apply (rule exI[where x = "?xs"])


1257 
apply (rule exI[where x = "?ys"])


1258 
using ls l unfolding foldl_add_append


1259 
by (auto simp add: natpermute_def)}


1260 
ultimately show ?thesis by blast


1261 
qed


1262 


1263 
lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"


1264 
by (auto simp add: natpermute_def)


1265 
lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"


1266 
apply (auto simp add: set_replicate_conv_if natpermute_def)


1267 
apply (rule nth_equalityI)


1268 
by simp_all


1269 


1270 
lemma natpermute_finite: "finite (natpermute n k)"


1271 
proof(induct k arbitrary: n)


1272 
case 0 thus ?case


1273 
apply (subst natpermute_split[of 0 0, simplified])


1274 
by (simp add: natpermute_0)


1275 
next


1276 
case (Suc k)


1277 
then show ?case unfolding natpermute_split[of k "Suc k", simplified]


1278 
apply 


1279 
apply (rule finite_UN_I)


1280 
apply simp


1281 
unfolding One_nat_def[symmetric] natlist_trivial_1


1282 
apply simp


1283 
unfolding image_Collect[symmetric]


1284 
unfolding Collect_def mem_def


1285 
apply (rule finite_imageI)


1286 
apply blast


1287 
done


1288 
qed


1289 


1290 
lemma natpermute_contain_maximal:


1291 
"{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"


1292 
(is "?A = ?B")


1293 
proof


1294 
{fix xs assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"


1295 
from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H


1296 
unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)


1297 
have eqs: "({0..k}  {i}) \<union> {i} = {0..k}" using i by auto


1298 
have f: "finite({0..k}  {i})" "finite {i}" by auto


1299 
have d: "({0..k}  {i}) \<inter> {i} = {}" using i by auto


1300 
from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def)


1301 
unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost)


1302 
also have "\<dots> = n + setsum (nth xs) ({0..k}  {i})"


1303 
unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp


1304 
finally have zxs: "\<forall> j\<in> {0..k}  {i}. xs!j = 0" by auto


1305 
from H have xsl: "length xs = k+1" by (simp add: natpermute_def)


1306 
from i have i': "i < length (replicate (k+1) 0)" "i < k+1"


1307 
unfolding length_replicate by arith+


1308 
have "xs = replicate (k+1) 0 [i := n]"


1309 
apply (rule nth_equalityI)


1310 
unfolding xsl length_list_update length_replicate


1311 
apply simp


1312 
apply clarify


1313 
unfolding nth_list_update[OF i'(1)]


1314 
using i zxs


1315 
by (case_tac "ia=i", auto simp del: replicate.simps)


1316 
then have "xs \<in> ?B" using i by blast}


1317 
moreover


1318 
{fix i assume i: "i \<in> {0..k}"


1319 
let ?xs = "replicate (k+1) 0 [i:=n]"


1320 
have nxs: "n \<in> set ?xs"


1321 
apply (rule set_update_memI) using i by simp


1322 
have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update)


1323 
have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}"


1324 
unfolding foldl_add_setsum add_0 length_replicate length_list_update ..


1325 
also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"


1326 
apply (rule setsum_cong2) by (simp del: replicate.simps)


1327 
also have "\<dots> = n" using i by (simp add: setsum_delta)


1328 
finally


1329 
have "?xs \<in> natpermute n (k+1)" using xsl unfolding natpermute_def Collect_def mem_def


1330 
by blast


1331 
then have "?xs \<in> ?A" using nxs by blast}


1332 
ultimately show ?thesis by auto


1333 
qed


1334 


1335 
(* The general form *)


1336 
lemma fps_setprod_nth:


1337 
fixes m :: nat and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"


1338 
shows "(setprod a {0 .. m})$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"


1339 
(is "?P m n")


1340 
proof(induct m arbitrary: n rule: nat_less_induct)


1341 
fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"


1342 
{assume m0: "m = 0"


1343 
hence "?P m n" apply simp


1344 
unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp}


1345 
moreover


1346 
{fix k assume k: "m = Suc k"


1347 
have km: "k < m" using k by arith


1348 
have u0: "{0 .. k} \<union> {m} = {0..m}" using k apply (simp add: expand_set_eq) by presburger


1349 
have f0: "finite {0 .. k}" "finite {m}" by auto


1350 
have d0: "{0 .. k} \<inter> {m} = {}" using k by auto


1351 
have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n"


1352 
unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp


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))"


1354 
unfolding fps_mult_nth H[rule_format, OF km] ..


1355 
also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)"


1356 
apply (simp add: k)


1357 
unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k]


1358 
apply (subst setsum_UN_disjoint)


1359 
apply simp


1360 
apply simp


1361 
unfolding image_Collect[symmetric]


1362 
apply clarsimp


1363 
apply (rule finite_imageI)


1364 
apply (rule natpermute_finite)


1365 
apply (clarsimp simp add: expand_set_eq)


1366 
apply auto


1367 
apply (rule setsum_cong2)


1368 
unfolding setsum_left_distrib


1369 
apply (rule sym)


1370 
apply (rule_tac f="\<lambda>xs. xs @[n  x]" in setsum_reindex_cong)


1371 
apply (simp add: inj_on_def)


1372 
apply auto


1373 
unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]


1374 
apply (clarsimp simp add: natpermute_def nth_append)


1375 
apply (rule_tac f="\<lambda>x. x * a (Suc k) $ (n  foldl op + 0 aa)" in cong[OF refl])


1376 
apply (rule setprod_cong)


1377 
apply simp


1378 
apply simp


1379 
done


1380 
finally have "?P m n" .}


1381 
ultimately show "?P m n " by (cases m, auto)


1382 
qed


1383 


1384 
text{* The special form for powers *}


1385 
lemma fps_power_nth_Suc:


1386 
fixes m :: nat and a :: "('a::comm_ring_1) fps"


1387 
shows "(a ^ Suc m)$n = setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))"


1388 
proof


1389 
have f: "finite {0 ..m}" by simp


1390 
have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp


1391 
show ?thesis unfolding th0 fps_setprod_nth ..


1392 
qed


1393 
lemma fps_power_nth:


1394 
fixes m :: nat and a :: "('a::comm_ring_1) fps"


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))"


1396 
by (cases m, simp_all add: fps_power_nth_Suc)


1397 


1398 
lemma fps_nth_power_0:


1399 
fixes m :: nat and a :: "('a::{comm_ring_1, recpower}) fps"


1400 
shows "(a ^m)$0 = (a$0) ^ m"


1401 
proof


1402 
{assume "m=0" hence ?thesis by simp}


1403 
moreover


1404 
{fix n assume m: "m = Suc n"


1405 
have c: "m = card {0..n}" using m by simp


1406 
have "(a ^m)$0 = setprod (\<lambda>i. a$0) {0..n}"


1407 
apply (simp add: m fps_power_nth del: replicate.simps)


1408 
apply (rule setprod_cong)


1409 
by (simp_all del: replicate.simps)


1410 
also have "\<dots> = (a$0) ^ m"


1411 
unfolding c by (rule setprod_constant, simp)


1412 
finally have ?thesis .}


1413 
ultimately show ?thesis by (cases m, auto)


1414 
qed


1415 


1416 
lemma fps_compose_inj_right:


1417 
assumes a0: "a$0 = (0::'a::{recpower,idom})"


1418 
and a1: "a$1 \<noteq> 0"


1419 
shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")


1420 
proof


1421 
{assume ?rhs then have "?lhs" by simp}


1422 
moreover


1423 
{assume h: ?lhs


1424 
{fix n have "b$n = c$n"


1425 
proof(induct n rule: nat_less_induct)


1426 
fix n assume H: "\<forall>m<n. b$m = c$m"


1427 
{assume n0: "n=0"


1428 
from h have "(b oo a)$n = (c oo a)$n" by simp


1429 
hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)}


1430 
moreover


1431 
{fix n1 assume n1: "n = Suc n1"


1432 
have f: "finite {0 .. n1}" "finite {n}" by simp_all


1433 
have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
