src/HOL/Library/Formal_Power_Series.thy

A formalization of formal power series

(*  Title:      Formal_Power_Series.thy
    ID:         
    Author:     Amine Chaieb, University of Cambridge
*)

header{* A formalization of formal power series *}

theory Formal_Power_Series
  imports Main Fact Parity
begin

section {* The type of formal power series*}

typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
  by simp

text{* Definition of the basic elements 0 and 1 and the basic operations of addition, negation and multiplication *}

instantiation fps :: (zero)  zero
begin

definition fps_zero_def: "(0 :: 'a fps) \<equiv> Abs_fps (\<lambda>(n::nat). 0)"
instance ..
end

instantiation fps :: ("{one,zero}")  one
begin

definition fps_one_def: "(1 :: 'a fps) \<equiv> Abs_fps (\<lambda>(n::nat). if n = 0 then 1 else 0)"
instance ..
end

instantiation fps :: (plus)  plus
begin

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))"
instance ..
end

instantiation fps :: (minus)  minus
begin

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))"
instance ..
end

instantiation fps :: (uminus)  uminus
begin

definition fps_uminus_def: "uminus \<equiv> (\<lambda>(f::'a fps). Abs_fps (\<lambda>(n::nat). - Rep_fps (f) n))"
instance ..
end

instantiation fps :: ("{comm_monoid_add, times}")  times
begin

definition fps_times_def: 
  "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}))"
instance ..
end

text{* Some useful theorems to get rid of Abs and Rep *}

lemma mem_fps_set_trivial[intro, simp]: "f \<in> fps" unfolding fps_def by blast
lemma Rep_fps_Abs_fps[simp]: "Rep_fps (Abs_fps f) = f" 
  by (blast intro: Abs_fps_inverse) 
lemma Abs_fps_Rep_fps[simp]: "Abs_fps (Rep_fps f) = f" 
  by (blast intro: Rep_fps_inverse) 
lemma Abs_fps_eq[simp]: "Abs_fps f = Abs_fps g \<longleftrightarrow> f = g"
proof-
  {assume "f = g" hence "Abs_fps f = Abs_fps g" by simp}
  moreover
  {assume a: "Abs_fps f = Abs_fps g"
    from a have "Rep_fps (Abs_fps f) = Rep_fps (Abs_fps g)" by simp
    hence "f = g" by simp}
  ultimately show ?thesis by blast
qed

lemma Rep_fps_eq[simp]: "Rep_fps f = Rep_fps g \<longleftrightarrow> f = g"
proof-
  {assume "Rep_fps f = Rep_fps g" 
    hence "Abs_fps (Rep_fps f) = Abs_fps (Rep_fps g)" by simp hence "f=g" by simp}
  moreover
  {assume "f = g" hence "Rep_fps f = Rep_fps g" by simp}
  ultimately show ?thesis by blast
qed

declare atLeastAtMost_iff[presburger] 
declare Bex_def[presburger]
declare Ball_def[presburger]

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

section{* Formal power series form a commutative ring with unity, if the range of sequences 
  they represent is a commutative ring with unity*}

instantiation fps :: (semigroup_add) semigroup_add
begin

instance
proof
  fix a b c :: "'a fps" show "a + b + c = a + (b + c)"
    by (auto simp add: fps_plus_def expand_fun_eq add_assoc)
qed

end

instantiation fps :: (ab_semigroup_add) ab_semigroup_add
begin

instance by (intro_classes, simp add: fps_plus_def expand_fun_eq add_commute)
end

instantiation fps :: (semiring_1) semigroup_mult
begin

instance
proof
  fix a b c :: "'a fps"
  let ?a = "Rep_fps a"
  let ?b = "Rep_fps b"
  let ?c = "Rep_fps c"
  let ?x = "\<lambda> i k. if k \<le> i then (1::'a) else 0" 
  show "a*b*c = a* (b * c)"
  proof(auto simp add: fps_times_def setsum_right_distrib setsum_left_distrib, rule ext)
    fix n::nat
    let ?r = "\<lambda>i. n - i"
    have i: "inj_on ?r {0..n}" by (auto simp add: inj_on_def)
    have ri: "{0 .. n} = ?r ` {0..n}" apply (auto simp add: image_iff)
      by presburger
    let ?f = "\<lambda>i j. ?a j * ?b (i - j) * ?c (n -i)"
    let ?g = "\<lambda>i j. ?a i * (?b j * ?c (n - (i + j)))"
    have "setsum (\<lambda>i. setsum (?f i) {0..i}) {0..n} 
      = setsum (\<lambda>i. setsum (\<lambda>j. ?f i j * ?x i j) {0..i}) {0..n}"
      by (rule setsum_cong2)+ auto
    also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f i j * ?x i j) {0..n}) {0..n}"
    proof(rule setsum_cong2)
      fix i assume i: "i \<in> {0..n}"
      have eq: "{0 .. n} = {0 ..i} \<union> {i+1 .. n}" using i by auto
      have d: "{0 ..i} \<inter> {i+1 .. n} = {}" using i by auto
      have f: "finite {0..i}" "finite {i+1 ..n}" by auto
      have s0: "setsum (\<lambda>j. ?f i j * ?x i j) {i+1 ..n} = 0" by simp
      show "setsum (\<lambda>j. ?f i j * ?x i j) {0..i} = setsum (\<lambda>j. ?f i j * ?x i j) {0..n}"
	unfolding eq setsum_Un_disjoint[OF f d] s0
	by simp
    qed
    also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f j i * ?x j i) {0 .. n}) {0 .. n}"
      by (rule setsum_commute)
    also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f j i * ?x j i) {i .. n}) {0 .. n}"
      apply(rule setsum_cong2)
      apply (rule setsum_mono_zero_right)
      apply auto
      done
    also have "\<dots> = setsum (\<lambda>i. setsum (?g i) {0..n - i}) {0..n}"
      apply (rule setsum_cong2)
      apply (rule_tac f="\<lambda>i. i + x" in setsum_reindex_cong)
      apply (simp add: inj_on_def)
      apply (rule set_ext)
      apply (presburger add: image_iff)
      by (simp add: add_ac mult_assoc)
    finally  show "setsum (\<lambda>i. setsum (\<lambda>j. ?a j * ?b (i - j) * ?c (n -i)) {0..i}) {0..n} 
      = setsum (\<lambda>i. setsum (\<lambda>j. ?a i * (?b j * ?c (n - (i + j)))) {0..n - i}) {0..n}".
  qed
qed

end

instantiation fps :: (comm_semiring_1) ab_semigroup_mult
begin

instance
proof
  fix a b :: "'a fps"
  show "a*b = b*a"
  apply(auto simp add: fps_times_def setsum_right_distrib setsum_left_distrib, rule ext)
  apply (rule_tac f = "\<lambda>i. n - i" in setsum_reindex_cong)
  apply (simp add: inj_on_def)
  apply presburger
  apply (rule set_ext)
  apply (presburger add: image_iff)
  by (simp add: mult_commute)
qed
end

instantiation fps :: (monoid_add) monoid_add
begin

instance
proof
  fix a :: "'a fps" show "0 + a = a "
    by (auto simp add: fps_plus_def fps_zero_def intro: ext)
next
  fix a :: "'a fps" show "a + 0 = a "
    by (auto simp add: fps_plus_def fps_zero_def intro: ext)
qed

end
instantiation fps :: (comm_monoid_add) comm_monoid_add
begin

instance
proof
  fix a :: "'a fps" show "0 + a = a "
    by (auto simp add: fps_plus_def fps_zero_def intro: ext)
qed

end

instantiation fps :: (semiring_1) monoid_mult
begin

instance
proof
  fix a :: "'a fps" show "1 * a = a"
    apply (auto simp add: fps_one_def fps_times_def)
    apply (subst (2) Abs_fps_Rep_fps[of a, symmetric])
    unfolding Abs_fps_eq
    apply (rule ext)
    by (simp add: cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
next
  fix a :: "'a fps" show "a*1 = a"
    apply (auto simp add: fps_one_def fps_times_def)
    apply (subst (2) Abs_fps_Rep_fps[of a, symmetric])
    unfolding Abs_fps_eq
    apply (rule ext)
    by (simp add: cond_value_iff cond_application_beta setsum_delta' cong del: if_weak_cong)
qed
end

instantiation fps :: (cancel_semigroup_add) cancel_semigroup_add
begin

instance by (intro_classes) (auto simp add: fps_plus_def expand_fun_eq Rep_fps_eq[symmetric])
end

instantiation fps :: (group_add) group_add
begin

instance
proof
  fix a :: "'a fps" show "- a + a = 0"
    by (auto simp add: fps_plus_def fps_uminus_def fps_zero_def intro: ext)
next
  fix a b :: "'a fps" show "a - b = a + - b"
    by (auto simp add: fps_plus_def fps_uminus_def fps_zero_def 
      fps_minus_def expand_fun_eq diff_minus)
qed
end

context comm_ring_1
begin
subclass group_add proof qed
end

instantiation fps :: (zero_neq_one) zero_neq_one
begin
instance by (intro_classes, auto simp add: zero_neq_one 
  fps_one_def fps_zero_def expand_fun_eq)
end

instantiation fps :: (semiring_1) semiring
begin

instance
proof
  fix a b c :: "'a fps"
  show "(a + b) * c = a * c + b*c"
    apply (auto simp add: fps_plus_def fps_times_def, rule ext)
    unfolding setsum_addf[symmetric]
    apply (simp add: ring_simps)
    done
next
  fix a b c :: "'a fps"
  show "a * (b + c) = a * b + a*c"
    apply (auto simp add: fps_plus_def fps_times_def, rule ext)
    unfolding setsum_addf[symmetric]
    apply (simp add: ring_simps)
    done
qed
end

instantiation fps :: (semiring_1) semiring_0
begin

instance
proof
  fix a:: "'a fps" show "0 * a = 0" by (simp add: fps_zero_def fps_times_def)
next
  fix a:: "'a fps" show "a*0 = 0" by (simp add: fps_zero_def fps_times_def)
qed
end
  
section {* Selection of the nth power of the implicit variable in the infinite sum*}

definition fps_nth:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" (infixl "$" 75)
  where "f $ n = Rep_fps f n"

lemma fps_nth_Abs_fps[simp]: "Abs_fps a $ n = a n" by (simp add: fps_nth_def)
lemma fps_zero_nth[simp]: "0 $ n = 0" by (simp add: fps_zero_def)
lemma fps_one_nth[simp]: "1 $ n = (if n = 0 then 1 else 0)" 
  by (simp add: fps_one_def)
lemma fps_add_nth[simp]: "(f + g) $ n = f$n + g$n" by (simp add: fps_plus_def fps_nth_def)
lemma fps_mult_nth: "(f * g) $ n = setsum (\<lambda>i. f$i * g$(n - i)) {0..n}"
  by (simp add: fps_times_def fps_nth_def)
lemma fps_neg_nth[simp]: "(- f) $n = - (f $n)" by (simp add: fps_nth_def fps_uminus_def)
lemma fps_sub_nth[simp]: "(f - g)$n = f$n - g$n" by (simp add: fps_nth_def fps_minus_def)

lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)"
proof-
  {assume "f \<noteq> 0"
    hence "Rep_fps f \<noteq> Rep_fps 0" by simp 
    hence "\<exists>n. f $n \<noteq> 0" by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
  moreover
  {assume "\<exists>n. f$n \<noteq> 0" and "f = 0" 
    then have False by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
  ultimately show ?thesis by blast
qed

lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0))"
proof-
  let ?S = "{n. f$n \<noteq> 0}"
  {assume "\<exists>n. f$n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0)" and "f = 0" 
    then have False by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
  moreover
  {assume f0: "f \<noteq> 0"
    from f0 fps_nonzero_nth have ex: "\<exists>n. f$n \<noteq> 0" by blast
    hence Se: "?S\<noteq> {}" by blast
    from ex obtain n where n: "f$n \<noteq> 0" by blast
    from n have nS: "n \<in> ?S" by blast
        let ?U = "?S \<inter> {0..n}"
    have fU: "finite ?U" by auto
    from n have Ue: "?U \<noteq> {}" by auto
    let ?m = "Min ?U" 
    have mU: "?m \<in> ?U" using Min_in[OF fU Ue] .
    hence mn: "?m \<le> n" by simp
    from mU have mf: "f $ ?m \<noteq> 0" by blast
    {fix m assume m: "m < ?m" and f: "f $m \<noteq> 0"
      from m mn have mn': "m < n" by arith
      with f have mU': "m \<in> ?U" by simp
      from Min_le[OF fU mU'] m have False by arith}
    hence "\<forall>m <?m. f$m = 0" by blast
    with mf have "\<exists> n. f $n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0)" by blast}
  ultimately show ?thesis by blast
qed

lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)"
  by (auto simp add: fps_nth_def Rep_fps_eq[unfolded expand_fun_eq])

lemma fps_setsum_nth: "(setsum f S) $ n = setsum (\<lambda>k. (f k) $ n) S" 
proof-
  {assume "\<not> finite S" hence ?thesis by simp}
  moreover
  {assume fS: "finite S"
    have ?thesis by(induct rule: finite_induct[OF fS]) auto}
  ultimately show ?thesis by blast
qed

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

definition "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
lemma fps_const_0_eq_0[simp]: "fps_const 0 = 0" by (simp add: fps_const_def fps_eq_iff)
lemma fps_const_1_eq_1[simp]: "fps_const 1 = 1" by (simp add: fps_const_def fps_eq_iff)
lemma fps_const_neg[simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"
  by (simp add: fps_uminus_def fps_const_def fps_eq_iff)
lemma fps_const_add[simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)"
  by (simp add: fps_plus_def fps_const_def fps_eq_iff)
lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)"
  by (auto simp add: fps_times_def fps_const_def fps_eq_iff intro: setsum_0')

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)"
  unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)
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)"
  unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)

lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)"
  unfolding fps_eq_iff fps_mult_nth 
  by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f$n * c)"
  unfolding fps_eq_iff fps_mult_nth 
  by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)

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

lemma fps_mult_left_const_nth[simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n"
  by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)

lemma fps_mult_right_const_nth[simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c"
  by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)

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

instantiation fps :: (ring_1) ring_1
begin

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

instantiation fps :: (comm_ring_1) comm_ring_1
begin

instance by (intro_classes, auto simp add: diff_minus left_distrib)
end
instantiation fps :: ("{ring_no_zero_divisors, comm_ring_1}") ring_no_zero_divisors
begin

instance 
proof
  fix a b :: "'a fps"
  assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
  then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0"
    and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0" unfolding fps_nonzero_nth_minimal
    by blast+
  have eq: "({0..i+j} -{i}) \<union> {i} = {0..i+j}" by auto
  have d: "({0..i+j} -{i}) \<inter> {i} = {}" by auto
  have f: "finite ({0..i+j} -{i})" "finite {i}" by auto
  have th0: "setsum (\<lambda>k. a$k * b$(i+j - k)) ({0..i+j} -{i}) = 0"
    apply (rule setsum_0')
    apply auto
    apply (case_tac "aa < i")
    using i
    apply auto
    apply (subgoal_tac "b $ (i+j - aa) = 0")
    apply blast
    apply (rule j(2)[rule_format])
    by arith
  have "(a*b) $ (i+j) =  setsum (\<lambda>k. a$k * b$(i+j - k)) {0..i+j}"
    by (rule fps_mult_nth)
  hence "(a*b) $ (i+j) = a$i * b$j"
    unfolding setsum_Un_disjoint[OF f d, unfolded eq] th0 by simp
  with i j have "(a*b) $ (i+j) \<noteq> 0" by simp
  then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
qed
end

instantiation fps :: (idom) idom
begin

instance ..
end

section{* Inverses of formal power series *}

declare setsum_cong[fundef_cong]


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

fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where 
  "natfun_inverse f 0 = inverse (f$0)"
| "natfun_inverse f n = - inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}" 

definition fps_inverse_def: 
  "inverse f = (if f$0 = 0 then 0 else Abs_fps (natfun_inverse f))"
definition fps_divide_def: "divide \<equiv> (\<lambda>(f::'a fps) g. f * inverse g)"
instance ..
end

lemma fps_inverse_zero[simp]: 
  "inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0"
  by (simp add: fps_zero_def fps_inverse_def)

lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
  apply (auto simp add: fps_one_def fps_inverse_def expand_fun_eq)
  by (case_tac x, auto)

instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}")  division_by_zero
begin
instance
  apply (intro_classes)
  by (rule fps_inverse_zero)
end

lemma inverse_mult_eq_1[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
  shows "inverse f * f = 1"
proof-
  have c: "inverse f * f = f * inverse f" by (simp add: mult_commute)
  from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n" 
    by (simp add: fps_inverse_def)
  from f0 have th0: "(inverse f * f) $ 0 = 1"
    by (simp add: fps_inverse_def fps_one_def fps_mult_nth)
  {fix n::nat assume np: "n >0 "
    from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto
    have d: "{0} \<inter> {1 .. n} = {}" by auto
    have f: "finite {0::nat}" "finite {1..n}" by auto
    from f0 np have th0: "- (inverse f$n) = 
      (setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}) / (f$0)"
      by (cases n, simp_all add: divide_inverse fps_inverse_def fps_nth_def ring_simps)
    from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
    have th1: "setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n} = 
      - (f$0) * (inverse f)$n" 
      by (simp add: ring_simps)
    have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n - i))" 
      unfolding fps_mult_nth ifn ..
    also have "\<dots> = f$0 * natfun_inverse f n 
      + (\<Sum>i = 1..n. f$i * natfun_inverse f (n-i))"
      unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]]
      by simp
    also have "\<dots> = 0" unfolding th1 ifn by simp
    finally have "(inverse f * f)$n = 0" unfolding c . }
  with th0 show ?thesis by (simp add: fps_eq_iff)
qed

lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) \<longleftrightarrow> f$0 = 0"
  apply (simp add: fps_inverse_def)
  by (metis fps_nth_def fps_nth_def inverse_zero_imp_zero)

lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $0 = 0"
proof-
  {assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)}
  moreover
  {assume h: "inverse f = 0" and c: "f $0 \<noteq> 0"
    from inverse_mult_eq_1[OF c] h have False by simp}
  ultimately show ?thesis by blast
qed

lemma fps_inverse_idempotent[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
  shows "inverse (inverse f) = f"
proof-
  from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp
  from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0] 
  have th0: "inverse f * f = inverse f * inverse (inverse f)"   by (simp add: mult_ac)
  then show ?thesis using f0 unfolding mult_cancel_left by simp
qed

lemma fps_inverse_unique: assumes f0: "f$0 \<noteq> (0::'a::field)" and fg: "f*g = 1" 
  shows "inverse f = g"
proof-
  from inverse_mult_eq_1[OF f0] fg
  have th0: "inverse f * f = g * f" by (simp add: mult_ac)
  then show ?thesis using f0  unfolding mult_cancel_right
    unfolding Rep_fps_eq[of f 0, symmetric]
    by (auto simp add: fps_zero_def expand_fun_eq fps_nth_def)
qed

lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
  = Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
  apply (rule fps_inverse_unique)
  apply simp
  apply (simp add: fps_eq_iff fps_nth_def fps_times_def fps_one_def)
proof(clarsimp)
  fix n::nat assume n: "n > 0"
  let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n - i = 1 then - 1 else 0"
  let ?g = "\<lambda>i. if i = n then 1 else if i=n - 1 then - 1 else 0"
  let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
  have th1: "setsum ?f {0..n} = setsum ?g {0..n}"  
    by (rule setsum_cong2) auto
  have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"  
    using n apply - by (rule setsum_cong2) auto
  have eq: "{0 .. n} = {0.. n - 1} \<union> {n}" by auto
  from n have d: "{0.. n - 1} \<inter> {n} = {}" by auto 
  have f: "finite {0.. n - 1}" "finite {n}" by auto
  show "setsum ?f {0..n} = 0"
    unfolding th1 
    apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
    unfolding th2
    by(simp add: setsum_delta)
qed

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

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

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

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

lemma fps_deriv_mult[simp]: 
  fixes f :: "('a :: comm_ring_1) fps"
  shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
proof-
  let ?D = "fps_deriv"
  {fix n::nat
    let ?Zn = "{0 ..n}"
    let ?Zn1 = "{0 .. n + 1}"
    let ?f = "\<lambda>i. i + 1"
    have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)
    have eq: "{1.. n+1} = ?f ` {0..n}" by auto
    let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n - i) +
        of_nat (i+1)* f $ (i+1) * g $ (n - i)"
    let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1) - i) +
        of_nat i* f $ i * g $ ((n + 1) - i)"
    {fix k assume k: "k \<in> {0..n}"
      have "?h (k + 1) = ?g k" using k by auto}
    note th0 = this
    have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto
    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"
      apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
      apply (simp add: inj_on_def Ball_def)
      apply presburger
      apply (rule set_ext)
      apply (presburger add: image_iff)
      by simp
    have s1: "setsum (\<lambda>i. f $ i * g $ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. f $ (n + 1 - i) * g $ i) ?Zn1"
      apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
      apply (simp add: inj_on_def Ball_def)
      apply presburger
      apply (rule set_ext)
      apply (presburger add: image_iff)
      by simp
    have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute)
    also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
      by (simp add: fps_mult_nth setsum_addf[symmetric])
    also have "\<dots> = setsum ?h {1..n+1}"
      using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto
    also have "\<dots> = setsum ?h {0..n+1}"
      apply (rule setsum_mono_zero_left)
      apply simp
      apply (simp add: subset_eq)
      unfolding eq'
      by simp
    also have "\<dots> = (fps_deriv (f * g)) $ n"
      apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)
      unfolding s0 s1
      unfolding setsum_addf[symmetric] setsum_right_distrib
      apply (rule setsum_cong2)
      by (auto simp add: of_nat_diff ring_simps)
    finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .}
  then show ?thesis unfolding fps_eq_iff by auto 
qed

lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: ('a:: comm_ring_1) fps)) = - (fps_deriv f)"
  by (simp add: fps_uminus_def fps_eq_iff fps_deriv_def fps_nth_def)
lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g"
  using fps_deriv_linear[of 1 f 1 g] by simp

lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) - g) = fps_deriv f - fps_deriv g"
  unfolding diff_minus by simp 

lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
  by (simp add: fps_deriv_def fps_const_def fps_zero_def)

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

lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
  by (simp add: fps_deriv_def fps_eq_iff)

lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
  by (simp add: fps_deriv_def fps_eq_iff )

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

lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S"
proof-
  {assume "\<not> finite S" hence ?thesis by simp}
  moreover
  {assume fS: "finite S"
    have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
  ultimately show ?thesis by blast
qed

lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))"
proof-
  {assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp
    hence "fps_deriv f = 0" by simp }
  moreover
  {assume z: "fps_deriv f = 0"
    hence "\<forall>n. (fps_deriv f)$n = 0" by simp
    hence "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
    hence "f = fps_const (f$0)"
      apply (clarsimp simp add: fps_eq_iff fps_const_def)
      apply (erule_tac x="n - 1" in allE)
      by simp}
  ultimately show ?thesis by blast
qed

lemma fps_deriv_eq_iff: 
  fixes f:: "('a::{idom,semiring_char_0}) fps"
  shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0 - g$0) + g)"
proof-
  have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0" by simp
  also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f-g)$0)" unfolding fps_deriv_eq_0_iff ..
  finally show ?thesis by (simp add: ring_simps)
qed

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)"
  apply auto unfolding fps_deriv_eq_iff by blast
  

fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where
  "fps_nth_deriv 0 f = f"
| "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"

lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
  by (induct n arbitrary: f, auto)

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

lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n (- (f:: ('a:: comm_ring_1) fps)) = - (fps_nth_deriv n f)"
  by (induct n arbitrary: f, simp_all)

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

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"
  unfolding diff_minus fps_nth_deriv_add by simp 

lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
  by (induct n, simp_all )

lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
  by (induct n, simp_all )

lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
  by (cases n, simp_all)

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

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

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"
proof-
  {assume "\<not> finite S" hence ?thesis by simp}
  moreover
  {assume fS: "finite S"
    have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
  ultimately show ?thesis by blast
qed

lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)"
  by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult)

section {* Powers*}

instantiation fps :: (semiring_1) power
begin

fun fps_pow :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where
  "fps_pow 0 f = 1"
| "fps_pow (Suc n) f = f * fps_pow n f"

definition fps_power_def: "power (f::'a fps) n = fps_pow n f"
instance ..
end

instantiation fps :: (comm_ring_1) recpower
begin
instance
  apply (intro_classes)
  by (simp_all add: fps_power_def)
end

lemma eq_neg_iff_add_eq_0: "(a::'a::ring) = -b \<longleftrightarrow> a + b = 0"
proof-
  {assume "a = -b" hence "b + a = b + -b" by simp
    hence "a + b = 0" by (simp add: ring_simps)}
  moreover
  {assume "a + b = 0" hence "a + b - b = -b" by simp
    hence "a = -b" by simp}
  ultimately show ?thesis by blast
qed

lemma fps_sqrare_eq_iff: "(a:: 'a::idom fps)^ 2 = b^2  \<longleftrightarrow> (a = b \<or> a = -b)"
proof-
  {assume "a = b \<or> a = -b" hence "a^2 = b^2" by auto}
  moreover
  {assume "a^2 = b^2 "
    hence "a^2 - b^2 = 0" by simp
    hence "(a-b) * (a+b) = 0" by (simp add: power2_eq_square ring_simps)
    hence "a = b \<or> a = -b" by (simp add: eq_neg_iff_add_eq_0)}
  ultimately show ?thesis by blast
qed

lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)"
  by (induct n, auto simp add: fps_power_def fps_times_def fps_nth_def fps_one_def)

lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"
proof(induct n)
  case 0 thus ?case by (simp add: fps_power_def)
next
  case (Suc n)
  note h = Suc.hyps[OF `a$0 = 1`]
  show ?case unfolding power_Suc fps_mult_nth 
    using h `a$0 = 1`  fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: ring_simps)
qed

lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"
  by (induct n, auto simp add: fps_power_def fps_mult_nth)

lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"
  by (induct n, auto simp add: fps_power_def fps_mult_nth)

lemma startsby_power:"a $0 = (v::'a::{comm_ring_1, recpower}) \<Longrightarrow> a^n $0 = v^n"
  by (induct n, auto simp add: fps_power_def fps_mult_nth power_Suc)

lemma startsby_zero_power_iff[simp]:
  "a^n $0 = (0::'a::{idom, recpower}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)"
apply (rule iffI)
apply (induct n, auto simp add: power_Suc fps_mult_nth)
by (rule startsby_zero_power, simp_all)

lemma startsby_zero_power_prefix: 
  assumes a0: "a $0 = (0::'a::idom)"
  shows "\<forall>n < k. a ^ k $ n = 0"
  using a0 
proof(induct k rule: nat_less_induct)
  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)"
  let ?ths = "\<forall>m<k. a ^ k $ m = 0"
  {assume "k = 0" then have ?ths by simp}
  moreover
  {fix l assume k: "k = Suc l"
    {fix m assume mk: "m < k"
      {assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0 
	  by simp}
      moreover
      {assume m0: "m \<noteq> 0"
	have "a ^k $ m = (a^l * a) $m" by (simp add: k power_Suc mult_commute)
	also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))" by (simp add: fps_mult_nth)
	also have "\<dots> = 0" apply (rule setsum_0')
	  apply auto
	  apply (case_tac "aa = m")
	  using a0
	  apply simp
	  apply (rule H[rule_format])
	  using a0 k mk by auto 
	finally have "a^k $ m = 0" .}
    ultimately have "a^k $ m = 0" by blast}
    hence ?ths by blast}
  ultimately show ?ths by (cases k, auto)
qed

lemma startsby_zero_setsum_depends: 
  assumes a0: "a $0 = (0::'a::idom)" and kn: "n \<ge> k"
  shows "setsum (\<lambda>i. (a ^ i)$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)$k) {0 .. k}"
  apply (rule setsum_mono_zero_right)
  using kn apply auto
  apply (rule startsby_zero_power_prefix[rule_format, OF a0])
  by arith

lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{recpower, idom})"
  shows "a^n $ n = (a$1) ^ n"
proof(induct n)
  case 0 thus ?case by (simp add: power_0)
next
  case (Suc n)
  have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: ring_simps power_Suc)
  also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth)
  also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {n .. Suc n}"
    apply (rule setsum_mono_zero_right)
    apply simp
    apply clarsimp
    apply clarsimp
    apply (rule startsby_zero_power_prefix[rule_format, OF a0])
    apply arith
    done
  also have "\<dots> = a^n $ n * a$1" using a0 by simp
  finally show ?case using Suc.hyps by (simp add: power_Suc)
qed

lemma fps_inverse_power:
  fixes a :: "('a::{field, recpower}) fps"
  shows "inverse (a^n) = inverse a ^ n"
proof-
  {assume a0: "a$0 = 0"
    hence eq: "inverse a = 0" by (simp add: fps_inverse_def)
    {assume "n = 0" hence ?thesis by simp}
    moreover
    {assume n: "n > 0"
      from startsby_zero_power[OF a0 n] eq a0 n have ?thesis 
	by (simp add: fps_inverse_def)}
    ultimately have ?thesis by blast}
  moreover
  {assume a0: "a$0 \<noteq> 0"
    have ?thesis
      apply (rule fps_inverse_unique)
      apply (simp add: a0)
      unfolding power_mult_distrib[symmetric]
      apply (rule ssubst[where t = "a * inverse a" and s= 1])
      apply simp_all
      apply (subst mult_commute)
      by (rule inverse_mult_eq_1[OF a0])}
  ultimately show ?thesis by blast
qed

lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n - 1)"
  apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add)
  by (case_tac n, auto simp add: power_Suc ring_simps)

lemma fps_inverse_deriv: 
  fixes a:: "('a :: field) fps"
  assumes a0: "a$0 \<noteq> 0"
  shows "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2"
proof-
  from inverse_mult_eq_1[OF a0]
  have "fps_deriv (inverse a * a) = 0" by simp
  hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp
  hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"  by simp
  with inverse_mult_eq_1[OF a0]
  have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0"
    unfolding power2_eq_square
    apply (simp add: ring_simps)
    by (simp add: mult_assoc[symmetric])
  hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * inverse a ^ 2 = 0 - fps_deriv a * inverse a ^ 2"
    by simp
  then show "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2" by (simp add: ring_simps)
qed

lemma fps_inverse_mult: 
  fixes a::"('a :: field) fps"
  shows "inverse (a * b) = inverse a * inverse b"
proof-
  {assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
    from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
    have ?thesis unfolding th by simp}
  moreover
  {assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
    from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
    have ?thesis unfolding th by simp}
  moreover
  {assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0"
    from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
    from inverse_mult_eq_1[OF ab0] 
    have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
    then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
      by (simp add: ring_simps)
    then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp}
ultimately show ?thesis by blast
qed

lemma fps_inverse_deriv': 
  fixes a:: "('a :: field) fps"
  assumes a0: "a$0 \<noteq> 0"
  shows "fps_deriv (inverse a) = - fps_deriv a / a ^ 2"
  using fps_inverse_deriv[OF a0]
  unfolding power2_eq_square fps_divide_def
    fps_inverse_mult by simp

lemma inverse_mult_eq_1': assumes f0: "f$0 \<noteq> (0::'a::field)"
  shows "f * inverse f= 1"
  by (metis mult_commute inverse_mult_eq_1 f0)

lemma fps_divide_deriv:   fixes a:: "('a :: field) fps"
  assumes a0: "b$0 \<noteq> 0"
  shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b ^ 2"
  using fps_inverse_deriv[OF a0]
  by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
  
section{* The eXtractor series X*}

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

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

lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
  = 1 - X"
  by (simp add: fps_inverse_gp fps_eq_iff X_def fps_minus_def fps_one_def)

lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) $n = (if n = 0 then 0 else f $ (n - 1))"
proof-
  {assume n: "n \<noteq> 0"
    have fN: "finite {0 .. n}" by simp
    have "(X * f) $n = (\<Sum>i = 0..n. X $ i * f $ (n - i))" by (simp add: fps_mult_nth)
    also have "\<dots> = f $ (n - 1)" 
      using n by (simp add: X_def cond_value_iff cond_application_beta setsum_delta[OF fN] 
	del: One_nat_def cong del:  if_weak_cong)
  finally have ?thesis using n by simp }
  moreover
  {assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)}
  ultimately show ?thesis by blast
qed

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

lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"
proof(induct k)
  case 0 thus ?case by (simp add: X_def fps_power_def fps_eq_iff)
next
  case (Suc k)
  {fix m 
    have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m - 1))"
      by (simp add: power_Suc del: One_nat_def)
    then     have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)"
      using Suc.hyps by (auto cong del: if_weak_cong)}
  then show ?case by (simp add: fps_eq_iff)
qed

lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n - k))"
  apply (induct k arbitrary: n)
  apply (simp)
  unfolding power_Suc mult_assoc 
  by (case_tac n, auto)

lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n - k))"
  by (metis X_power_mult_nth mult_commute)
lemma fps_deriv_X[simp]: "fps_deriv X = 1"
  by (simp add: fps_deriv_def X_def fps_eq_iff)

lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)"
  by (cases "n", simp_all)

lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def)
lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))"
  by (simp add: X_power_iff)

lemma fps_inverse_X_plus1:
  "inverse (1 + X) = Abs_fps (\<lambda>n. (- (1::'a::{recpower, field})) ^ n)" (is "_ = ?r")
proof-
  have eq: "(1 + X) * ?r = 1"
    unfolding minus_one_power_iff
    apply (auto simp add: ring_simps fps_eq_iff)
    by presburger+
  show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
qed

  
section{* Integration *}
definition "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"

lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a"
  by (simp add: fps_integral_def fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)

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")
proof-
  have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)
  moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def)
  ultimately show ?thesis
    unfolding fps_deriv_eq_iff by auto
qed
  
section {* Composition of FPSs *}
definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where
  fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})"

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

lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"
  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)
 
lemma fps_const_compose[simp]: 
  "fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)"
  apply (auto simp add: fps_eq_iff fps_compose_nth fps_mult_nth
  cond_application_beta cond_value_iff cong del: if_weak_cong)
  by (simp add: setsum_delta )

lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"
  apply (auto simp add: fps_compose_def fps_eq_iff cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
  apply (simp add: power_Suc)
  apply (subgoal_tac "n = 0")
  by simp_all


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

subsection {* Rule 1 *}
  (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)

lemma fps_power_mult_eq_shift: 
  "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")
proof-
  {fix n:: nat
    have "?lhs $ n = (if n < Suc k then 0 else a n)" 
      unfolding X_power_mult_nth by auto
    also have "\<dots> = ?rhs $ n"
    proof(induct k)
      case 0 thus ?case by (simp add: fps_setsum_nth power_Suc)
    next
      case (Suc k)
      note th = Suc.hyps[symmetric]
      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)
      also  have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
	using th 
	unfolding fps_sub_nth by simp
      also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
	unfolding X_power_mult_right_nth
	apply (auto simp add: not_less fps_const_def)
	apply (rule cong[of a a, OF refl])
	by arith
      finally show ?case by simp
    qed
    finally have "?lhs $ n = ?rhs $ n"  .}
  then show ?thesis by (simp add: fps_eq_iff)
qed

subsection{* Rule 2*}

  (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
  (* If f reprents {a_n} and P is a polynomial, then 
        P(xD) f represents {P(n) a_n}*)

definition "XD = op * X o fps_deriv"

lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"
  by (simp add: XD_def ring_simps)

lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
  by (simp add: XD_def ring_simps)

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)"
  by simp


fun funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
  "funpow 0 f = id"
  | "funpow (Suc n) f = f o funpow n f"

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)"
  by (induct n, simp_all)

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)

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

subsection{* Rule 3 is trivial and is given by fps_times_def*}
subsection{* Rule 5 --- summation and "division" by (1 - X)*}

lemma fps_divide_X_minus1_setsum_lemma:
  "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
proof-
  let ?X = "X::('a::comm_ring_1) fps"
  let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
  have th0: "\<And>i. (1 - (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp
  {fix n:: nat
    {assume "n=0" hence "a$n = ((1 - ?X) * ?sa) $ n" 
	by (simp add: fps_mult_nth)}
    moreover
    {assume n0: "n \<noteq> 0"
      then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}"
	"{0..n - 1}\<union>{n} = {0..n}"
	apply (simp_all add: expand_set_eq) by presburger+
      have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}" 
	"{0..n - 1}\<inter>{n} ={}" using n0
	by (simp_all add: expand_set_eq, presburger+)
      have f: "finite {0}" "finite {1}" "finite {2 .. n}" 
	"finite {0 .. n - 1}" "finite {n}" by simp_all 
    have "((1 - ?X) * ?sa) $ n = setsum (\<lambda>i. (1 - ?X)$ i * ?sa $ (n - i)) {0 .. n}"
      by (simp add: fps_mult_nth)
    also have "\<dots> = a$n" unfolding th0
      unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
      unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]
      apply (simp)
      unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]
      by simp
    finally have "a$n = ((1 - ?X) * ?sa) $ n" by simp}
  ultimately have "a$n = ((1 - ?X) * ?sa) $ n" by blast}
then show ?thesis 
  unfolding fps_eq_iff by blast
qed

lemma fps_divide_X_minus1_setsum:
  "a /((1::('a::field) fps) - X)  = Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
proof-
  let ?X = "1 - (X::('a::field) fps)"
  have th0: "?X $ 0 \<noteq> 0" by simp
  have "a /?X = ?X *  Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) * inverse ?X"
    using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
    by (simp add: fps_divide_def mult_assoc)
  also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) "
    by (simp add: mult_ac)
  finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
qed

subsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary 
  finite product of FPS, also the relvant instance of powers of a FPS*}

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

lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
  apply (auto simp add: natpermute_def)
  apply (case_tac x, auto)
  done

lemma foldl_add_start0: 
  "foldl op + x xs = x + foldl op + (0::nat) xs"
  apply (induct xs arbitrary: x)
  apply simp
  unfolding foldl.simps
  apply atomize
  apply (subgoal_tac "\<forall>x\<Colon>nat. foldl op + x xs = x + foldl op + (0\<Colon>nat) xs")
  apply (erule_tac x="x + a" in allE)
  apply (erule_tac x="a" in allE)
  apply simp
  apply assumption
  done

lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys"
  apply (induct ys arbitrary: x xs)
  apply auto
  apply (subst (2) foldl_add_start0)
  apply simp
  apply (subst (2) foldl_add_start0)
  by simp

lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}"
proof(induct xs arbitrary: x)
  case Nil thus ?case by simp
next
  case (Cons a as x)
  have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}"
    apply (rule setsum_reindex_cong [where f=Suc])
    by (simp_all add: inj_on_def)
  have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all
  have d: "{0} \<inter> {1..<length (a#as)} = {}" by simp
  have seq: "{0} \<union> {1..<length(a#as)} = {0 ..<length (a#as)}" by auto
  have "foldl op + x (a#as) = x + foldl op + a as "
    apply (subst foldl_add_start0)    by simp
  also have "\<dots> = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp
  also have "\<dots> = x + setsum (op ! (a#as)) {0..<length (a#as)}"
    unfolding eq[symmetric] 
    unfolding setsum_Un_disjoint[OF f d, unfolded seq]
    by simp
  finally show ?case  .
qed


lemma append_natpermute_less_eq:
  assumes h: "xs@ys \<in> natpermute n k" shows "foldl op + 0 xs \<le> n" and "foldl op + 0 ys \<le> n"
proof-
  {from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def)
    hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .}
  note th = this
  {from th show "foldl op + 0 xs \<le> n" by simp}
  {from th show "foldl op + 0 ys \<le> n" by simp}
qed

lemma natpermute_split:
  assumes mn: "h \<le> k"
  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)")
proof-
  {fix l assume l: "l \<in> ?R" 
    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
    from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def)
    from ys have ys': "foldl op + 0 ys = n - m" by (simp add: natpermute_def)
    have "l \<in> ?L" using leq xs ys h 
      apply simp
      apply (clarsimp simp add: natpermute_def simp del: foldl_append)
      apply (simp add: foldl_add_append[unfolded foldl_append])
      unfolding xs' ys'
      using mn xs ys 
      unfolding natpermute_def by simp}
  moreover
  {fix l assume l: "l \<in> natpermute n k"
    let ?xs = "take h l"
    let ?ys = "drop h l"
    let ?m = "foldl op + 0 ?xs"
    from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def)
    have xs: "?xs \<in> natpermute ?m h" using l mn by (simp add: natpermute_def)     
    have ys: "?ys \<in> natpermute (n - ?m) (k - h)" using l mn ls[unfolded foldl_add_append]
      by (simp add: natpermute_def)
    from ls have m: "?m \<in> {0..n}"  unfolding foldl_add_append by simp
    from xs ys ls have "l \<in> ?R" 
      apply auto
      apply (rule bexI[where x = "?m"])
      apply (rule exI[where x = "?xs"])
      apply (rule exI[where x = "?ys"])
      using ls l unfolding foldl_add_append 
      by (auto simp add: natpermute_def)}
  ultimately show ?thesis by blast
qed

lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
  by (auto simp add: natpermute_def)
lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
  apply (auto simp add: set_replicate_conv_if natpermute_def)
  apply (rule nth_equalityI)
  by simp_all

lemma natpermute_finite: "finite (natpermute n k)"
proof(induct k arbitrary: n)
  case 0 thus ?case 
    apply (subst natpermute_split[of 0 0, simplified])
    by (simp add: natpermute_0)
next
  case (Suc k)
  then show ?case unfolding natpermute_split[of k "Suc k", simplified]
    apply -
    apply (rule finite_UN_I)
    apply simp
    unfolding One_nat_def[symmetric] natlist_trivial_1
    apply simp
    unfolding image_Collect[symmetric]
    unfolding Collect_def mem_def
    apply (rule finite_imageI)
    apply blast
    done
qed

lemma natpermute_contain_maximal:
  "{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"
  (is "?A = ?B")
proof-
  {fix xs assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"
    from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H
      unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def) 
    have eqs: "({0..k} - {i}) \<union> {i} = {0..k}" using i by auto
    have f: "finite({0..k} - {i})" "finite {i}" by auto
    have d: "({0..k} - {i}) \<inter> {i} = {}" using i by auto
    from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def)
      unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost)
    also have "\<dots> = n + setsum (nth xs) ({0..k} - {i})"
      unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp
    finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0" by auto
    from H have xsl: "length xs = k+1" by (simp add: natpermute_def)
    from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
      unfolding length_replicate  by arith+
    have "xs = replicate (k+1) 0 [i := n]"
      apply (rule nth_equalityI)
      unfolding xsl length_list_update length_replicate
      apply simp
      apply clarify
      unfolding nth_list_update[OF i'(1)]
      using i zxs
      by (case_tac "ia=i", auto simp del: replicate.simps)
    then have "xs \<in> ?B" using i by blast}
  moreover
  {fix i assume i: "i \<in> {0..k}"
    let ?xs = "replicate (k+1) 0 [i:=n]"
    have nxs: "n \<in> set ?xs"
      apply (rule set_update_memI) using i by simp
    have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update)
    have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}"
      unfolding foldl_add_setsum add_0 length_replicate length_list_update ..
    also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
      apply (rule setsum_cong2) by (simp del: replicate.simps)
    also have "\<dots> = n" using i by (simp add: setsum_delta)
    finally 
    have "?xs \<in> natpermute n (k+1)" using xsl unfolding natpermute_def Collect_def mem_def
      by blast
    then have "?xs \<in> ?A"  using nxs  by blast}
  ultimately show ?thesis by auto
qed

    (* The general form *)	
lemma fps_setprod_nth:
  fixes m :: nat and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"
  shows "(setprod a {0 .. m})$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"
  (is "?P m n")
proof(induct m arbitrary: n rule: nat_less_induct)
  fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
  {assume m0: "m = 0"
    hence "?P m n" apply simp
      unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp}
  moreover
  {fix k assume k: "m = Suc k"
    have km: "k < m" using k by arith
    have u0: "{0 .. k} \<union> {m} = {0..m}" using k apply (simp add: expand_set_eq) by presburger
    have f0: "finite {0 .. k}" "finite {m}" by auto
    have d0: "{0 .. k} \<inter> {m} = {}" using k by auto
    have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n"
      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp
    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))"
      unfolding fps_mult_nth H[rule_format, OF km] ..
    also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)"
      apply (simp add: k)
      unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k]
      apply (subst setsum_UN_disjoint)
      apply simp 
      apply simp
      unfolding image_Collect[symmetric]
      apply clarsimp
      apply (rule finite_imageI)
      apply (rule natpermute_finite)
      apply (clarsimp simp add: expand_set_eq)
      apply auto
      apply (rule setsum_cong2)
      unfolding setsum_left_distrib
      apply (rule sym)
      apply (rule_tac f="\<lambda>xs. xs @[n - x]" in  setsum_reindex_cong)
      apply (simp add: inj_on_def)
      apply auto
      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]
      apply (clarsimp simp add: natpermute_def nth_append)
      apply (rule_tac f="\<lambda>x. x * a (Suc k) $ (n - foldl op + 0 aa)" in cong[OF refl])
      apply (rule setprod_cong)
      apply simp
      apply simp
      done
    finally have "?P m n" .}
  ultimately show "?P m n " by (cases m, auto)
qed

text{* The special form for powers *}
lemma fps_power_nth_Suc:
  fixes m :: nat and a :: "('a::comm_ring_1) fps"
  shows "(a ^ Suc m)$n = setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))"
proof-
  have f: "finite {0 ..m}" by simp
  have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp
  show ?thesis unfolding th0 fps_setprod_nth ..
qed
lemma fps_power_nth:
  fixes m :: nat and a :: "('a::comm_ring_1) fps"
  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))"
  by (cases m, simp_all add: fps_power_nth_Suc)

lemma fps_nth_power_0: 
  fixes m :: nat and a :: "('a::{comm_ring_1, recpower}) fps"
  shows "(a ^m)$0 = (a$0) ^ m"
proof-
  {assume "m=0" hence ?thesis by simp}
  moreover
  {fix n assume m: "m = Suc n"
    have c: "m = card {0..n}" using m by simp
   have "(a ^m)$0 = setprod (\<lambda>i. a$0) {0..n}"
     apply (simp add: m fps_power_nth del: replicate.simps)
     apply (rule setprod_cong)
     by (simp_all del: replicate.simps)
   also have "\<dots> = (a$0) ^ m"
     unfolding c by (rule setprod_constant, simp)
   finally have ?thesis .}
 ultimately show ?thesis by (cases m, auto)
qed

lemma fps_compose_inj_right: 
  assumes a0: "a$0 = (0::'a::{recpower,idom})"
  and a1: "a$1 \<noteq> 0"
  shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")
proof-
  {assume ?rhs then have "?lhs" by simp}
  moreover
  {assume h: ?lhs
    {fix n have "b$n = c$n" 
      proof(induct n rule: nat_less_induct)
	fix n assume H: "\<forall>m<n. b$m = c$m"
	{assume n0: "n=0"
	  from h have "(b oo a)$n = (c oo a)$n" by simp
	  hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)}
	moreover
	{fix n1 assume n1: "n = Suc n1"
	  have f: "finite {0 .. n1}" "finite {n}" by simp_all
	  have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto