(*  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)