Quelle  Formal_Power_Series.thy

(*  Title:      HOL/Computational_Algebra/Formal_Power_Series.thy
  Author: Amine Chaieb, University of Cambridge
  Author: Jeremy Sylvestre, University of Alberta (Augustana Campus)
  Author: Manuel Eberl, TU München
*)

section ‹A formalization of formal power series›

theory Formal_Power_Series
imports
  Complex_Main
  Euclidean_Algorithm
  Primes
  "HOL-Library.FuncSet"
  "HOL-Library.Multiset"
begin


subsection ‹The type of formal power series›

typedef 'a fps = "{f :: nat ==> 'a. True}"
  morphisms fps_nth Abs_fps
  by simp

notation fps_nth (infixl ‹$› 75)

lemma expand_fps_eq: "p = q ⟷ (∀n. p $ n = q $ n)"
  by (simp add: fps_nth_inject [symmetric] fun_eq_iff)

lemmas fps_eq_iff = expand_fps_eq

lemma fps_ext: "(∧n. p $ n = q $ n) ==> p = q"
  by (simp add: expand_fps_eq)

lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n"
  by (simp add: Abs_fps_inverse)

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 = Abs_fps (λn. 0)"
  instance ..
end

lemma fps_zero_nth [simp]: "0 $ n = 0"
  unfolding fps_zero_def by simp

lemma fps_nonzero_nth: "f ≠ 0 ⟷ (∃ n. f $ n ≠ 0)"
  by (simp add: expand_fps_eq)

lemma fps_nonzero_nth_minimal: "f ≠ 0 ⟷ (∃n. f $ n ≠ 0 ∧ (∀m < n. f $ m = 0))"
  (is "?lhs ⟷ ?rhs")
proof
  let ?n = "LEAST n. f $ n ≠ 0"
  show ?rhs if ?lhs
  proof -
    from that have "∃n. f $ n ≠ 0"
      by (simp add: fps_nonzero_nth)
    then have "f $ ?n ≠ 0"
      by (rule LeastI_ex)
    moreover have "∀m
      by (auto dest: not_less_Least)
    ultimately show ?thesis by metis
  qed
qed (auto simp: expand_fps_eq)

lemma fps_nonzeroI: "f$n ≠ 0 ==> f ≠ 0"
  by auto

instantiation fps :: ("{one, zero}") one
begin
  definition fps_one_def: "1 = Abs_fps (λn. if n = 0 then 1 else 0)"
  instance ..
end

lemma fps_one_nth [simp]: "1 $ n = (if n = 0 then 1 else 0)"
  unfolding fps_one_def by simp

instantiation fps :: (plus) plus
begin
  definition fps_plus_def: "(+) = (λf g. Abs_fps (λn. f $ n + g $ n))"
  instance ..
end

lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n"
  unfolding fps_plus_def by simp

instantiation fps :: (minus) minus
begin
  definition fps_minus_def: "(-) = (λf g. Abs_fps (λn. f $ n - g $ n))"
  instance ..
end

lemma fps_sub_nth [simp]: "(f - g) $ n = f $ n - g $ n"
  unfolding fps_minus_def by simp

instantiation fps :: (uminus) uminus
begin
  definition fps_uminus_def: "uminus = (λf. Abs_fps (λn. - (f $ n)))"
  instance ..
end

lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)"
  unfolding fps_uminus_def by simp

lemma fps_neg_0 [simp]: "-(0::'a::group_add fps) = 0"
  by (rule iffD2, rule fps_eq_iff, auto)

instantiation fps :: ("{comm_monoid_add, times}") times
begin
  definition fps_times_def: "(*) = (\f g. Abs_fps (\n. \i=0..n. f $ i * g $ (n - i)))"
  instance ..
end

lemma fps_mult_nth: "(f * g) $ n = (∑i=0..n. f$i * g$(n - i))"
  unfolding fps_times_def by simp

lemma fps_mult_nth_0 [simp]: "(f * g) $ 0 = f $ 0 * g $ 0"
  unfolding fps_times_def by simp

lemma fps_mult_nth_1: "(f * g) $ 1 = f$0 * g$1 + f$1 * g$0"
  by (simp add: fps_mult_nth)

lemma fps_mult_nth_1' [simp]: "(f * g) $ Suc 0 = f$0 * g$Suc 0 + f$Suc 0 * g$0"
  by (simp add: fps_mult_nth)

lemmas mult_nth_0 = fps_mult_nth_0
lemmas mult_nth_1 = fps_mult_nth_1

instance fps :: ("{comm_monoid_add, mult_zero}") mult_zero
proof
  fix a :: "'a fps"
  show "0 * a = 0" by (simp add: fps_ext fps_mult_nth)
  show "a * 0 = 0" by (simp add: fps_ext fps_mult_nth)
qed

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

lemma mult_delta_left:
  fixes x y :: "'a::mult_zero"
  shows "(if b then x else 0) * y = (if b then x * y else 0)"
  by simp

lemma mult_delta_right:
  fixes x y :: "'a::mult_zero"
  shows "x * (if b then y else 0) = (if b then x * y else 0)"
  by simp

lemma fps_one_mult:
  fixes f :: "'a::{comm_monoid_add, mult_zero, monoid_mult} fps"
  shows "1 * f = f"
  and   "f * 1 = f"
  by    (simp_all add: fps_ext fps_mult_nth mult_delta_left mult_delta_right)


subsection ‹Subdegrees›

definition subdegree :: "('a::zero) fps ==> nat" where
  "subdegree f = (if f = 0 then 0 else LEAST n. f$n ≠ 0)"

lemma subdegreeI:
  assumes "f $ d ≠ 0" and "∧i. i < d ==> f $ i = 0"
  shows   "subdegree f = d"
  by (smt (verit) LeastI_ex assms fps_zero_nth linorder_cases not_less_Least subdegree_def)

lemma nth_subdegree_nonzero [simp,intro]: "f ≠ 0 ==> f $ subdegree f ≠ 0"
  using fps_nonzero_nth_minimal subdegreeI by blast

lemma nth_less_subdegree_zero [dest]: "n < subdegree f ==> f $ n = 0"
  by (metis fps_nonzero_nth_minimal fps_zero_nth subdegreeI)

lemma subdegree_geI:
  assumes "f ≠ 0" "∧i. i < n ==> f$i = 0"
  shows   "subdegree f ≥ n"
  by (meson assms leI nth_subdegree_nonzero)

lemma subdegree_greaterI:
  assumes "f ≠ 0" "∧i. i ≤ n ==> f$i = 0"
  shows   "subdegree f > n"
  by (meson assms leI nth_subdegree_nonzero)

lemma subdegree_leI:
  "f $ n ≠ 0 ==> subdegree f ≤ n"
  using linorder_not_less by blast

lemma subdegree_0 [simp]: "subdegree 0 = 0"
  by (simp add: subdegree_def)

lemma subdegree_1 [simp]: "subdegree 1 = 0"
  by (metis fps_one_nth nth_subdegree_nonzero subdegree_0)

lemma subdegree_eq_0_iff: "subdegree f = 0 ⟷ f = 0 ∨ f $ 0 ≠ 0"
  using nth_subdegree_nonzero subdegree_leI by fastforce

lemma subdegree_eq_0 [simp]: "f $ 0 ≠ 0 ==> subdegree f = 0"
  by (simp add: subdegree_eq_0_iff)

lemma nth_subdegree_zero_iff [simp]: "f $ subdegree f = 0 ⟷ f = 0"
  by (cases "f = 0") auto

lemma fps_nonzero_subdegree_nonzeroI: "subdegree f > 0 ==> f ≠ 0"
 by auto

lemma subdegree_uminus [simp]:
  "subdegree (-(f::('a::group_add) fps)) = subdegree f"
proof (cases "f=0")
  case False thus ?thesis by (force intro: subdegreeI)
qed simp

lemma subdegree_minus_commute [simp]:
  fixes f :: "'a::group_add fps"
  shows "subdegree (f-g) = subdegree (g - f)"
proof (cases "g-f=0")
  case True then show ?thesis
    by (metis fps_sub_nth nth_subdegree_nonzero right_minus_eq)
next
  case False show ?thesis
    using nth_subdegree_nonzero[OF False] by (fastforce intro: subdegreeI)
qed

lemma subdegree_add_ge':
  fixes   f g :: "'a::monoid_add fps"
  assumes "f + g ≠ 0"
  shows   "subdegree (f + g) ≥ min (subdegree f) (subdegree g)"
  using   assms
  by      (force intro: subdegree_geI)

lemma subdegree_add_ge:
  assumes "f ≠ -(g :: ('a :: group_add) fps)"
  shows   "subdegree (f + g) ≥ min (subdegree f) (subdegree g)"
proof (rule subdegree_add_ge')
  have "f + g = 0 ==> False"
  proof-
    assume fg: "f + g = 0"
    have "∧n. f $ n = - g $ n"
      by (metis add_eq_0_iff equation_minus_iff fg fps_add_nth fps_neg_nth fps_zero_nth)
    with assms show False by (auto intro: fps_ext)
  qed
  thus "f + g ≠ 0" by fast
qed

lemma subdegree_add_eq1:
  assumes "f ≠ 0"
  and     "subdegree f < subdegree (g :: 'a::monoid_add fps)"
  shows   "subdegree (f + g) = subdegree f"
  using assms by(auto intro: subdegreeI simp: nth_less_subdegree_zero)

lemma subdegree_add_eq2:
  assumes "g ≠ 0"
  and     "subdegree g < subdegree (f :: 'a :: monoid_add fps)"
  shows   "subdegree (f + g) = subdegree g"
  using assms by (auto intro: subdegreeI simp: nth_less_subdegree_zero)

lemma subdegree_diff_eq1:
  assumes "f ≠ 0"
  and     "subdegree f < subdegree (g :: 'a :: group_add fps)"
  shows   "subdegree (f - g) = subdegree f"
  using assms by (auto intro: subdegreeI simp: nth_less_subdegree_zero)

lemma subdegree_diff_eq1_cancel:
  assumes "f ≠ 0"
  and     "subdegree f < subdegree (g :: 'a :: cancel_comm_monoid_add fps)"
  shows   "subdegree (f - g) = subdegree f"
  using assms by (auto intro: subdegreeI simp: nth_less_subdegree_zero)

lemma subdegree_diff_eq2:
  assumes "g ≠ 0"
  and     "subdegree g < subdegree (f :: 'a :: group_add fps)"
  shows   "subdegree (f - g) = subdegree g"
  using assms by (auto intro: subdegreeI simp: nth_less_subdegree_zero)

lemma subdegree_diff_ge [simp]:
  assumes "f ≠ (g :: 'a :: group_add fps)"
  shows   "subdegree (f - g) ≥ min (subdegree f) (subdegree g)"
proof-
  have "f ≠ - (- g)"
    using assms expand_fps_eq by fastforce
  moreover have "f + - g = f - g" by (simp add: fps_ext)
  ultimately show ?thesis
    using subdegree_add_ge[of f "-g"] by simp
qed

lemma subdegree_diff_ge':
  fixes   f g :: "'a :: comm_monoid_diff fps"
  assumes "f - g ≠ 0"
  shows   "subdegree (f - g) ≥ subdegree f"
  using assms by (auto intro: subdegree_geI simp: nth_less_subdegree_zero)

lemma nth_subdegree_mult_left [simp]:
  fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
  shows "(f * g) $ (subdegree f) = f $ subdegree f * g $ 0"
  by    (cases "subdegree f") (simp_all add: fps_mult_nth nth_less_subdegree_zero)

lemma nth_subdegree_mult_right [simp]:
  fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
  shows "(f * g) $ (subdegree g) = f $ 0 * g $ subdegree g"
  by    (cases "subdegree g") (simp_all add: fps_mult_nth nth_less_subdegree_zero sum.atLeast_Suc_atMost)

lemma nth_subdegree_mult [simp]:
  fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
  shows "(f * g) $ (subdegree f + subdegree g) = f $ subdegree f * g $ subdegree g"
proof-
  let ?n = "subdegree f + subdegree g"
  have "(f * g) $ ?n = (∑i=0..?n. f$i * g$(?n-i))"
    by (simp add: fps_mult_nth)
  also have "... = (∑i=0..?n. if i = subdegree f then f$i * g$(?n-i) else 0)"
  proof (intro sum.cong)
    fix x assume x: "x ∈ {0..?n}"
    hence "x = subdegree f ∨ x < subdegree f ∨ ?n - x < subdegree g" by auto
    thus "f $ x * g $ (?n - x) = (if x = subdegree f then f $ x * g $ (?n - x) else 0)"
      by (elim disjE conjE) auto
  qed auto
  also have "... = f $ subdegree f * g $ subdegree g" by simp
  finally show ?thesis .
qed

lemma fps_mult_nth_eq0:
  fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
  assumes "n < subdegree f + subdegree g"
  shows   "(f*g) $ n = 0"
proof-
  have "∧i. i∈{0..n} ==> f$i * g$(n - i) = 0"
  proof-
    fix i assume i: "i∈{0..n}"
    show "f$i * g$(n - i) = 0"
    proof (cases "i < subdegree f ∨ n - i < subdegree g")
      case False with assms i show ?thesis by auto
    qed (auto simp: nth_less_subdegree_zero)
  qed
  thus "(f * g) $ n = 0" by (simp add: fps_mult_nth)
qed

lemma fps_mult_subdegree_ge:
  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
  assumes "f*g ≠ 0"
  shows   "subdegree (f*g) ≥ subdegree f + subdegree g"
  using   assms fps_mult_nth_eq0
  by      (intro subdegree_geI) simp

lemma subdegree_mult':
  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
  assumes "f $ subdegree f * g $ subdegree g ≠ 0"
  shows   "subdegree (f*g) = subdegree f + subdegree g"
proof-
  from assms have "(f * g) $ (subdegree f + subdegree g) ≠ 0" by simp
  hence "f*g ≠ 0" by fastforce
  hence "subdegree (f*g) ≥ subdegree f + subdegree g" using fps_mult_subdegree_ge by fast
  moreover from assms have "subdegree (f*g) ≤ subdegree f + subdegree g"
    by (intro subdegree_leI) simp
  ultimately show ?thesis by simp
qed

lemma subdegree_mult [simp]:
  fixes   f g :: "'a :: {semiring_no_zero_divisors} fps"
  assumes "f ≠ 0" "g ≠ 0"
  shows   "subdegree (f * g) = subdegree f + subdegree g"
  using   assms
  by      (intro subdegree_mult') simp

lemma fps_mult_nth_conv_upto_subdegree_left:
  fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
  shows "(f * g) $ n = (∑i=subdegree f..n. f $ i * g $ (n - i))"
proof (cases "subdegree f ≤ n")
  case True
  hence "{0..n} = {0..∪ {subdegree f..n}" by auto
  moreover have "{0..∩ {subdegree f..n} = {}" by auto
  ultimately show ?thesis
    using nth_less_subdegree_zero[of _ f]
    by    (simp add: fps_mult_nth sum.union_disjoint)
qed (simp add: fps_mult_nth nth_less_subdegree_zero)

lemma fps_mult_nth_conv_upto_subdegree_right:
  fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
  shows "(f * g) $ n = (∑i=0..n - subdegree g. f $ i * g $ (n - i))"
proof-
  have "{0..n} = {0..n - subdegree g} ∪ {n - subdegree g<..n}" by auto
  moreover have "{0..n - subdegree g} ∩ {n - subdegree g<..n} = {}" by auto
  moreover have "∀i∈{n - subdegree g<..n}. g $ (n - i) = 0"
    using nth_less_subdegree_zero[of _ g] by auto
  ultimately show ?thesis by (simp add: fps_mult_nth sum.union_disjoint)
qed

lemma fps_mult_nth_conv_inside_subdegrees:
  fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
  shows "(f * g) $ n = (∑i=subdegree f..n - subdegree g. f $ i * g $ (n - i))"
proof (cases "subdegree f ≤ n - subdegree g")
  case True
  hence "{subdegree f..n} = {subdegree f..n - subdegree g} ∪ {n - subdegree g<..n}"
    by auto
  moreover have "{subdegree f..n - subdegree g} ∩ {n - subdegree g<..n} = {}" by auto
  moreover have "∀i∈{n - subdegree g<..n}. f $ i * g $ (n - i) = 0"
    using nth_less_subdegree_zero[of _ g] by auto
  ultimately show ?thesis
    using fps_mult_nth_conv_upto_subdegree_left[of f g n]
    by    (simp add: sum.union_disjoint)
next
  case False
  hence 1: "subdegree f > n - subdegree g" by simp
  show ?thesis
  proof (cases "f*g = 0")
    case False
    with 1 have "n < subdegree (f*g)" using fps_mult_subdegree_ge[of f g] by simp
    with 1 show ?thesis by auto
  qed (simp add: 1)
qed

lemma fps_mult_nth_outside_subdegrees:
  fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
  shows "n < subdegree f ==> (f * g) $ n = 0"
  and   "n < subdegree g ==> (f * g) $ n = 0"
  by    (auto simp: fps_mult_nth_conv_inside_subdegrees)


subsection ‹Ring structure›

instance fps :: (semigroup_add) semigroup_add
proof
  fix a b c :: "'a fps"
  show "a + b + c = a + (b + c)"
    by (simp add: fps_ext add.assoc)
qed

instance fps :: (ab_semigroup_add) ab_semigroup_add
proof
  fix a b :: "'a fps"
  show "a + b = b + a"
    by (simp add: fps_ext add.commute)
qed

instance fps :: (monoid_add) monoid_add
proof
  fix a :: "'a fps"
  show "0 + a = a" by (simp add: fps_ext)
  show "a + 0 = a" by (simp add: fps_ext)
qed

instance fps :: (comm_monoid_add) comm_monoid_add
proof
  fix a :: "'a fps"
  show "0 + a = a" by (simp add: fps_ext)
qed

instance fps :: (cancel_semigroup_add) cancel_semigroup_add
proof
  fix a b c :: "'a fps"
  show "b = c" if "a + b = a + c"
    using that by (simp add: expand_fps_eq)
  show "b = c" if "b + a = c + a"
    using that by (simp add: expand_fps_eq)
qed

instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
proof
  fix a b c :: "'a fps"
  show "a + b - a = b"
    by (simp add: expand_fps_eq)
  show "a - b - c = a - (b + c)"
    by (simp add: expand_fps_eq diff_diff_eq)
qed

instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..

instance fps :: (group_add) group_add
proof
  fix a b :: "'a fps"
  show "- a + a = 0" by (simp add: fps_ext)
  show "a + - b = a - b" by (simp add: fps_ext)
qed

instance fps :: (ab_group_add) ab_group_add
proof
  fix a b :: "'a fps"
  show "- a + a = 0" by (simp add: fps_ext)
  show "a - b = a + - b" by (simp add: fps_ext)
qed

instance fps :: (zero_neq_one) zero_neq_one
  by standard (simp add: expand_fps_eq)

lemma fps_mult_assoc_lemma:
  fixes k :: nat
    and f :: "nat ==> nat ==> nat ==> 'a::comm_monoid_add"
  shows "(∑j=0..k. ∑i=0..j. f i (j - i) (n - j)) =
         (∑j=0..k. ∑i=0..k - j. f j i (n - j - i))"
  by (induct k) (simp_all add: Suc_diff_le sum.distrib add.assoc)

instance fps :: (semiring_0) semiring_0
proof
  fix a b c :: "'a fps"
  show "(a + b) * c = a * c + b * c"
    by (simp add: expand_fps_eq fps_mult_nth distrib_right sum.distrib)
  show "a * (b + c) = a * b + a * c"
    by (simp add: expand_fps_eq fps_mult_nth distrib_left sum.distrib)
  show "(a * b) * c = a * (b * c)"
  proof (rule fps_ext)
    fix n :: nat
    have "(∑j=0..n. ∑i=0..j. a$i * b$(j - i) * c$(n - j)) =
          (∑j=0..n. ∑i=0..n - j. a$j * b$i * c$(n - j - i))"
      by (rule fps_mult_assoc_lemma)
    then show "((a * b) * c) $ n = (a * (b * c)) $ n"
      by (simp add: fps_mult_nth sum_distrib_left sum_distrib_right mult.assoc)
  qed
qed

instance fps :: (semiring_0_cancel) semiring_0_cancel ..

lemma fps_mult_commute_lemma:
  fixes n :: nat
    and f :: "nat ==> nat ==> 'a::comm_monoid_add"
  shows "(∑i=0..n. f i (n - i)) = (∑i=0..n. f (n - i) i)"
  by (rule sum.reindex_bij_witness[where i="(-) n" and j="(-) n"]) auto

instance fps :: (comm_semiring_0) comm_semiring_0
proof
  fix a b c :: "'a fps"
  show "a * b = b * a"
  proof (rule fps_ext)
    fix n :: nat
    have "(∑i=0..n. a$i * b$(n - i)) = (∑i=0..n. a$(n - i) * b$i)"
      by (rule fps_mult_commute_lemma)
    then show "(a * b) $ n = (b * a) $ n"
      by (simp add: fps_mult_nth mult.commute)
  qed 
qed (simp add: distrib_right)

instance fps :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..

instance fps :: (semiring_1) semiring_1
proof
  fix a :: "'a fps"
  show "1 * a = a" "a * 1 = a" by (simp_all add: fps_one_mult)
qed

instance fps :: (comm_semiring_1) comm_semiring_1
  by standard simp

instance fps :: (semiring_1_cancel) semiring_1_cancel ..

lemma fps_square_nth: "(f^2) $ n = (∑k≤n. f $ k * f $ (n - k))"
  by (simp add: power2_eq_square fps_mult_nth atLeast0AtMost)

lemma fps_sum_nth: "sum f S $ n = sum (λk. (f k) $ n) S"
proof (cases "finite S")
  case True
  then show ?thesis by (induct set: finite) auto
next
  case False
  then show ?thesis by simp
qed


definition "fps_const c = Abs_fps (λn. if n = 0 then c else 0)"

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

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

lemma fps_const_nonzero_eq_nonzero: "c ≠ 0 ==> fps_const c ≠ 0"
  using fps_nonzeroI[of "fps_const c" 0] by simp

lemma fps_const_eq_0_iff [simp]: "fps_const c = 0 ⟷ c = 0"
  by (auto simp: fps_eq_iff)

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

lemma fps_const_eq_1_iff [simp]: "fps_const c = 1 ⟷ c = 1"
  by (auto simp: fps_eq_iff)

lemma subdegree_fps_const [simp]: "subdegree (fps_const c) = 0"
  by (cases "c = 0") (auto intro!: subdegreeI)

lemma fps_const_neg [simp]: "- (fps_const (c::'a::group_add)) = fps_const (- c)"
  by (simp add: fps_ext)

lemma fps_const_add [simp]: "fps_const (c::'a::monoid_add) + fps_const d = fps_const (c + d)"
  by (simp add: fps_ext)

lemma fps_const_add_left: "fps_const (c::'a::monoid_add) + f =
    Abs_fps (λn. if n = 0 then c + f$0 else f$n)"
  by (simp add: fps_ext)

lemma fps_const_add_right: "f + fps_const (c::'a::monoid_add) =
    Abs_fps (λn. if n = 0 then f$0 + c else f$n)"
  by (simp add: fps_ext)

lemma fps_const_sub [simp]: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
  by (simp add: fps_ext)

lemmas fps_const_minus = fps_const_sub

lemma fps_const_mult[simp]:
  fixes c d :: "'a::{comm_monoid_add,mult_zero}"
  shows "fps_const c * fps_const d = fps_const (c * d)"
  by    (simp add: fps_eq_iff fps_mult_nth sum.neutral)

lemma fps_const_mult_left:
  "fps_const (c::'a::{comm_monoid_add,mult_zero}) * f = Abs_fps (λn. c * f$n)"
  unfolding fps_eq_iff fps_mult_nth
  by (simp add: fps_const_def mult_delta_left)

lemma fps_const_mult_right:
  "f * fps_const (c::'a::{comm_monoid_add,mult_zero}) = Abs_fps (λn. f$n * c)"
  unfolding fps_eq_iff fps_mult_nth
  by (simp add: fps_const_def mult_delta_right)

lemma fps_mult_left_const_nth [simp]:
  "(fps_const (c::'a::{comm_monoid_add,mult_zero}) * f)$n = c* f$n"
  by (simp add: fps_mult_nth mult_delta_left)

lemma fps_mult_right_const_nth [simp]:
  "(f * fps_const (c::'a::{comm_monoid_add,mult_zero}))$n = f$n * c"
  by (simp add: fps_mult_nth mult_delta_right)

lemma fps_const_power [simp]: "fps_const c ^ n = fps_const (c^n)"
  by (induct n) auto


instance fps :: (ring) ring ..

instance fps :: (comm_ring) comm_ring ..

instance fps :: (ring_1) ring_1 ..

instance fps :: (comm_ring_1) comm_ring_1 ..

instance fps :: (semiring_no_zero_divisors) semiring_no_zero_divisors
proof
  fix a b :: "'a fps"
  assume "a ≠ 0" and "b ≠ 0"
  hence "(a * b) $ (subdegree a + subdegree b) ≠ 0" by simp
  thus "a * b ≠ 0" using fps_nonzero_nth by fast
qed

instance fps :: (semiring_1_no_zero_divisors) semiring_1_no_zero_divisors ..

instance fps :: ("{cancel_semigroup_add,semiring_no_zero_divisors_cancel}")
  semiring_no_zero_divisors_cancel
proof
  fix a b c :: "'a fps"
  show "(a * c = b * c) = (c = 0 ∨ a = b)"
  proof
    assume ab: "a * c = b * c"
    have "c ≠ 0 ==> a = b"
    proof (rule fps_ext)
      fix n
      assume c: "c ≠ 0"
      show "a $ n = b $ n"
      proof (induct n rule: nat_less_induct)
        case (1 n)
        with ab c show ?case
          using fps_mult_nth_conv_upto_subdegree_right[of a c "subdegree c + n"]
                fps_mult_nth_conv_upto_subdegree_right[of b c "subdegree c + n"]
          by    (cases n) auto
      qed
    qed
    thus "c = 0 ∨ a = b" by fast
  qed auto
  show "(c * a = c * b) = (c = 0 ∨ a = b)"
  proof
    assume ab: "c * a = c * b"
    have "c ≠ 0 ==> a = b"
    proof (rule fps_ext)
      fix n
      assume c: "c ≠ 0"
      show "a $ n = b $ n"
      proof (induct n rule: nat_less_induct)
        case (1 n)
        moreover have "∀i∈{Suc (subdegree c)..subdegree c + n}. subdegree c + n - i < n" by auto
        ultimately show ?case
          using ab c fps_mult_nth_conv_upto_subdegree_left[of c a "subdegree c + n"]
                fps_mult_nth_conv_upto_subdegree_left[of c b "subdegree c + n"]
          by    (simp add: sum.atLeast_Suc_atMost)
      qed
    qed    
    thus "c = 0 ∨ a = b" by fast
  qed auto
qed

instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors ..

instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..

instance fps :: (idom) idom ..

lemma fps_of_nat: "fps_const (of_nat c) = of_nat c"
  by (induction c) (simp_all add: fps_const_add [symmetric] del: fps_const_add)

lemma fps_of_int: "fps_const (of_int c) = of_int c"
  by (induction c) (simp_all add: fps_const_minus [symmetric] fps_of_nat fps_const_neg [symmetric] 
                             del: fps_const_minus fps_const_neg)

lemma semiring_char_fps [simp]: "CHAR('a :: comm_semiring_1 fps) = CHAR('a)"
  by (rule CHAR_eqI) (auto simp flip: fps_of_nat simp: of_nat_eq_0_iff_char_dvd)

instance fps :: ("{semiring_prime_char,comm_semiring_1}") semiring_prime_char
  by (rule semiring_prime_charI) auto
instance fps :: ("{comm_semiring_prime_char,comm_semiring_1}") comm_semiring_prime_char
  by standard
instance fps :: ("{comm_ring_prime_char,comm_semiring_1}") comm_ring_prime_char
  by standard
instance fps :: ("{idom_prime_char,comm_semiring_1}") idom_prime_char
  by standard

lemma fps_numeral_fps_const: "numeral k = fps_const (numeral k)"
  by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1 fps_const_add [symmetric])

lemmas numeral_fps_const = fps_numeral_fps_const

lemma neg_numeral_fps_const:
  "(- numeral k :: 'a :: ring_1 fps) = fps_const (- numeral k)"
  by (simp add: numeral_fps_const)

lemma fps_numeral_nth: "numeral n $ i = (if i = 0 then numeral n else 0)"
  by (simp add: numeral_fps_const)

lemma fps_numeral_nth_0 [simp]: "numeral n $ 0 = numeral n"
  by (simp add: numeral_fps_const)

lemma subdegree_numeral [simp]: "subdegree (numeral n) = 0"
  by (simp add: numeral_fps_const)

lemma fps_nth_of_nat [simp]:
  "(of_nat c) $ n = (if n=0 then of_nat c else 0)"
  by (simp add: fps_of_nat[symmetric])

lemma fps_nth_of_int [simp]:
  "(of_int c) $ n = (if n=0 then of_int c else 0)"
  by (simp add: fps_of_int[symmetric])

lemma fps_mult_of_nat_nth [simp]:
  shows "(of_nat k * f) $ n = of_nat k * f$n"
  and   "(f * of_nat k ) $ n = f$n * of_nat k"
  by    (simp_all add: fps_of_nat[symmetric])

lemma fps_mult_of_int_nth [simp]:
  shows "(of_int k * f) $ n = of_int k * f$n"
  and   "(f * of_int k ) $ n = f$n * of_int k"
  by    (simp_all add: fps_of_int[symmetric])

lemma numeral_neq_fps_zero [simp]: "(numeral f :: 'a :: field_char_0 fps) ≠ 0"
proof
  assume "numeral f = (0 :: 'a fps)"
  from arg_cong[of _ _ "λF. F $ 0", OF this] show False by simp
qed 

instance fps :: (semiring_char_0) semiring_char_0
proof
  show "inj (of_nat :: nat ==> 'a fps)"
  proof
    fix m n :: nat
    assume "of_nat m = (of_nat n :: 'a fps)"
    hence "fps_nth (of_nat m) 0 = (fps_nth (of_nat n) 0 :: 'a)"
      by (simp only: )
    thus "m = n"
      by simp
  qed
qed

lemma subdegree_power_ge:
  "f^n ≠ 0 ==> subdegree (f^n) ≥ n * subdegree f"
proof (induct n)
  case (Suc n) thus ?case using fps_mult_subdegree_ge by fastforce
qed simp

lemma fps_pow_nth_below_subdegree:
  "k < n * subdegree f ==> (f^n) $ k = 0"
proof (cases "f^n = 0")
  case False
  assume "k < n * subdegree f"
  with False have "k < subdegree (f^n)" using subdegree_power_ge[of f n] by simp
  thus "(f^n) $ k = 0" by auto
qed simp

lemma fps_pow_base [simp]:
  "(f ^ n) $ (n * subdegree f) = (f $ subdegree f) ^ n"
proof (induct n)
  case (Suc n)
  show ?case
  proof (cases "Suc n * subdegree f < subdegree f + subdegree (f^n)")
    case True with Suc show ?thesis
      by (auto simp: fps_mult_nth_eq0 distrib_right)
  next
    case False
    hence "∀i∈{Suc (subdegree f)..Suc n * subdegree f - subdegree (f ^ n)}.
            f ^ n $ (Suc n * subdegree f - i) = 0"
     by (auto simp: fps_pow_nth_below_subdegree)
   with False Suc show ?thesis
      using fps_mult_nth_conv_inside_subdegrees[of f "f^n" "Suc n * subdegree f"]
            sum.atLeast_Suc_atMost[of
              "subdegree f"
              "Suc n * subdegree f - subdegree (f ^ n)"
              "λi. f $ i * f ^ n $ (Suc n * subdegree f - i)"
            ]
      by    simp
  qed
qed simp

lemma subdegree_power_eqI:
  fixes f :: "'a::semiring_1 fps"
  shows "(f $ subdegree f) ^ n ≠ 0 ==> subdegree (f ^ n) = n * subdegree f"
proof (induct n)
  case (Suc n)
  from Suc have 1: "subdegree (f ^ n) = n * subdegree f" by fastforce
  with Suc(2) have "f $ subdegree f * f ^ n $ subdegree (f ^ n) ≠ 0" by simp
  with 1 show ?case using subdegree_mult'[of f "f^n"] by simp
qed simp

lemma subdegree_power [simp]:
  "subdegree ((f :: ('a :: semiring_1_no_zero_divisors) fps) ^ n) = n * subdegree f"
  by (cases "f = 0"; induction n) simp_all

lemma subdegree_prod:
  fixes f :: "'a ==> 'b :: idom fps"
  assumes "∧x. x ∈ A ==> f x ≠ 0"
  shows   "subdegree (∏x∈A. f x) = (∑x∈A. subdegree (f x))"
  using assms by (induction A rule: infinite_finite_induct) auto


lemma minus_one_power_iff: "(- (1::'a::ring_1)) ^ n = (if even n then 1 else - 1)"
  by (induct n) auto

definition "fps_X = Abs_fps (λn. if n = 1 then 1 else 0)"

lemma subdegree_fps_X [simp]: "subdegree (fps_X :: ('a :: zero_neq_one) fps) = 1"
  by (auto intro!: subdegreeI simp: fps_X_def)

lemma fps_X_mult_nth [simp]:
  fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  shows "(fps_X * f) $ n = (if n = 0 then 0 else f $ (n - 1))"
proof (cases n)
  case (Suc m)
  moreover have "(fps_X * f) $ Suc m = f $ (Suc m - 1)"
  proof (cases m)
    case 0 thus ?thesis using fps_mult_nth_1[of "fps_X" f] by (simp add: fps_X_def)
  next
    case (Suc k) thus ?thesis by (simp add: fps_mult_nth fps_X_def sum.atLeast_Suc_atMost)
  qed
  ultimately show ?thesis by simp
qed (simp add: fps_X_def)

lemma fps_X_mult_right_nth [simp]:
  fixes a :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  shows "(a * fps_X) $ n = (if n = 0 then 0 else a $ (n - 1))"
proof (cases n)
  case (Suc m)
  moreover have "(a * fps_X) $ Suc m = a $ (Suc m - 1)"
  proof (cases m)
    case 0 thus ?thesis using fps_mult_nth_1[of a "fps_X"] by (simp add: fps_X_def)
  next
    case (Suc k)
    hence "(a * fps_X) $ Suc m = (∑i=0..k. a$i * fps_X$(Suc m - i)) + a$(Suc k)"
      by (simp add: fps_mult_nth fps_X_def)
    moreover have "∀i∈{0..k}. a$i * fps_X$(Suc m - i) = 0" by (auto simp: Suc fps_X_def)
    ultimately show ?thesis by (simp add: Suc)
  qed
  ultimately show ?thesis by simp
qed (simp add: fps_X_def)

lemma fps_mult_fps_X_commute:
  fixes a :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  shows "fps_X * a = a * fps_X" 
  by (simp add: fps_eq_iff)

lemma fps_mult_fps_X_power_commute: "fps_X ^ k * a = a * fps_X ^ k"
proof (induct k)
  case (Suc k)
  hence "fps_X ^ Suc k * a = a * fps_X * fps_X ^ k"
    by (simp add: mult.assoc fps_mult_fps_X_commute[symmetric])
  thus ?case by (simp add: mult.assoc)
qed simp

lemma fps_subdegree_mult_fps_X:
  fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  assumes "f ≠ 0"
  shows   "subdegree (fps_X * f) = subdegree f + 1"
  and     "subdegree (f * fps_X) = subdegree f + 1"
proof-
  show "subdegree (fps_X * f) = subdegree f + 1"
  proof (intro subdegreeI)
    fix i :: nat assume i: "i < subdegree f + 1"
    show "(fps_X * f) $ i = 0"
    proof (cases "i=0")
      case False with i show ?thesis by (simp add: nth_less_subdegree_zero)
    next
      case True thus ?thesis using fps_X_mult_nth[of f i] by simp
    qed
  qed (simp add: assms)
  thus "subdegree (f * fps_X) = subdegree f + 1"
    by (simp add: fps_mult_fps_X_commute)
qed

lemma fps_mult_fps_X_nonzero:
  fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  assumes "f ≠ 0"
  shows   "fps_X * f ≠ 0"
  and     "f * fps_X ≠ 0"
  using   assms fps_subdegree_mult_fps_X[of f]
          fps_nonzero_subdegree_nonzeroI[of "fps_X * f"]
          fps_nonzero_subdegree_nonzeroI[of "f * fps_X"]
  by      auto

lemma fps_mult_fps_X_power_nonzero:
  assumes "f ≠ 0"
  shows   "fps_X ^ n * f ≠ 0"
  and     "f * fps_X ^ n ≠ 0"
proof -
  show "fps_X ^ n * f ≠ 0"
    by (induct n) (simp_all add: assms mult.assoc fps_mult_fps_X_nonzero(1))
  thus "f * fps_X ^ n ≠ 0"
    by (simp add: fps_mult_fps_X_power_commute)
qed

lemma fps_X_power_iff: "fps_X ^ n = Abs_fps (λm. if m = n then 1 else 0)"
  by (induction n) (auto simp: fps_eq_iff)

lemma fps_X_nth[simp]: "fps_X$n = (if n = 1 then 1 else 0)"
  by (simp add: fps_X_def)

lemma fps_X_power_nth[simp]: "(fps_X^k) $n = (if n = k then 1 else 0)"
  by (simp add: fps_X_power_iff)

lemma fps_X_power_subdegree: "subdegree (fps_X^n) = n"
  by (auto intro: subdegreeI)

lemma fps_X_power_mult_nth:
  "(fps_X^k * f) $ n = (if n < k then 0 else f $ (n - k))"
  by  (cases "n)
      (simp_all add: fps_mult_nth_conv_upto_subdegree_left fps_X_power_subdegree sum.atLeast_Suc_atMost)

lemma fps_X_power_mult_right_nth:
  "(f * fps_X^k) $ n = (if n < k then 0 else f $ (n - k))"
  using fps_mult_fps_X_power_commute[of k f] fps_X_power_mult_nth[of k f] by simp

lemma fps_subdegree_mult_fps_X_power:
  assumes "f ≠ 0"
  shows   "subdegree (fps_X ^ n * f) = subdegree f + n"
  and     "subdegree (f * fps_X ^ n) = subdegree f + n"
proof -
  from assms show "subdegree (fps_X ^ n * f) = subdegree f + n"
    by (induct n)
       (simp_all add: algebra_simps fps_subdegree_mult_fps_X(1) fps_mult_fps_X_power_nonzero(1))
  thus "subdegree (f * fps_X ^ n) = subdegree f + n"
    by (simp add: fps_mult_fps_X_power_commute)
qed

lemma fps_mult_fps_X_plus_1_nth:
  "((1+fps_X)*a) $n = (if n = 0 then (a$n :: 'a::semiring_1) else a$n + a$(n - 1))"
proof (cases n)
  case 0
  then show ?thesis
    by (simp add: fps_mult_nth)
next
  case (Suc m)
  have "((1 + fps_X)*a) $ n = sum (λi. (1 + fps_X) $ i * a $ (n - i)) {0..n}"
    by (simp add: fps_mult_nth)
  also have "… = sum (λi. (1+fps_X)$i * a$(n-i)) {0.. 1}"
    unfolding Suc by (rule sum.mono_neutral_right) auto
  also have "… = (if n = 0 then a$n else a$n + a$(n - 1))"
    by (simp add: Suc)
  finally show ?thesis .
qed

lemma fps_mult_right_fps_X_plus_1_nth:
  fixes a :: "'a :: semiring_1 fps"
  shows "(a*(1+fps_X)) $ n = (if n = 0 then a$n else a$n + a$(n - 1))"
  using fps_mult_fps_X_plus_1_nth
  by    (simp add: distrib_left fps_mult_fps_X_commute distrib_right)

lemma fps_X_neq_fps_const [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ fps_const c"
proof
  assume "(fps_X::'a fps) = fps_const (c::'a)"
  hence "fps_X$1 = (fps_const (c::'a))$1" by (simp only:)
  thus False by auto
qed

lemma fps_X_neq_zero [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ 0"
  by (simp only: fps_const_0_eq_0[symmetric] fps_X_neq_fps_const) simp

lemma fps_X_neq_one [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ 1"
  by (simp only: fps_const_1_eq_1[symmetric] fps_X_neq_fps_const) simp

lemma fps_X_neq_numeral [simp]: "fps_X ≠ numeral c"
  by (simp only: numeral_fps_const fps_X_neq_fps_const) simp

lemma fps_X_pow_eq_fps_X_pow_iff [simp]: "fps_X ^ m = fps_X ^ n ⟷ m = n"
proof
  assume "(fps_X :: 'a fps) ^ m = fps_X ^ n"
  hence "(fps_X :: 'a fps) ^ m $ m = fps_X ^ n $ m" by (simp only:)
  thus "m = n" by (simp split: if_split_asm)
qed simp_all


subsection ‹Shifting and slicing›

definition fps_shift :: "nat ==> 'a fps ==> 'a fps" where
  "fps_shift n f = Abs_fps (λi. f $ (i + n))"

lemma fps_shift_nth [simp]: "fps_shift n f $ i = f $ (i + n)"
  by (simp add: fps_shift_def)

lemma fps_shift_0 [simp]: "fps_shift 0 f = f"
  by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_zero [simp]: "fps_shift n 0 = 0"
  by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_one: "fps_shift n 1 = (if n = 0 then 1 else 0)"
  by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_fps_const: "fps_shift n (fps_const c) = (if n = 0 then fps_const c else 0)"
  by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_numeral: "fps_shift n (numeral c) = (if n = 0 then numeral c else 0)"
  by (simp add: numeral_fps_const fps_shift_fps_const)

lemma fps_shift_fps_X [simp]:
  "n ≥ 1 ==> fps_shift n fps_X = (if n = 1 then 1 else 0)"
  by (intro fps_ext) (auto simp: fps_X_def)

lemma fps_shift_fps_X_power [simp]:
  "n ≤ m ==> fps_shift n (fps_X ^ m) = fps_X ^ (m - n)"
 by (intro fps_ext) auto

lemma fps_shift_subdegree [simp]:
  "n ≤ subdegree f ==> subdegree (fps_shift n f) = subdegree f - n"
  by (cases "f=0") (auto intro: subdegreeI simp: nth_less_subdegree_zero)

lemma fps_shift_fps_shift:
  "fps_shift (m + n) f = fps_shift m (fps_shift n f)"
  by (rule fps_ext) (simp add: add_ac)

lemma fps_shift_fps_shift_reorder:
  "fps_shift m (fps_shift n f) = fps_shift n (fps_shift m f)"
  using fps_shift_fps_shift[of m n f] fps_shift_fps_shift[of n m f] by (simp add: add.commute)

lemma fps_shift_rev_shift:
  "m ≤ n ==> fps_shift n (Abs_fps (λk. if k
  "m > n ==> fps_shift n (Abs_fps (λk. if k
    Abs_fps (λk. if k
proof -
  assume "m ≤ n"
  thus "fps_shift n (Abs_fps (λk. if k
    by (intro fps_ext) auto
next
  assume mn: "m > n"
  hence "∧k. k ≥ m-n ==> k+n-m = k - (m-n)" by auto
  thus
    "fps_shift n (Abs_fps (λk. if k
      Abs_fps (λk. if k
    by (intro fps_ext) auto
qed

lemma fps_shift_add:
  "fps_shift n (f + g) = fps_shift n f + fps_shift n g"
  by (simp add: fps_eq_iff)

lemma fps_shift_diff:
  "fps_shift n (f - g) = fps_shift n f - fps_shift n g"
  by (auto intro: fps_ext)

lemma fps_shift_uminus:
  "fps_shift n (-f) = - fps_shift n f"
  by (auto intro: fps_ext)

lemma fps_shift_mult:
  assumes "n ≤ subdegree (g :: 'b :: {comm_monoid_add, mult_zero} fps)"
  shows "fps_shift n (h*g) = h * fps_shift n g"
proof-
  have case1: "∧a b::'b fps. 1 ≤ subdegree b ==> fps_shift 1 (a*b) = a * fps_shift 1 b"
  proof (rule fps_ext)
    fix a b :: "'b fps"
    and n :: nat
    assume b: "1 ≤ subdegree b"
    have "∧i. i ≤ n ==> n + 1 - i = (n-i) + 1"
      by (simp add: algebra_simps)
    with b show "fps_shift 1 (a*b) $ n = (a * fps_shift 1 b) $ n"
      by (simp add: fps_mult_nth nth_less_subdegree_zero)
  qed
  have "n ≤ subdegree g ==> fps_shift n (h*g) = h * fps_shift n g"
  proof (induct n)
    case (Suc n)
    have "fps_shift (Suc n) (h*g) = fps_shift 1 (fps_shift n (h*g))"
      by (simp add: fps_shift_fps_shift[symmetric])
    also have "… = h * (fps_shift 1 (fps_shift n g))"
      using Suc case1 by force
    finally show ?case by (simp add: fps_shift_fps_shift[symmetric])
  qed simp
  with assms show ?thesis by fast
qed

lemma fps_shift_mult_right_noncomm:
  assumes "n ≤ subdegree (g :: 'b :: {comm_monoid_add, mult_zero} fps)"
  shows "fps_shift n (g*h) = fps_shift n g * h"
proof-
  have case1: "∧a b::'b fps. 1 ≤ subdegree a ==> fps_shift 1 (a*b) = fps_shift 1 a * b"
  proof (rule fps_ext)
    fix a b :: "'b fps"
    and n :: nat
    assume "1 ≤ subdegree a"
    hence "fps_shift 1 (a*b) $ n = (∑i=Suc 0..Suc n. a$i * b$(n+1-i))"
      using sum.atLeast_Suc_atMost[of 0 "n+1" "λi. a$i * b$(n+1-i)"]
      by    (simp add: fps_mult_nth nth_less_subdegree_zero)
    thus "fps_shift 1 (a*b) $ n = (fps_shift 1 a * b) $ n"
      using sum.shift_bounds_cl_Suc_ivl[of "λi. a$i * b$(n+1-i)" 0 n]
      by    (simp add: fps_mult_nth)
  qed
  have "n ≤ subdegree g ==> fps_shift n (g*h) = fps_shift n g * h"
  proof (induct n)
    case (Suc n)
    have "fps_shift (Suc n) (g*h) = fps_shift 1 (fps_shift n (g*h))"
      by (simp add: fps_shift_fps_shift[symmetric])
    also have "… = (fps_shift 1 (fps_shift n g)) * h"
      using Suc case1 by force
    finally show ?case by (simp add: fps_shift_fps_shift[symmetric])
  qed simp
  with assms show ?thesis by fast
qed

lemma fps_shift_mult_right:
  assumes "n ≤ subdegree (g :: 'b :: comm_semiring_0 fps)"
  shows   "fps_shift n (g*h) = h * fps_shift n g"
  by      (simp add: assms fps_shift_mult_right_noncomm mult.commute)

lemma fps_shift_mult_both:
  fixes   f g :: "'a::{comm_monoid_add, mult_zero} fps"
  assumes "m ≤ subdegree f" "n ≤ subdegree g"
  shows   "fps_shift m f * fps_shift n g = fps_shift (m+n) (f*g)"
  using   assms
  by      (simp add: fps_shift_mult fps_shift_mult_right_noncomm fps_shift_fps_shift)

lemma fps_shift_subdegree_zero_iff [simp]:
  "fps_shift (subdegree f) f = 0 ⟷ f = 0"
  by (subst (1) nth_subdegree_zero_iff[symmetric], cases "f = 0")
     (simp_all del: nth_subdegree_zero_iff)

lemma fps_shift_times_fps_X:
  fixes f g :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  shows "1 ≤ subdegree f ==> fps_shift 1 f * fps_X = f"
  by (intro fps_ext) (simp add: nth_less_subdegree_zero)

lemma fps_shift_times_fps_X' [simp]:
  fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  shows "fps_shift 1 (f * fps_X) = f"
  by (intro fps_ext) (simp add: nth_less_subdegree_zero)

lemma fps_shift_times_fps_X'':
  fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
  shows "1 ≤ n ==> fps_shift n (f * fps_X) = fps_shift (n - 1) f"
  by (intro fps_ext) (simp add: nth_less_subdegree_zero)

lemma fps_shift_times_fps_X_power:
  "n ≤ subdegree f ==> fps_shift n f * fps_X ^ n = f"
  by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero)

lemma fps_shift_times_fps_X_power' [simp]:
  "fps_shift n (f * fps_X^n) = f"
  by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero)

lemma fps_shift_times_fps_X_power'':
  "m ≤ n ==> fps_shift n (f * fps_X^m) = fps_shift (n - m) f"
  by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero)

lemma fps_shift_times_fps_X_power''':
  "m > n ==> fps_shift n (f * fps_X^m) = f * fps_X^(m - n)"
proof (cases "f=0")
  case False
  assume m: "m>n"
  hence "m = n + (m-n)" by auto
  with False m show ?thesis
    using power_add[of "fps_X::'a fps" n "m-n"]
          fps_shift_mult_right_noncomm[of n "f * fps_X^n" "fps_X^(m-n)"]
    by    (simp add: mult.assoc fps_subdegree_mult_fps_X_power(2))
qed simp

lemma subdegree_decompose:
  "f = fps_shift (subdegree f) f * fps_X ^ subdegree f"
  by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth)

lemma subdegree_decompose':
  "n ≤ subdegree f ==> f = fps_shift n f * fps_X^n"
  by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth intro!: nth_less_subdegree_zero)

instantiation fps :: (zero) unit_factor
begin
definition fps_unit_factor_def [simp]:
  "unit_factor f = fps_shift (subdegree f) f"
instance ..
end

lemma fps_unit_factor_zero_iff: "unit_factor (f::'a::zero fps) = 0 ⟷ f = 0"
  by simp

lemma fps_unit_factor_nth_0: "f ≠ 0 ==> unit_factor f $ 0 ≠ 0"
  by simp

lemma fps_X_unit_factor: "unit_factor (fps_X :: 'a :: zero_neq_one fps) = 1"
 by (intro fps_ext) auto

lemma fps_X_power_unit_factor: "unit_factor (fps_X ^ n) = 1"
proof-
  define X :: "'a fps" where "X ≡ fps_X"
  hence "unit_factor (X^n) = fps_shift n (X^n)"
    by (simp add: fps_X_power_subdegree)
  moreover have "fps_shift n (X^n) = 1"
    by (auto intro: fps_ext simp: fps_X_power_iff X_def)
  ultimately show ?thesis by (simp add: X_def)
qed

lemma fps_unit_factor_decompose:
  "f = unit_factor f * fps_X ^ subdegree f"
  by (simp add: subdegree_decompose)

lemma fps_unit_factor_decompose':
  "f = fps_X ^ subdegree f * unit_factor f"
  using fps_unit_factor_decompose by (simp add: fps_mult_fps_X_power_commute)

lemma fps_unit_factor_uminus:
  "unit_factor (-f) = - unit_factor (f::'a::group_add fps)"
  by    (simp add: fps_shift_uminus)

lemma fps_unit_factor_shift:
  assumes "n ≤ subdegree f"
  shows   "unit_factor (fps_shift n f) = unit_factor f"
  by      (simp add: assms fps_shift_fps_shift[symmetric])

lemma fps_unit_factor_mult_fps_X:
  fixes f :: "'a::{comm_monoid_add,monoid_mult,mult_zero} fps"
  shows "unit_factor (fps_X * f) = unit_factor f"
  and   "unit_factor (f * fps_X) = unit_factor f"
proof -
  show "unit_factor (fps_X * f) = unit_factor f"
    by (cases "f=0") (auto intro: fps_ext simp: fps_subdegree_mult_fps_X(1))
  thus "unit_factor (f * fps_X) = unit_factor f" by (simp add: fps_mult_fps_X_commute)
qed

lemma fps_unit_factor_mult_fps_X_power:
  shows "unit_factor (fps_X ^ n * f) = unit_factor f"
  and   "unit_factor (f * fps_X ^ n) = unit_factor f"
proof -
  show "unit_factor (fps_X ^ n * f) = unit_factor f"
  proof (induct n)
    case (Suc m) thus ?case
      using fps_unit_factor_mult_fps_X(1)[of "fps_X ^ m * f"] by (simp add: mult.assoc)
  qed simp
  thus "unit_factor (f * fps_X ^ n) = unit_factor f"
    by (simp add: fps_mult_fps_X_power_commute)
qed

lemma fps_unit_factor_mult_unit_factor:
  fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
  shows "unit_factor (f * unit_factor g) = unit_factor (f * g)"
  and   "unit_factor (unit_factor f * g) = unit_factor (f * g)"
proof -
  show "unit_factor (f * unit_factor g) = unit_factor (f * g)"
  proof (cases "f*g = 0")
    case False thus ?thesis
      using fps_mult_subdegree_ge[of f g] fps_unit_factor_shift[of "subdegree g" "f*g"]
      by    (simp add: fps_shift_mult)
  next
    case True
    moreover have "f * unit_factor g = fps_shift (subdegree g) (f*g)"
      by (simp add: fps_shift_mult)
    ultimately show ?thesis by simp
  qed
  show "unit_factor (unit_factor f * g) = unit_factor (f * g)"
  proof (cases "f*g = 0")
    case False thus ?thesis
      using fps_mult_subdegree_ge[of f g] fps_unit_factor_shift[of "subdegree f" "f*g"]
      by    (simp add: fps_shift_mult_right_noncomm)
  next
    case True
    moreover have "unit_factor f * g = fps_shift (subdegree f) (f*g)"
      by (simp add: fps_shift_mult_right_noncomm)
    ultimately show ?thesis by simp
  qed
qed

lemma fps_unit_factor_mult_both_unit_factor:
  fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
  shows "unit_factor (unit_factor f * unit_factor g) = unit_factor (f * g)"
  using fps_unit_factor_mult_unit_factor(1)[of "unit_factor f" g]
        fps_unit_factor_mult_unit_factor(2)[of f g]
  by    simp

lemma fps_unit_factor_mult':
  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
  assumes "f $ subdegree f * g $ subdegree g ≠ 0"
  shows   "unit_factor (f * g) = unit_factor f * unit_factor g"
  using   assms
  by      (simp add: subdegree_mult' fps_shift_mult_both)

lemma fps_unit_factor_mult:
  fixes f g :: "'a::semiring_no_zero_divisors fps"
  shows "unit_factor (f * g) = unit_factor f * unit_factor g"
  using fps_unit_factor_mult'[of f g]
  by    (cases "f=0 ∨ g=0") auto

definition "fps_cutoff n f = Abs_fps (λi. if i < n then f$i else 0)"

lemma fps_cutoff_nth [simp]: "fps_cutoff n f $ i = (if i < n then f$i else 0)"
  unfolding fps_cutoff_def by simp

lemma fps_cutoff_zero_iff: "fps_cutoff n f = 0 ⟷ (f = 0 ∨ n ≤ subdegree f)"
proof
  assume A: "fps_cutoff n f = 0"
  thus "f = 0 ∨ n ≤ subdegree f"
  proof (cases "f = 0")
    assume "f ≠ 0"
    with A have "n ≤ subdegree f"
      by (intro subdegree_geI) (simp_all add: fps_eq_iff split: if_split_asm)
    thus ?thesis ..
  qed simp
qed (auto simp: fps_eq_iff intro: nth_less_subdegree_zero)

lemma fps_cutoff_0 [simp]: "fps_cutoff 0 f = 0"
  by (simp add: fps_eq_iff)

lemma fps_cutoff_zero [simp]: "fps_cutoff n 0 = 0"
  by (simp add: fps_eq_iff)

lemma fps_cutoff_one: "fps_cutoff n 1 = (if n = 0 then 0 else 1)"
  by (simp add: fps_eq_iff)

lemma fps_cutoff_fps_const: "fps_cutoff n (fps_const c) = (if n = 0 then 0 else fps_const c)"
  by (simp add: fps_eq_iff)

lemma fps_cutoff_numeral: "fps_cutoff n (numeral c) = (if n = 0 then 0 else numeral c)"
  by (simp add: numeral_fps_const fps_cutoff_fps_const)

lemma fps_shift_cutoff:
  "fps_shift n f * fps_X^n + fps_cutoff n f = f"
  by (simp add: fps_eq_iff fps_X_power_mult_right_nth)

lemma fps_shift_cutoff':
  "fps_X^n * fps_shift n f + fps_cutoff n f = f"
  by (simp add: fps_eq_iff fps_X_power_mult_nth)

lemma fps_cutoff_left_mult_nth:
  "k < n ==> (fps_cutoff n f * g) $ k = (f * g) $ k"
  by (simp add: fps_mult_nth)

lemma fps_cutoff_add: "fps_cutoff n (f + g :: 'a :: monoid_add fps) = fps_cutoff n f + fps_cutoff n g"
  by (auto simp: fps_eq_iff)

lemma fps_cutoff_diff: "fps_cutoff n (f - g :: 'a :: group_add fps) = fps_cutoff n f - fps_cutoff n g"
  by (auto simp: fps_eq_iff)

lemma fps_cutoff_uminus: "fps_cutoff n (-f :: 'a :: group_add fps) = -fps_cutoff n f"
  by (auto simp: fps_eq_iff)

lemma fps_cutoff_right_mult_nth:
  assumes "k < n"
  shows   "(f * fps_cutoff n g) $ k = (f * g) $ k"
proof-
  from assms have "∀i∈{0..k}. fps_cutoff n g $ (k - i) = g $ (k - i)" by auto
  thus ?thesis by (simp add: fps_mult_nth)
qed

lemma fps_cutoff_eq_fps_cutoff_iff:
  "fps_cutoff n f = fps_cutoff n g ⟷ (∀k
  by (subst fps_eq_iff) auto

lemma fps_conv_fps_X_power_mult_fps_shift: 
  assumes "f = 0 ∨ subdegree f ≥ n"
  shows   "f = fps_X ^ n * fps_shift n f"
proof -
  have "f = fps_X ^ n * fps_shift n f + fps_cutoff n f"
    by (auto simp: fps_eq_iff fps_X_power_mult_nth)
  also have "fps_cutoff n f = 0"
    by (subst fps_cutoff_zero_iff) (use assms in auto)
  finally show ?thesis by simp
qed


subsection ‹Metrizability›

instantiation fps :: ("{minus,zero}") dist
begin

definition
  dist_fps_def: "dist (a :: 'a fps) b = (if a = b then 0 else inverse (2 ^ subdegree (a - b)))"

lemma dist_fps_ge0: "dist (a :: 'a fps) b ≥ 0"
  by (simp add: dist_fps_def)

instance ..

end

instantiation fps :: (group_add) metric_space
begin

definition uniformity_fps_def [code del]:
  "(uniformity :: ('a fps × 'a fps) filter) = (INF e∈{0 <..}. principal {(x, y). dist x y < e})"

definition open_fps_def' [code del]:
  "open (U :: 'a fps set) ⟷ (∀x∈U. eventually (λ(x', y). x' = x ⟶ y ∈ U) uniformity)"

lemma dist_fps_sym: "dist (a :: 'a fps) b = dist b a"
  by (simp add: dist_fps_def)

instance
proof
  show th: "dist a b = 0 ⟷ a = b" for a b :: "'a fps"
    by (simp add: dist_fps_def split: if_split_asm)
  then have th'[simp]: "dist a a = 0" for a :: "'a fps" by simp

  fix a b c :: "'a fps"
  consider "a = b" | "c = a ∨ c = b" | "a ≠ b" "a ≠ c" "b ≠ c" by blast
  then show "dist a b ≤ dist a c + dist b c"
  proof cases
    case 1
    then show ?thesis by (simp add: dist_fps_def)
  next
    case 2
    then show ?thesis
      by (cases "c = a") (simp_all add: th dist_fps_sym)
  next
    case neq: 3
    have False if "dist a b > dist a c + dist b c"
    proof -
      let ?n = "subdegree (a - b)"
      from neq have "dist a b > 0" "dist b c > 0" and "dist a c > 0" by (simp_all add: dist_fps_def)
      with that have "dist a b > dist a c" and "dist a b > dist b c" by simp_all
      with neq have "?n < subdegree (a - c)" and "?n < subdegree (b - c)"
        by (simp_all add: dist_fps_def field_simps)
      hence "(a - c) $ ?n = 0" and "(b - c) $ ?n = 0"
        by (simp_all only: nth_less_subdegree_zero)
      hence "(a - b) $ ?n = 0" by simp
      moreover from neq have "(a - b) $ ?n ≠ 0" by (intro nth_subdegree_nonzero) simp_all
      ultimately show False by contradiction
    qed
    thus ?thesis by (auto simp add: not_le[symmetric])
  qed
qed (rule open_fps_def' uniformity_fps_def)+

end

declare uniformity_Abort[where 'a="'a :: group_add fps", code]

lemma open_fps_def: "open (S :: 'a::group_add fps set) = (∀a ∈ S. ∃r. r >0 ∧ {y. dist y a < r} ⊆ S)"
  unfolding open_dist subset_eq by simp


text ‹Topology›

subsection ‹The topology of formal power series›

text ‹
  A set of formal power series is open iff for any power series $f$ in it, there exists some
  number $n$ such that all power series that agree with $f$ on the first $n$ components are
  also in it.
 ›
lemma open_fps_iff:
  "open A ⟷ (∀F∈A. ∃n. {G. fps_cutoff n G = fps_cutoff n F} ⊆ A)"
proof
  assume "open A"
  show "∀F∈A. ∃n. {G. fps_cutoff n G = fps_cutoff n F} ⊆ A"
  proof
    fix F :: "'a fps"
    assume F: "F ∈ A"
    with ‹open A› obtain e where e: "e > 0" "∧G. dist G F < e ==> G ∈ A"
      by (force simp: open_fps_def)
    thm dist_fps_def
    have "filterlim (λn. (1/2)^n :: real) (nhds 0) at_top"
      by (intro LIMSEQ_realpow_zero) auto
    from order_tendstoD(2)[OF this e(1)] have "eventually (λn. 1 / 2 ^ n < e) at_top"
      by (simp add: power_divide)
    then obtain n where n: "1 / 2 ^ n < e"
      by (auto simp: eventually_sequentially)
    show "∃n. {G. fps_cutoff n G = fps_cutoff n F} ⊆ A"
    proof (rule exI[of _ n], safe)
      fix G assume *: "fps_cutoff n G = fps_cutoff n F"
      show "G ∈ A"
      proof (cases "G = F")
        case False
        hence "dist G F = inverse (2 ^ subdegree (G - F))"
          by (auto simp: dist_fps_def)
        also have "subdegree (G - F) ≥ n"
        proof (rule subdegree_geI)
          fix i assume "i < n"
          hence "fps_nth (G - F) i = fps_nth (fps_cutoff n G - fps_cutoff n F) i"
            by (auto simp: fps_eq_iff)
          also from * have "… = 0"
            by simp
          finally show "fps_nth (G - F) i = 0" .
        qed (use False in auto)
        hence "inverse (2 ^ subdegree (G - F) :: real) ≤ inverse (2 ^ n)"
          by (intro le_imp_inverse_le power_increasing) auto
        also have "… < e"
          using n by (simp add: field_simps)
        finally show "G ∈ A"
          using e(2)[of G] by auto
      qed (use ‹F ∈ A› in auto)
    qed
  qed
next
  assume *: "∀F∈A. ∃n. {G. fps_cutoff n G = fps_cutoff n F} ⊆ A"
  show "open A"
    unfolding open_fps_def
  proof safe
    fix F assume F: "F ∈ A"
    with * obtain n where n: "∧G. fps_cutoff n G = fps_cutoff n F ==> G ∈ A"
      by blast
    show "∃r>0. {G. dist G F < r} ⊆ A"
    proof (rule exI[of _ "1 / 2 ^ n"], safe)
      fix G assume dist: "dist G F < 1 / 2 ^ n"
      show "G ∈ A"
      proof (cases "G = F")
        case False
        hence "dist G F = inverse (2 ^ subdegree (F - G))"
          by (simp add: dist_fps_def)
        with dist have "n < subdegree (F - G)"
          by (auto simp: field_simps)
        hence "fps_nth (F - G) i = 0" if "i ≤ n" for i
          using that nth_less_subdegree_zero[of i "F - G"] by simp
        hence "fps_cutoff n G = fps_cutoff n F"
          by (auto simp: fps_eq_iff fps_cutoff_def)
        thus "G ∈ A"
          by (rule n)
      qed (use ‹F ∈ A› in auto)
    qed auto
  qed
qed

lemma open_fps_cutoff: "open {H. fps_cutoff N H = fps_cutoff N G}"
  unfolding open_fps_iff
proof safe
  fix F assume F: "fps_cutoff N F = fps_cutoff N G"
  show "∃n. {G. fps_cutoff n G = fps_cutoff n F}
             ⊆ {H. fps_cutoff N H = fps_cutoff N G}"
    by (rule exI[of _ N]) (use F in ‹auto simp: fps_eq_iff›)
qed

lemma eventually_fps_nth_eq_nhds_fps_strong:
  "eventually (λg. ∀k≤n. fps_nth g k = fps_nth f k) (nhds f)"
proof -
  have "eventually (λg. g ∈ {g. fps_cutoff (n+1) g = fps_cutoff (n+1) f}) (nhds f)"
    by (rule eventually_nhds_in_open, rule open_fps_cutoff) auto
  thus ?thesis
    by eventually_elim (auto simp: fps_cutoff_eq_fps_cutoff_iff)
qed

lemma eventually_fps_nth_eq_nhds_fps: "eventually (λg. fps_nth g k = fps_nth f k) (nhds f)"
  using eventually_fps_nth_eq_nhds_fps_strong[of k] by eventually_elim auto

text ‹
  A family of formal power series $f_x$ tends to a limit series $g$ at some filter $F$
  iff for any $N\geq 0$, the set of $x$ for which $f_x$ and $G$ agree on the first $N$ coefficients
  is in $F$.
 
  For a sequence $(f_i)_{n\geq 0}$ this means that $f_i \longrightarrow G$ iff for any
  $N\geq 0$, $f_x$ and $G$ agree for all but finitely many $x$.
 ›

lemma tendsto_fps_iff:
  "filterlim f (nhds (g :: 'a :: group_add fps)) F ⟷
     (∀n. eventually (λx. fps_nth (f x) n = fps_nth g n) F)"
proof safe
  assume lim: "filterlim f (nhds (g :: 'a :: group_add fps)) F"
  show "eventually (λx. fps_nth (f x) n = fps_nth g n) F" for n
  proof -
    define S where "S = {H. fps_cutoff (n+1) H = fps_cutoff (n+1) g}"
    have S: "open S" "g ∈ S"
      unfolding S_def using open_fps_cutoff[of "n+1" g] by (auto simp: S_def)
    from lim and S have "eventually (λx. f x ∈ S) F"
      using topological_tendstoD by blast
    thus "eventually (λx. fps_nth (f x) n = fps_nth g n) F"
      by eventually_elim (auto simp: S_def fps_cutoff_eq_fps_cutoff_iff)
  qed
next
  assume *: "∀n. eventually (λx. fps_nth (f x) n = fps_nth g n) F"
  show "filterlim f (nhds (g :: 'a :: group_add fps)) F"
  proof (rule topological_tendstoI)
    fix S :: "'a fps set"
    assume S: "open S" "g ∈ S"
    then obtain N where N: "{H. fps_cutoff N H = fps_cutoff N g} ⊆ S"
      unfolding open_fps_iff by blast
    have "eventually (λx. ∀n∈{..
      by (subst eventually_ball_finite_distrib) (use * in auto)
    hence "eventually (λx. f x ∈ {H. fps_cutoff N H = fps_cutoff N g}) F"
      by eventually_elim (auto simp: fps_cutoff_eq_fps_cutoff_iff)
    thus "eventually (λx. f x ∈ S) F"
      by eventually_elim (use N in auto)
  qed
qed

lemma tendsto_fpsI:
  assumes "∧n. eventually (λx. fps_nth (f x) n = fps_nth G n) F"
  shows   "filterlim f (nhds (G :: 'a :: group_add fps)) F"
  unfolding tendsto_fps_iff using assms by blast


text ‹The infinite sums and justification of the notation in textbooks.›

lemma reals_power_lt_ex:
  fixes x y :: real
  assumes xp: "x > 0"
    and y1: "y > 1"
  shows "∃k>0. (1/y)^k < x"
proof -
  have yp: "y > 0"
    using y1 by simp
  from reals_Archimedean2[of "max 0 (- log y x) + 1"]
  obtain k :: nat where k: "real k > max 0 (- log y x) + 1"
    by blast
  from k have kp: "k > 0"
    by simp
  from k have "real k > - log y x"
    by simp
  then have "ln y * real k > - ln x"
    unfolding log_def
    using ln_gt_zero_iff[OF yp] y1
    by (simp add: minus_divide_left field_simps del: minus_divide_left[symmetric])
  then have "ln y * real k + ln x > 0"
    by simp
  then have "exp (real k * ln y + ln x) > exp 0"
    by (simp add: ac_simps)
  then have "y ^ k * x > 1"
    unfolding exp_zero exp_add exp_of_nat_mult exp_ln [OF xp] exp_ln [OF yp]
    by simp
  then have "x > (1 / y)^k" using yp
    by (simp add: field_simps)
  then show ?thesis
    using kp by blast
qed

lemma fps_sum_rep_nth: "(sum (λi. fps_const(a$i)*fps_X^i) {0..m})$n = (if n ≤ m then a$n else 0)"
  by (simp add: fps_sum_nth if_distrib cong del: if_weak_cong)

lemma fps_notation: "(λn. sum (λi. fps_const(a$i) * fps_X^i) {0..n}) <---- a"
  (is "?s <---- a")
proof -
  have "∃n0. ∀n ≥ n0. dist (?s n) a < r" if "r > 0" for r
  proof -
    obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0"
      using reals_power_lt_ex[OF ‹r > 0›, of 2] by auto
    show ?thesis
    proof -
      have "dist (?s n) a < r" if nn0: "n ≥ n0" for n
      proof -
        from that have thnn0: "(1/2)^n ≤ (1/2 :: real)^n0"
          by (simp add: field_split_simps)
        show ?thesis
        proof (cases "?s n = a")
          case True
          then show ?thesis
            unfolding dist_eq_0_iff[of "?s n" a, symmetric]
            using ‹r > 0› by (simp del: dist_eq_0_iff)
        next
          case False
          from False have dth: "dist (?s n) a = (1/2)^subdegree (?s n - a)"
            by (simp add: dist_fps_def field_simps)
          from False have kn: "subdegree (?s n - a) > n"
            by (intro subdegree_greaterI) (simp_all add: fps_sum_rep_nth)
          then have "dist (?s n) a < (1/2)^n"
            by (simp add: field_simps dist_fps_def)
          also have "… ≤ (1/2)^n0"
            using nn0 by (simp add: field_split_simps)
          also have "… < r"
            using n0 by simp
          finally show ?thesis .
        qed
      qed
      then show ?thesis by blast
    qed
  qed
  then show ?thesis
    unfolding lim_sequentially by blast
qed


subsection ‹Division›

declare sum.cong[fundef_cong]

fun fps_left_inverse_constructor ::
  "'a::{comm_monoid_add,times,uminus} fps ==> 'a ==> nat ==> 'a"
where
  "fps_left_inverse_constructor f a 0 = a"
| "fps_left_inverse_constructor f a (Suc n) =
    - sum (λi. fps_left_inverse_constructor f a i * f$(Suc n - i)) {0..n} * a"

🍋 ‹This will construct a left inverse for f in case that 🍋‹x * f$0 = 1›\›
abbreviation "fps_left_inverse ≡ (λf x. Abs_fps (fps_left_inverse_constructor f x))"

fun fps_right_inverse_constructor ::
  "'a::{comm_monoid_add,times,uminus} fps ==> 'a ==> nat ==> 'a"
where
  "fps_right_inverse_constructor f a 0 = a"
| "fps_right_inverse_constructor f a n =
    - a * sum (λi. f$i * fps_right_inverse_constructor f a (n - i)) {1..n}"

🍋 ‹This will construct a right inverse for f in case that 🍋‹f$0 * y = 1›\›
abbreviation "fps_right_inverse ≡ (λf y. Abs_fps (fps_right_inverse_constructor f y))"

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

🍋 ‹For backwards compatibility.›
abbreviation natfun_inverse:: "'a fps ==> nat ==> 'a"
  where "natfun_inverse f ≡ fps_right_inverse_constructor f (inverse (f$0))"

definition fps_inverse_def: "inverse f = Abs_fps (natfun_inverse f)"
🍋 ‹
  With scalars from a (possibly non-commutative) ring, this defines a right inverse.
  Furthermore, if scalars are of class @{class mult_zero} and satisfy
  condition @{term "inverse 0 = 0"}, then this will evaluate to zero when
  the zeroth term is zero.
 ›

definition fps_divide_def: "f div g = fps_shift (subdegree g) (f * inverse (unit_factor g))"
🍋 ‹
  If scalars are of class @{class mult_zero} and satisfy condition
  @{term "inverse 0 = 0"}, then div by zero will equal zero.
 ›

instance ..

end

lemma fps_lr_inverse_0_iff:
  "(fps_left_inverse f x) $ 0 = 0 ⟷ x = 0"
  "(fps_right_inverse f x) $ 0 = 0 ⟷ x = 0"
  by auto

lemma fps_inverse_0_iff': "(inverse f) $ 0 = 0 ⟷ inverse (f $ 0) = 0"
  by (simp add: fps_inverse_def fps_lr_inverse_0_iff(2))

lemma fps_inverse_0_iff[simp]: "(inverse f) $ 0 = (0::'a::division_ring) ⟷ f $ 0 = 0"
  by (simp add: fps_inverse_0_iff')

lemma fps_lr_inverse_nth_0:
  "(fps_left_inverse f x) $ 0 = x" "(fps_right_inverse f x) $ 0 = x"
  by auto

lemma fps_inverse_nth_0 [simp]: "(inverse f) $ 0 = inverse (f $ 0)"
  by (simp add: fps_inverse_def)

lemma fps_lr_inverse_starting0:
  fixes f :: "'a::{comm_monoid_add,mult_zero,uminus} fps"
  and   g :: "'b::{ab_group_add,mult_zero} fps"
  shows "fps_left_inverse f 0 = 0"
  and   "fps_right_inverse g 0 = 0"
proof-
  show "fps_left_inverse f 0 = 0"
  proof (rule fps_ext)
    fix n show "fps_left_inverse f 0 $ n = 0 $ n"
      by (cases n) (simp_all add: fps_inverse_def)
  qed
  show "fps_right_inverse g 0 = 0"
  proof (rule fps_ext)
    fix n show "fps_right_inverse g 0 $ n = 0 $ n"
      by (cases n) (simp_all add: fps_inverse_def)
  qed
qed

lemma fps_lr_inverse_eq0_imp_starting0:
  "fps_left_inverse f x = 0 ==> x = 0"
  "fps_right_inverse f x = 0 ==> x = 0"
proof-
  assume A: "fps_left_inverse f x = 0"
  have "0 = fps_left_inverse f x $ 0" by (subst A) simp
  thus "x = 0" by simp
next
  assume A: "fps_right_inverse f x = 0"
  have "0 = fps_right_inverse f x $ 0" by (subst A) simp
  thus "x = 0" by simp
qed

lemma fps_lr_inverse_eq_0_iff:
  fixes x :: "'a::{comm_monoid_add,mult_zero,uminus}"
  and   y :: "'b::{ab_group_add,mult_zero}"
  shows "fps_left_inverse f x = 0 ⟷ x = 0"
  and   "fps_right_inverse g y = 0 ⟷ y = 0"
  using fps_lr_inverse_starting0 fps_lr_inverse_eq0_imp_starting0
  by    auto

lemma fps_inverse_eq_0_iff':
  fixes f :: "'a::{ab_group_add,inverse,mult_zero} fps"
  shows "inverse f = 0 ⟷ inverse (f $ 0) = 0"
  by    (simp add: fps_inverse_def fps_lr_inverse_eq_0_iff(2))

lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::division_ring) fps) ⟷ f $ 0 = 0"
  using fps_inverse_eq_0_iff'[of f] by simp

lemmas fps_inverse_eq_0' = iffD2[OF fps_inverse_eq_0_iff']
lemmas fps_inverse_eq_0  = iffD2[OF fps_inverse_eq_0_iff]

lemma fps_const_lr_inverse:
  fixes a :: "'a::{ab_group_add,mult_zero}"
  and   b :: "'b::{comm_monoid_add,mult_zero,uminus}"
  shows "fps_left_inverse (fps_const a) x = fps_const x"
  and   "fps_right_inverse (fps_const b) y = fps_const y"
proof-
  show "fps_left_inverse (fps_const a) x = fps_const x"
  proof (rule fps_ext)
    fix n show "fps_left_inverse (fps_const a) x $ n = fps_const x $ n"
      by (cases n) auto
  qed
  show "fps_right_inverse (fps_const b) y = fps_const y"
  proof (rule fps_ext)
    fix n show "fps_right_inverse (fps_const b) y $ n = fps_const y $ n"
      by (cases n) auto
  qed
qed

lemma fps_const_inverse:
  fixes     a :: "'a::{comm_monoid_add,inverse,mult_zero,uminus}"
  shows     "inverse (fps_const a) = fps_const (inverse a)"
  unfolding fps_inverse_def
  by        (simp add: fps_const_lr_inverse(2))

lemma fps_lr_inverse_zero:
  fixes x :: "'a::{ab_group_add,mult_zero}"
  and   y :: "'b::{comm_monoid_add,mult_zero,uminus}"
  shows "fps_left_inverse 0 x = fps_const x"
  and   "fps_right_inverse 0 y = fps_const y"
  using fps_const_lr_inverse[of 0]
  by    simp_all

lemma fps_inverse_zero_conv_fps_const:
  "inverse (0::'a::{comm_monoid_add,mult_zero,uminus,inverse} fps) = fps_const (inverse 0)"
  using fps_lr_inverse_zero(2)[of "inverse (0::'a)"] by (simp add: fps_inverse_def)

lemma fps_inverse_zero':
  assumes "inverse (0::'a::{comm_monoid_add,inverse,mult_zero,uminus}) = 0"
  shows   "inverse (0::'a fps) = 0"
  by      (simp add: assms fps_inverse_zero_conv_fps_const)

lemma fps_inverse_zero [simp]:
  "inverse (0::'a::division_ring fps) = 0"
  by (rule fps_inverse_zero'[OF inverse_zero])

lemma fps_lr_inverse_one:
  fixes x :: "'a::{ab_group_add,mult_zero,one}"
  and   y :: "'b::{comm_monoid_add,mult_zero,uminus,one}"
  shows "fps_left_inverse 1 x = fps_const x"
  and   "fps_right_inverse 1 y = fps_const y"
  using fps_const_lr_inverse[of 1]
  by    simp_all

lemma fps_lr_inverse_one_one:
  "fps_left_inverse 1 1 = (1::'a::{ab_group_add,mult_zero,one} fps)"
  "fps_right_inverse 1 1 = (1::'b::{comm_monoid_add,mult_zero,uminus,one} fps)"
  by (simp_all add: fps_lr_inverse_one)

lemma fps_inverse_one':
  assumes "inverse (1::'a::{comm_monoid_add,inverse,mult_zero,uminus,one}) = 1"
  shows   "inverse (1 :: 'a fps) = 1"
  using   assms fps_lr_inverse_one_one(2)
  by      (simp add: fps_inverse_def)

lemma fps_inverse_one [simp]: "inverse (1 :: 'a :: division_ring fps) = 1"
  by (rule fps_inverse_one'[OF inverse_1])

lemma fps_lr_inverse_minus:
  fixes f :: "'a::ring_1 fps"
  shows "fps_left_inverse (-f) (-x) = - fps_left_inverse f x"
  and   "fps_right_inverse (-f) (-x) = - fps_right_inverse f x"
proof-

  show "fps_left_inverse (-f) (-x) = - fps_left_inverse f x"
  proof (intro fps_ext)
    fix n show "fps_left_inverse (-f) (-x) $ n = - fps_left_inverse f x $ n"
    proof (induct n rule: nat_less_induct)
      case (1 n) thus ?case by (cases n) (simp_all add: sum_negf algebra_simps)
    qed
  qed

  show "fps_right_inverse (-f) (-x) = - fps_right_inverse f x"
  proof (intro fps_ext)
    fix n show "fps_right_inverse (-f) (-x) $ n = - fps_right_inverse f x $ n"
    proof (induct n rule: nat_less_induct)
      case (1 n) show ?case
      proof (cases n)
        case (Suc m)
        with 1 have
          "∀i∈{1..Suc m}. fps_right_inverse (-f) (-x) $ (Suc m - i) =
            - fps_right_inverse f x $ (Suc m - i)"
          by auto
        with Suc show ?thesis by (simp add: sum_negf algebra_simps)
      qed simp
    qed
  qed

qed

lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: division_ring fps)"
  by (simp add: fps_inverse_def fps_lr_inverse_minus(2))

lemma fps_left_inverse:
  fixes   f :: "'a::ring_1 fps"
  assumes f0: "x * f$0 = 1"
  shows   "fps_left_inverse f x * f = 1"
proof (rule fps_ext)
  fix n show "(fps_left_inverse f x * f) $ n = 1 $ n"
    by (cases n) (simp_all add: f0 fps_mult_nth mult.assoc)
qed

lemma fps_right_inverse:
  fixes   f :: "'a::ring_1 fps"
  assumes f0: "f$0 * y = 1"
  shows   "f * fps_right_inverse f y = 1"
proof (rule fps_ext)
  fix n
  show "(f * fps_right_inverse f y) $ n = 1 $ n"
  proof (cases n)
    case (Suc k)
    moreover from Suc have "fps_right_inverse f y $ n =
            - y * sum (λi. f$i * fps_right_inverse_constructor f y (n - i)) {1..n}"
      by simp
    hence
      "(f * fps_right_inverse f y) $ n =
        - 1 * sum (λi. f$i * fps_right_inverse_constructor f y (n - i)) {1..n} +
        sum (λi. f$i * (fps_right_inverse_constructor f y (n - i))) {1..n}"
      by (simp add: fps_mult_nth sum.atLeast_Suc_atMost mult.assoc f0[symmetric])
    thus "(f * fps_right_inverse f y) $ n = 1 $ n" by (simp add: Suc)
  qed (simp add: f0 fps_inverse_def)
qed

text ‹
  It is possible in a ring for an element to have a left inverse but not a right inverse, or
  vice versa. But when an element has both, they must be the same.
 ›
lemma fps_left_inverse_eq_fps_right_inverse:
  fixes   f :: "'a::ring_1 fps"
  assumes f0: "x * f$0 = 1" "f $ 0 * y = 1"
  🍋 ‹These assumptions imply that $x$ equals $y$, but no need to assume that.›
  shows   "fps_left_inverse f x = fps_right_inverse f y"
proof-
  from f0(2) have "f * fps_right_inverse f y = 1"
      by (simp add: fps_right_inverse)
  hence "fps_left_inverse f x * f * fps_right_inverse f y = fps_left_inverse f x"
    by (simp add: mult.assoc)
  moreover from f0(1) have
    "fps_left_inverse f x * f * fps_right_inverse f y = fps_right_inverse f y"
    by (simp add: fps_left_inverse)
  ultimately show ?thesis by simp
qed

lemma fps_left_inverse_eq_fps_right_inverse_comm:
  fixes   f :: "'a::comm_ring_1 fps"
  assumes f0: "x * f$0 = 1"
  shows   "fps_left_inverse f x = fps_right_inverse f x"
  using   assms fps_left_inverse_eq_fps_right_inverse[of x f x]
  by      (simp add: mult.commute)

lemma fps_left_inverse':
  fixes   f :: "'a::ring_1 fps"
  assumes "x * f$0 = 1" "f$0 * y = 1"
  🍋 ‹These assumptions imply $x$ equals $y$, but no need to assume that.›
  shows   "fps_right_inverse f y * f = 1"
  using   assms fps_left_inverse_eq_fps_right_inverse[of x f y] fps_left_inverse[of x f]
  by      simp

lemma fps_right_inverse':
  fixes   f :: "'a::ring_1 fps"
  assumes "x * f$0 = 1" "f$0 * y = 1"
  🍋 ‹These assumptions imply $x$ equals $y$, but no need to assume that.›
  shows   "f * fps_left_inverse f x = 1"
  using   assms fps_left_inverse_eq_fps_right_inverse[of x f y] fps_right_inverse[of f y]
  by      simp

lemma inverse_mult_eq_1 [intro]:
  assumes "f$0 ≠ (0::'a::division_ring)"
  shows   "inverse f * f = 1"
  using   fps_left_inverse'[of "inverse (f$0)"]
  by      (simp add: assms fps_inverse_def)

lemma inverse_mult_eq_1':
  assumes "f$0 ≠ (0::'a::division_ring)"
  shows   "f * inverse f = 1"
  using   assms fps_right_inverse
  by      (force simp: fps_inverse_def)

lemma fps_mult_left_inverse_unit_factor:
  fixes   f :: "'a::ring_1 fps"
  assumes "x * f $ subdegree f = 1"
  shows   "fps_left_inverse (unit_factor f) x * f = fps_X ^ subdegree f"
proof-
  have
    "fps_left_inverse (unit_factor f) x * f =
      fps_left_inverse (unit_factor f) x * unit_factor f * fps_X ^ subdegree f"
    using fps_unit_factor_decompose[of f] by (simp add: mult.assoc)
  with assms show ?thesis by (simp add: fps_left_inverse)
qed

lemma fps_mult_right_inverse_unit_factor:
  fixes   f :: "'a::ring_1 fps"
  assumes "f $ subdegree f * y = 1"
  shows   "f * fps_right_inverse (unit_factor f) y = fps_X ^ subdegree f"
proof-
  have
    "f * fps_right_inverse (unit_factor f) y =
      fps_X ^ subdegree f * (unit_factor f * fps_right_inverse (unit_factor f) y)"
    using fps_unit_factor_decompose'[of f] by (simp add: mult.assoc[symmetric])
  with assms show ?thesis by (simp add: fps_right_inverse)
qed

lemma fps_mult_right_inverse_unit_factor_divring:
  "(f :: 'a::division_ring fps) ≠ 0 ==> f * inverse (unit_factor f) = fps_X ^ subdegree f"
  using   fps_mult_right_inverse_unit_factor[of f]
  by      (simp add: fps_inverse_def)

lemma fps_left_inverse_idempotent_ring1:
  fixes   f :: "'a::ring_1 fps"
  assumes "x * f$0 = 1" "y * x = 1"
  🍋 ‹These assumptions imply $y$ equals ‹f$0›, but no need to assume that.›
  shows   "fps_left_inverse (fps_left_inverse f x) y = f"
proof-
  from assms(1) have
    "fps_left_inverse (fps_left_inverse f x) y * fps_left_inverse f x * f =
      fps_left_inverse (fps_left_inverse f x) y"
    by (simp add: fps_left_inverse mult.assoc)
  moreover from assms(2) have
    "fps_left_inverse (fps_left_inverse f x) y * fps_left_inverse f x = 1"
    by (simp add: fps_left_inverse)
  ultimately show ?thesis by simp
qed

lemma fps_left_inverse_idempotent_comm_ring1:
  fixes   f :: "'a::comm_ring_1 fps"
  assumes "x * f$0 = 1"
  shows   "fps_left_inverse (fps_left_inverse f x) (f$0) = f"
  using   assms fps_left_inverse_idempotent_ring1[of x f "f$0"]
  by      (simp add: mult.commute)

lemma fps_right_inverse_idempotent_ring1:
  fixes   f :: "'a::ring_1 fps"
  assumes "f$0 * x = 1" "x * y = 1"
  🍋 ‹These assumptions imply $y$ equals ‹f$0›, but no need to assume that.›
  shows   "fps_right_inverse (fps_right_inverse f x) y = f"
proof-
  from assms(1) have "f * (fps_right_inverse f x * fps_right_inverse (fps_right_inverse f x) y) =
      fps_right_inverse (fps_right_inverse f x) y"
    by (simp add: fps_right_inverse mult.assoc[symmetric])
  moreover from assms(2) have
    "fps_right_inverse f x * fps_right_inverse (fps_right_inverse f x) y = 1"
    by (simp add: fps_right_inverse)
  ultimately show ?thesis by simp
qed

lemma fps_right_inverse_idempotent_comm_ring1:
  fixes   f :: "'a::comm_ring_1 fps"
  assumes "f$0 * x = 1"
  shows   "fps_right_inverse (fps_right_inverse f x) (f$0) = f"
  using   assms fps_right_inverse_idempotent_ring1[of f x "f$0"]
  by      (simp add: mult.commute)

lemma fps_inverse_idempotent[intro, simp]:
  "f$0 ≠ (0::'a::division_ring) ==> inverse (inverse f) = f"
  using fps_right_inverse_idempotent_ring1[of f]
  by    (simp add: fps_inverse_def)

lemma fps_lr_inverse_unique_ring1:
  fixes   f g :: "'a :: ring_1 fps"
  assumes fg: "f * g = 1" "g$0 * f$0 = 1"
  shows   "fps_left_inverse g (f$0) = f"
  and     "fps_right_inverse f (g$0) = g"
proof-

  show "fps_left_inverse g (f$0) = f"
  proof (intro fps_ext)
    fix n show "fps_left_inverse g (f$0) $ n = f $ n"
    proof (induct n rule: nat_less_induct)
      case (1 n) show ?case
      proof (cases n)
        case (Suc k)
        hence "∀i∈{0..k}. fps_left_inverse g (f$0) $ i = f $ i" using 1 by simp
        hence "fps_left_inverse g (f$0) $ Suc k = f $ Suc k - 1 $ Suc k * f$0"
          by (simp add: fps_mult_nth fg(1)[symmetric] distrib_right mult.assoc fg(2))
        with Suc show ?thesis by simp
      qed simp
    qed
  qed

  show "fps_right_inverse f (g$0) = g"
  proof (intro fps_ext)
    fix n show "fps_right_inverse f (g$0) $ n = g $ n"
    proof (induct n rule: nat_less_induct)
      case (1 n) show ?case
      proof (cases n)
        case (Suc k)
        hence "∀i∈{1..Suc k}. fps_right_inverse f (g$0) $ (Suc k - i) = g $ (Suc k - i)"
          using 1 by auto
        hence
          "fps_right_inverse f (g$0) $ Suc k = 1 * g $ Suc k - g$0 * 1 $ Suc k"
          by (simp add: fps_mult_nth fg(1)[symmetric] algebra_simps fg(2)[symmetric] sum.atLeast_Suc_atMost)
        with Suc show ?thesis by simp
      qed simp
    qed
  qed

qed

lemma fps_lr_inverse_unique_divring:
  fixes   f g :: "'a ::division_ring fps"
  assumes fg: "f * g = 1"
  shows   "fps_left_inverse g (f$0) = f"
  and     "fps_right_inverse f (g$0) = g"
proof-
  from fg have "f$0 * g$0 = 1" using fps_mult_nth_0[of f g] by simp
  hence "g$0 * f$0 = 1" using inverse_unique[of "f$0"] left_inverse[of "f$0"] by force
  thus "fps_left_inverse g (f$0) = f" "fps_right_inverse f (g$0) = g"
    using fg fps_lr_inverse_unique_ring1 by auto
qed

lemma fps_inverse_unique:
  fixes   f g :: "'a :: division_ring fps"
  assumes fg: "f * g = 1"
  shows   "inverse f = g"
proof -
  from fg have if0: "inverse (f$0) = g$0" "f$0 ≠ 0"
    using inverse_unique[of "f$0"] fps_mult_nth_0[of f g] by auto
  with fg have "fps_right_inverse f (g$0) = g"
    using left_inverse[of "f$0"] by (intro fps_lr_inverse_unique_ring1(2)) simp_all
  with if0(1) show ?thesis by (simp add: fps_inverse_def)
qed

lemma inverse_fps_numeral:
  "inverse (numeral n :: ('a :: field_char_0) fps) = fps_const (inverse (numeral n))"
  by (intro fps_inverse_unique fps_ext) (simp_all add: fps_numeral_nth)

lemma inverse_fps_of_nat:
  "inverse (of_nat n :: 'a :: {semiring_1,times,uminus,inverse} fps) =
    fps_const (inverse (of_nat n))"
  by (simp add: fps_of_nat fps_const_inverse[symmetric])

lemma fps_lr_inverse_mult_ring1:
  fixes   f g :: "'a::ring_1 fps"
  assumes x: "x * f$0 = 1" "f$0 * x = 1"
  and     y: "y * g$0 = 1" "g$0 * y = 1"
  shows   "fps_left_inverse (f * g) (y*x) = fps_left_inverse g y * fps_left_inverse f x"
  and     "fps_right_inverse (f * g) (y*x) = fps_right_inverse g y * fps_right_inverse f x"
proof -
  define h where "h ≡ fps_left_inverse g y * fps_left_inverse f x"
  hence h0: "h$0 = y*x" by simp
  have "fps_left_inverse (f*g) (h$0) = h"
  proof (intro fps_lr_inverse_unique_ring1(1))
    from h_def
      have  "h * (f * g) = fps_left_inverse g y * (fps_left_inverse f x * f) * g"
      by    (simp add: mult.assoc)
    thus "h * (f * g) = 1"
      using fps_left_inverse[OF x(1)] fps_left_inverse[OF y(1)] by simp
    from h_def have "(f*g)$0 * h$0 = f$0 * 1 * x"
      by (simp add: mult.assoc y(2)[symmetric])
    with x(2) show "(f * g) $ 0 * h $ 0 = 1" by simp
  qed
  with h_def
    show  "fps_left_inverse (f * g) (y*x) = fps_left_inverse g y * fps_left_inverse f x"
    by    simp
next
  define h where "h ≡ fps_right_inverse g y * fps_right_inverse f x"
  hence h0: "h$0 = y*x" by simp
  have "fps_right_inverse (f*g) (h$0) = h"
  proof (intro fps_lr_inverse_unique_ring1(2))
    from h_def
      have  "f * g * h = f * (g * fps_right_inverse g y) * fps_right_inverse f x"
      by    (simp add: mult.assoc)
    thus "f * g * h = 1"
      using fps_right_inverse[OF x(2)] fps_right_inverse[OF y(2)] by simp
    from h_def have "h$0 * (f*g)$0 = y * 1 * g$0"
      by (simp add: mult.assoc x(1)[symmetric])
    with y(1) show "h$0 * (f*g)$0 = 1" by simp
  qed
  with h_def
    show  "fps_right_inverse (f * g) (y*x) = fps_right_inverse g y * fps_right_inverse f x"
    by    simp
qed

lemma fps_lr_inverse_mult_divring:
  fixes f g :: "'a::division_ring fps"
  shows "fps_left_inverse (f * g) (inverse ((f*g)$0)) =
          fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))"
  and   "fps_right_inverse (f * g) (inverse ((f*g)$0)) =
          fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))"
proof-
  show "fps_left_inverse (f * g) (inverse ((f*g)$0)) =
          fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))"
  proof (cases "f$0 = 0 ∨ g$0 = 0")
    case True
    hence "fps_left_inverse (f * g) (inverse ((f*g)$0)) = 0"
      by (simp add: fps_lr_inverse_eq_0_iff(1))
    moreover from True have
      "fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0)) = 0"
      by (auto simp: fps_lr_inverse_eq_0_iff(1))
    ultimately show ?thesis by simp
  next
    case False
    hence "fps_left_inverse (f * g) (inverse (g$0) * inverse (f$0)) =
            fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))"
      by  (intro fps_lr_inverse_mult_ring1(1)) simp_all
    with False show ?thesis by (simp add: nonzero_inverse_mult_distrib)
  qed
  show "fps_right_inverse (f * g) (inverse ((f*g)$0)) =
          fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))"
  proof (cases "f$0 = 0 ∨ g$0 = 0")
    case True
    from True have "fps_right_inverse (f * g) (inverse ((f*g)$0)) = 0"
      by (simp add: fps_lr_inverse_eq_0_iff(2))
    moreover from True have
      "fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0)) = 0"
      by (auto simp: fps_lr_inverse_eq_0_iff(2))
    ultimately show ?thesis by simp
  next
    case False
    hence "fps_right_inverse (f * g) (inverse (g$0) * inverse (f$0)) =
            fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))"
      by  (intro fps_lr_inverse_mult_ring1(2)) simp_all
    with False show ?thesis by (simp add: nonzero_inverse_mult_distrib)
  qed
qed

lemma fps_inverse_mult_divring:
  "inverse (f * g) = inverse g * inverse (f :: 'a::division_ring fps)"
  using fps_lr_inverse_mult_divring(2) by (simp add: fps_inverse_def)

lemma fps_inverse_mult: "inverse (f * g :: 'a::field fps) = inverse f * inverse g"
  by (simp add: fps_inverse_mult_divring)

lemma inverse_prod_fps: "inverse (prod f A) = (∏x∈A. inverse (f x) :: 'a :: field fps)"
  by (induction A rule: infinite_finite_induct) (auto simp: fps_inverse_mult)

lemma fps_lr_inverse_gp_ring1:
  fixes   ones ones_inv :: "'a :: ring_1 fps"
  defines "ones ≡ Abs_fps (λn. 1)"
  and     "ones_inv ≡ Abs_fps (λn. if n=0 then 1 else if n=1 then - 1 else 0)"
  shows   "fps_left_inverse ones 1 = ones_inv"
  and     "fps_right_inverse ones 1 = ones_inv"
proof-
  show "fps_left_inverse ones 1 = ones_inv"
  proof (rule fps_ext)
    fix n
    show "fps_left_inverse ones 1 $ n = ones_inv $ n"
    proof (induct n rule: nat_less_induct)
      case (1 n) show ?case
      proof (cases n)
        case (Suc m)
        have m: "n = Suc m" by fact
        moreover have "fps_left_inverse ones 1 $ Suc m = ones_inv $ Suc m"
        proof (cases m)
          case (Suc k) thus ?thesis
            using Suc m 1 by (simp add: ones_def ones_inv_def sum.atLeast_Suc_atMost)
        qed (simp add: ones_def ones_inv_def)
        ultimately show ?thesis by simp
      qed (simp add: ones_inv_def)
    qed
  qed
  moreover have "fps_right_inverse ones 1 = fps_left_inverse ones 1"
    by (auto intro: fps_left_inverse_eq_fps_right_inverse[symmetric] simp: ones_def)
  ultimately show "fps_right_inverse ones 1 = ones_inv" by simp
qed

lemma fps_lr_inverse_gp_ring1':
  fixes   ones :: "'a :: ring_1 fps"
  defines "ones ≡ Abs_fps (λn. 1)"
  shows   "fps_left_inverse ones 1 = 1 - fps_X"
  and     "fps_right_inverse ones 1 = 1 - fps_X"
proof-
  define ones_inv :: "'a :: ring_1 fps" 
    where "ones_inv ≡ Abs_fps (λn. if n=0 then 1 else if n=1 then - 1 else 0)"
  hence "fps_left_inverse ones 1 = ones_inv"
  and   "fps_right_inverse ones 1 = ones_inv"
    using ones_def fps_lr_inverse_gp_ring1 by auto
  thus "fps_left_inverse ones 1 = 1 - fps_X"
  and   "fps_right_inverse ones 1 = 1 - fps_X"
    by (auto intro: fps_ext simp: ones_inv_def)
qed

lemma fps_inverse_gp:
  "inverse (Abs_fps(λn. (1::'a::division_ring))) =
    Abs_fps (λn. if n= 0 then 1 else if n=1 then - 1 else 0)"
  using fps_lr_inverse_gp_ring1(2) by (simp add: fps_inverse_def)

lemma fps_inverse_gp': "inverse (Abs_fps (λn. 1::'a::division_ring)) = 1 - fps_X"
  by (simp add: fps_inverse_def fps_lr_inverse_gp_ring1'(2))

lemma fps_lr_inverse_one_minus_fps_X:
  fixes   ones :: "'a :: ring_1 fps"
  defines "ones ≡ Abs_fps (λn. 1)"
  shows "fps_left_inverse (1 - fps_X) 1 = ones"
  and   "fps_right_inverse (1 - fps_X) 1 = ones"
proof-
  have "fps_left_inverse ones 1 = 1 - fps_X"
    using fps_lr_inverse_gp_ring1'(1) by (simp add: ones_def)
  thus "fps_left_inverse (1 - fps_X) 1 = ones"
    using fps_left_inverse_idempotent_ring1[of 1 ones 1] by (simp add: ones_def)
  have "fps_right_inverse ones 1 = 1 - fps_X"
    using fps_lr_inverse_gp_ring1'(2) by (simp add: ones_def)
  thus "fps_right_inverse (1 - fps_X) 1 = ones"
    using fps_right_inverse_idempotent_ring1[of ones 1 1] by (simp add: ones_def)
qed

lemma fps_inverse_one_minus_fps_X:
  fixes   ones :: "'a :: division_ring fps"
  defines "ones ≡ Abs_fps (λn. 1)"
  shows   "inverse (1 - fps_X) = ones"
  by      (simp add: fps_inverse_def assms fps_lr_inverse_one_minus_fps_X(2))

lemma fps_lr_one_over_one_minus_fps_X_squared:
  shows   "fps_left_inverse ((1 - fps_X)^2) (1::'a::ring_1) = Abs_fps (λn. of_nat (n+1))"
          "fps_right_inverse ((1 - fps_X)^2) (1::'a) = Abs_fps (λn. of_nat (n+1))"
proof-
  define  f invf2 :: "'a fps"
    where "f ≡ (1 - fps_X)"
    and   "invf2 ≡ Abs_fps (λn. of_nat (n+1))"

  have f2_nth_simps:
    "f^2 $ 1 = - of_nat 2" "f^2 $ 2 = 1" "∧n. n>2 ==> f^2 $ n = 0"
      by (simp_all add: power2_eq_square f_def fps_mult_nth sum.atLeast_Suc_atMost)

  show "fps_left_inverse (f^2) 1 = invf2"
  proof (intro fps_ext)
    fix n show "fps_left_inverse (f^2) 1 $ n = invf2 $ n"
    proof (induct n rule: nat_less_induct)
      case (1 t)
      hence induct_assm:
        "∧m. m < t ==> fps_left_inverse (f🪙2) 1 $ m = invf2 $ m"
        by fast
      show ?case
      proof (cases t)
        case (Suc m)
        have m: "t = Suc m" by fact
        moreover have "fps_left_inverse (f^2) 1 $ Suc m = invf2 $ Suc m"
        proof (cases m)
          case 0 thus ?thesis using f2_nth_simps(1) by (simp add: invf2_def)
        next
          case (Suc l)
          have l: "m = Suc l" by fact
          moreover have "fps_left_inverse (f^2) 1 $ Suc (Suc l) = invf2 $ Suc (Suc l)"
          proof (cases l)
            case 0 thus ?thesis using f2_nth_simps(1,2) by (simp add: Suc_1[symmetric] invf2_def)
          next
            case (Suc k)
            from Suc l m
              have A: "fps_left_inverse (f🪙2) 1 $ Suc (Suc k) = invf2 $ Suc (Suc k)"
              and  B: "fps_left_inverse (f🪙2) 1 $ Suc k = invf2 $ Suc k"
              using induct_assm[of "Suc k"] induct_assm[of "Suc (Suc k)"]
              by    auto
            have times2: "∧a::nat. 2*a = a + a" by simp
            have "∀i∈{0..k}. (f^2)$(Suc (Suc (Suc k)) - i) = 0"
              using f2_nth_simps(3) by auto
            hence
              "fps_left_inverse (f^2) 1 $ Suc (Suc (Suc k)) =
                fps_left_inverse (f🪙2) 1 $ Suc (Suc k) * of_nat 2 -
                fps_left_inverse (f🪙2) 1 $ Suc k"
              using sum.ub_add_nat f2_nth_simps(1,2) by simp
            also have "… = of_nat (2 * Suc (Suc (Suc k))) - of_nat (Suc (Suc k))"
              by (subst A, subst B) (simp add: invf2_def mult.commute)
            also have "… = of_nat (Suc (Suc (Suc k)) + 1)"
              by (subst times2[of "Suc (Suc (Suc k))"]) simp
            finally have
              "fps_left_inverse (f^2) 1 $ Suc (Suc (Suc k)) = invf2 $ Suc (Suc (Suc k))"
               by (simp add: invf2_def)
            with Suc show ?thesis by simp
          qed
          ultimately show ?thesis by simp
        qed
        ultimately show ?thesis by simp
      qed (simp add: invf2_def)
    qed
  qed

  moreover have "fps_right_inverse (f^2) 1 = fps_left_inverse (f^2) 1"
    by  (auto
          intro: fps_left_inverse_eq_fps_right_inverse[symmetric]
          simp: f_def power2_eq_square
        )
  ultimately show "fps_right_inverse (f^2) 1 = invf2"
    by simp

qed

lemma fps_one_over_one_minus_fps_X_squared':
  assumes "inverse (1::'a::{ring_1,inverse}) = 1"
  shows   "inverse ((1 - fps_X)^2 :: 'a fps) = Abs_fps (λn. of_nat (n+1))"
  using   assms fps_lr_one_over_one_minus_fps_X_squared(2)
  by      (simp add: fps_inverse_def power2_eq_square)

lemma fps_one_over_one_minus_fps_X_squared:
  "inverse ((1 - fps_X)^2 :: 'a :: division_ring fps) = Abs_fps (λn. of_nat (n+1))"
  by (rule fps_one_over_one_minus_fps_X_squared'[OF inverse_1])

lemma fps_lr_inverse_fps_X_plus1:
  "fps_left_inverse (1 + fps_X) (1::'a::ring_1) = Abs_fps (λn. (-1)^n)"
  "fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (λn. (-1)^n)"
proof-

  show "fps_left_inverse (1 + fps_X) (1::'a) = Abs_fps (λn. (-1)^n)"
  proof (rule fps_ext)
    fix n show "fps_left_inverse (1 + fps_X) (1::'a) $ n = Abs_fps (λn. (-1)^n) $ n"
    proof (induct n rule: nat_less_induct)
      case (1 n) show ?case
      proof (cases n)
        case (Suc m)
        have m: "n = Suc m" by fact
        from Suc 1 have
          A:  "fps_left_inverse (1 + fps_X) (1::'a) $ n =
                - (∑i=0..m. (- 1)^i * (1 + fps_X) $ (Suc m - i))"
          by simp
        show ?thesis
        proof (cases m)
          case (Suc l)
          have "∀i∈{0..l}. ((1::'a fps) + fps_X) $ (Suc (Suc l) - i) = 0" by auto
          with Suc A m show ?thesis by simp
        qed (simp add: m A)
      qed simp
    qed
  qed

  moreover have
    "fps_right_inverse (1 + fps_X) (1::'a) = fps_left_inverse (1 + fps_X) 1"
    by (intro fps_left_inverse_eq_fps_right_inverse[symmetric]) simp_all
  ultimately show "fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (λn. (-1)^n)" by simp

qed

lemma fps_inverse_fps_X_plus1':
  assumes "inverse (1::'a::{ring_1,inverse}) = 1"
  shows   "inverse (1 + fps_X) = Abs_fps (λn. (- (1::'a)) ^ n)"
  using   assms fps_lr_inverse_fps_X_plus1(2)
  by      (simp add: fps_inverse_def)

lemma fps_inverse_fps_X_plus1:
  "inverse (1 + fps_X) = Abs_fps (λn. (- (1::'a::division_ring)) ^ n)"
  by (rule fps_inverse_fps_X_plus1'[OF inverse_1])

lemma subdegree_lr_inverse:
  fixes x :: "'a::{comm_monoid_add,mult_zero,uminus}"
  and   y :: "'b::{ab_group_add,mult_zero}"
  shows "subdegree (fps_left_inverse f x) = 0"
  and   "subdegree (fps_right_inverse g y) = 0"
proof-
  show "subdegree (fps_left_inverse f x) = 0"
    using fps_lr_inverse_eq_0_iff(1) subdegree_eq_0_iff by fastforce
  show "subdegree (fps_right_inverse g y) = 0"
    using fps_lr_inverse_eq_0_iff(2) subdegree_eq_0_iff by fastforce
qed

lemma subdegree_inverse [simp]:
  fixes f :: "'a::{ab_group_add,inverse,mult_zero} fps"
  shows "subdegree (inverse f) = 0"
  using subdegree_lr_inverse(2)
  by    (simp add: fps_inverse_def)

lemma fps_right_inverse_constructor_rec:
  "n > 0 ==> fps_right_inverse_constructor f a n =
               -a * sum (λi. fps_nth f i * fps_right_inverse_constructor f a (n - i)) {1..n}"
  by (cases n) auto

lemma fps_right_inverse_constructor_cong:
  assumes "∧k. k ≤ n ==> fps_nth f k = fps_nth g k"
  shows   "fps_right_inverse_constructor f c n = fps_right_inverse_constructor g c n"
  using assms
proof (induction n rule: less_induct)
  case (less n)
  show ?case
  proof (cases "n > 0")
    case n: True
    have "fps_right_inverse_constructor f c n =
            -c * sum (λi. fps_nth f i * fps_right_inverse_constructor f c (n - i)) {1..n}"
      by (subst fps_right_inverse_constructor_rec) (use n in auto)
    also have "sum (λi. fps_nth f i * fps_right_inverse_constructor f c (n - i)) {1..n} =
               sum (λi. fps_nth g i * fps_right_inverse_constructor g c (n - i)) {1..n}"
      by (intro sum.cong refl arg_cong2[of _ _ _ _ "(*)"] less) (use assms in auto)
    also have "-c * … = fps_right_inverse_constructor g c n"
      by (subst (2) fps_right_inverse_constructor_rec) (use n in auto)
    finally show ?thesis .
  qed auto
qed

lemma fps_cutoff_inverse:
  fixes f :: "'a :: field fps"
  assumes "fps_nth f 0 ≠ 0"
  shows "fps_cutoff n (inverse (fps_cutoff n f)) = fps_cutoff n (inverse f)"
proof (cases "n = 0")
  case True
  show ?thesis
    by (simp add: True)
next
  case False
  show ?thesis
  proof (subst fps_cutoff_eq_fps_cutoff_iff, safe)
    fix k assume "k < n"
    have "fps_nth (inverse (fps_cutoff n f)) k =
            fps_right_inverse_constructor (fps_cutoff n f) (inverse (fps_nth f 0)) k"
      using False by (simp add: fps_inverse_def)
    also have "… = fps_right_inverse_constructor f (inverse (fps_nth f 0)) k"
      by (rule fps_right_inverse_constructor_cong) (use ‹k < n› in auto)
    also have "… = fps_nth (inverse f) k"
      using False by (simp add: fps_inverse_def)
    finally show "fps_nth (inverse (fps_cutoff n f)) k = fps_nth (inverse f) k" .
  qed
qed

lemma tendsto_inverse_fps_aux:
  fixes f :: "'a :: field fps"
  assumes "fps_nth f 0 ≠ 0"
  shows "((λf. inverse f) ---> inverse f) (at f)"
  unfolding tendsto_fps_iff
proof
  fix n :: nat
  have "eventually (λg. ∀k≤n. fps_nth g k = fps_nth f k) (nhds f)"
    by (rule eventually_fps_nth_eq_nhds_fps_strong)
  hence "eventually (λg. ∀k≤n. fps_nth g k = fps_nth f k) (at f)"
    using eventually_nhds_conv_at by blast
  thus "eventually (λg. fps_nth (inverse g) n = fps_nth (inverse f) n) (at f)"
  proof eventually_elim
    case (elim g)
    from elim have "fps_nth g 0 = fps_nth f 0"
      by auto
    with assms have [simp]: "fps_nth g 0 ≠ 0"
      by simp
    have "fps_cutoff (n+1) (inverse f) = fps_cutoff (n+1) (inverse (fps_cutoff (n+1) f))"
      by (rule fps_cutoff_inverse [symmetric]) fact
    also have "fps_cutoff (n+1) f = fps_cutoff (n+1) g"
      by (subst fps_cutoff_eq_fps_cutoff_iff) (use elim in auto)
    also have "fps_cutoff (n+1) (inverse …) = fps_cutoff (n+1) (inverse g)"
      by (rule fps_cutoff_inverse) auto
    finally show ?case
      by (subst (asm) fps_cutoff_eq_fps_cutoff_iff) auto
  qed
qed

lemma tendsto_inverse_fps [tendsto_intros]:
  fixes g :: "'a :: field fps"
  assumes "(f ---> g) F"
  assumes "fps_nth g 0 ≠ 0"
  shows "((λx. inverse (f x)) ---> inverse g) F"
  by (rule tendsto_compose[OF tendsto_inverse_fps_aux assms(1)]) fact

lemma fps_div_zero [simp]:
  "0 div (g :: 'a :: {comm_monoid_add,inverse,mult_zero,uminus} fps) = 0"
  by (simp add: fps_divide_def)

lemma fps_div_by_zero':
  fixes g :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fps"
  assumes "inverse (0::'a) = 0"
  shows "g div 0 = 0"
  by (simp add: fps_divide_def assms fps_inverse_zero')

lemma fps_div_by_zero [simp]: "(g::'a::division_ring fps) div 0 = 0"
  by (rule fps_div_by_zero'[OF inverse_zero])

lemma fps_divide_unit': "subdegree g = 0 ==> f div g = f * inverse g"
  by (simp add: fps_divide_def)

lemma fps_divide_unit: "g$0 ≠ 0 ==> f div g = f * inverse g"
  by (intro fps_divide_unit') (simp add: subdegree_eq_0_iff)

lemma fps_divide_nth_0':
  "subdegree (g::'a::division_ring fps) = 0 ==> (f div g) $ 0 = f $ 0 / (g $ 0)"
  by (simp add: fps_divide_unit' divide_inverse)

lemma fps_divide_nth_0 [simp]:
  "g $ 0 ≠ 0 ==> (f div g) $ 0 = f $ 0 / (g $ 0 :: _ :: division_ring)"
  by (simp add: fps_divide_nth_0')

lemma fps_divide_nth_below:
  fixes f g :: "'a::{comm_monoid_add,uminus,mult_zero,inverse} fps"
  shows "n < subdegree f - subdegree g ==> (f div g) $ n = 0"
  by (simp add: fps_divide_def fps_mult_nth_eq0)

lemma fps_divide_nth_base:
  fixes f g :: "'a::division_ring fps"
  assumes "subdegree g ≤ subdegree f"
  shows "(f div g) $ (subdegree f - subdegree g) = f $ subdegree f * inverse (g $ subdegree g)"
  by (simp add: assms fps_divide_def fps_divide_unit')

lemma fps_divide_subdegree_ge:
  fixes f g :: "'a::{comm_monoid_add,uminus,mult_zero,inverse} fps"
  assumes "f / g ≠ 0"
  shows "subdegree (f / g) ≥ subdegree f - subdegree g"
  by (intro subdegree_geI) (simp_all add: assms fps_divide_nth_below)

lemma fps_divide_subdegree:
  fixes f g :: "'a::division_ring fps"
  assumes "f ≠ 0" "g ≠ 0" "subdegree g ≤ subdegree f"
  shows "subdegree (f / g) = subdegree f - subdegree g"
proof (intro antisym)
  from assms have 1: "(f div g) $ (subdegree f - subdegree g) ≠ 0"
    using fps_divide_nth_base[of g f] by simp
  thus "subdegree (f / g) ≤ subdegree f - subdegree g" by (intro subdegree_leI) simp
  from 1 have "f / g ≠ 0" by (auto intro: fps_nonzeroI)
  thus "subdegree f - subdegree g ≤ subdegree (f / g)" by (rule fps_divide_subdegree_ge)
qed

lemma fps_divide_shift_numer:
  fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
  assumes "n ≤ subdegree f"
  shows "fps_shift n f / g = fps_shift n (f/g)"
  using assms fps_shift_mult_right_noncomm[of n f "inverse (unit_factor g)"]
          fps_shift_fps_shift_reorder[of "subdegree g" n "f * inverse (unit_factor g)"]
  by (simp add: fps_divide_def)

lemma fps_divide_shift_denom:
  fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
  assumes "n ≤ subdegree g" "subdegree g ≤ subdegree f"
  shows "f / fps_shift n g = Abs_fps (λk. if k<n then 0 else (f/g) $ (k-n))"
proof (intro fps_ext)
  fix k
  from assms(1) have LHS:
    "(f / fps_shift n g) $ k = (f * inverse (unit_factor g)) $ (k + (subdegree g - n))"
    using fps_unit_factor_shift[of n g]
    by (simp add: fps_divide_def)
  show "(f / fps_shift n g) $ k = Abs_fps (λk. if k<n then 0 else (f/g) $ (k-n)) $ k"
  proof (cases "k<n")
    case True with assms LHS show ?thesis using fps_mult_nth_eq0[of _ f] by simp
  next
    case False
    hence "(f/g) $ (k-n) = (f * inverse (unit_factor g)) $ ((k-n) + subdegree g)"
      by (simp add: fps_divide_def)
    with False LHS assms(1) show ?thesis by auto
  qed
qed

lemma fps_divide_unit_factor_numer:
  fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
  shows "unit_factor f / g = fps_shift (subdegree f) (f/g)"
  by (simp add: fps_divide_shift_numer)

lemma fps_divide_unit_factor_denom:
  fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
  assumes "subdegree g ≤ subdegree f"
  shows
    "f / unit_factor g = Abs_fps (λk. if k<subdegree g then 0 else (f/g) $ (k-subdegree g))"
  by (simp add: assms fps_divide_shift_denom)

lemma fps_divide_unit_factor_both':
  fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
  assumes "subdegree g ≤ subdegree f"
  shows "unit_factor f / unit_factor g = fps_shift (subdegree f - subdegree g) (f / g)"
  using assms fps_divide_unit_factor_numer[of f "unit_factor g"]
          fps_divide_unit_factor_denom[of g f]
          fps_shift_rev_shift(1)[of "subdegree g" "subdegree f" "f/g"]
  by simp

lemma fps_divide_unit_factor_both:
  fixes f g :: "'a::division_ring fps"
  assumes "subdegree g ≤ subdegree f"
  shows "unit_factor f / unit_factor g = unit_factor (f / g)"
  using assms fps_divide_unit_factor_both'[of g f] fps_divide_subdegree[of f g]
  by (cases "f=0 ∨ g=0") auto

lemma fps_divide_self:
  "(f::'a::division_ring fps) ≠ 0 ==> f / f = 1"
  using fps_mult_right_inverse_unit_factor_divring[of f]
  by (simp add: fps_divide_def)

lemma fps_divide_add:
  fixes f g h :: "'a::{semiring_0,inverse,uminus} fps"
  shows "(f + g) / h = f / h + g / h"
  by (simp add: fps_divide_def algebra_simps fps_shift_add)

lemma fps_divide_diff:
  fixes f g h :: "'a::{ring,inverse} fps"
  shows "(f - g) / h = f / h - g / h"
  by (simp add: fps_divide_def algebra_simps fps_shift_diff)

lemma fps_divide_uminus:
  fixes f g h :: "'a::{ring,inverse} fps"
  shows "(- f) / g = - (f / g)"
  by (simp add: fps_divide_def algebra_simps fps_shift_uminus)

lemma fps_divide_uminus':
  fixes f g h :: "'a::division_ring fps"
  shows "f / (- g) = - (f / g)"
  by (simp add: fps_divide_def fps_unit_factor_uminus fps_shift_uminus)

lemma fps_divide_times:
  fixes f g h :: "'a::{semiring_0,inverse,uminus} fps"
  assumes "subdegree h ≤ subdegree g"
  shows "(f * g) / h = f * (g / h)"
  using assms fps_mult_subdegree_ge[of g "inverse (unit_factor h)"]
          fps_shift_mult[of "subdegree h" "g * inverse (unit_factor h)" f]
  by (fastforce simp add: fps_divide_def mult.assoc)

lemma fps_divide_times2:
  fixes f g h :: "'a::{comm_semiring_0,inverse,uminus} fps"
  assumes "subdegree h ≤ subdegree f"
  shows "(f * g) / h = (f / h) * g"
  using assms fps_divide_times[of h f g]
  by (simp add: mult.commute)

lemma fps_times_divide_eq:
  fixes f g :: "'a::field fps"
  assumes "g ≠ 0" and "subdegree f ≥ subdegree g"
  shows "f div g * g = f"
  using assms fps_divide_times2[of g f g]
  by (simp add: fps_divide_times fps_divide_self)

lemma fps_divide_times_eq:
  "(g :: 'a::division_ring fps) ≠ 0 ==> (f * g) div g = f"
  by (simp add: fps_divide_times fps_divide_self)

lemma fps_divide_by_mult':
  fixes f g h :: "'a :: division_ring fps"
  assumes "subdegree h ≤ subdegree f"
  shows "f / (g * h) = f / h / g"
proof (cases "f=0 ∨ g=0 ∨ h=0")
  case False with assms show ?thesis
    using fps_unit_factor_mult[of g h]
    by (auto simp:
            fps_divide_def fps_shift_fps_shift fps_inverse_mult_divring mult.assoc
            fps_shift_mult_right_noncomm
          )
qed auto

lemma fps_divide_by_mult:
  fixes f g h :: "'a :: field fps"
  assumes "subdegree g ≤ subdegree f"
  shows "f / (g * h) = f / g / h"
proof-
  have "f / (g * h) = f / (h * g)" by (simp add: mult.commute)
  also have "… = f / g / h" using fps_divide_by_mult'[OF assms] by simp
  finally show ?thesis by simp
qed

lemma fps_divide_cancel:
  fixes f g h :: "'a :: division_ring fps"
  shows "h ≠ 0 ==> (f * h) div (g * h) = f div g"
  by (cases "f=0")
        (auto simp: fps_divide_by_mult' fps_divide_times_eq)

lemma fps_divide_1':
  fixes a :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fps"
  assumes "inverse (1::'a) = 1"
  shows "a / 1 = a"
  using assms fps_inverse_one' fps_one_mult(2)[of a]
  by (force simp: fps_divide_def)

lemma fps_divide_1 [simp]: "(a :: 'a::division_ring fps) / 1 = a"
  by (rule fps_divide_1'[OF inverse_1])

lemma fps_divide_X':
  fixes f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fps"
  assumes "inverse (1::'a) = 1"
  shows "f / fps_X = fps_shift 1 f"
  using assms fps_one_mult(2)[of f]
  by (simp add: fps_divide_def fps_X_unit_factor fps_inverse_one')

lemma fps_divide_X [simp]: "a / fps_X = fps_shift 1 (a::'a::division_ring fps)"
  by (rule fps_divide_X'[OF inverse_1])

lemma fps_divide_X_power':
  fixes f :: "'a::{semiring_1,inverse,uminus} fps"
  assumes "inverse (1::'a) = 1"
  shows "f / (fps_X ^ n) = fps_shift n f"
  using fps_inverse_one'[OF assms] fps_one_mult(2)[of f]
  by (simp add: fps_divide_def fps_X_power_subdegree)

lemma fps_divide_X_power [simp]: "a / (fps_X ^ n) = fps_shift n (a::'a::division_ring fps)"
  by (rule fps_divide_X_power'[OF inverse_1])

lemma fps_divide_shift_denom_conv_times_fps_X_power:
  fixes f g :: "'a::{semiring_1,inverse,uminus} fps"
  assumes "n ≤ subdegree g" "subdegree g ≤ subdegree f"
  shows "f / fps_shift n g = f / g * fps_X ^ n"
  using assms
  by (intro fps_ext) (simp_all add: fps_divide_shift_denom fps_X_power_mult_right_nth)

lemma fps_divide_unit_factor_denom_conv_times_fps_X_power:
  fixes f g :: "'a::{semiring_1,inverse,uminus} fps"
  assumes "subdegree g ≤ subdegree f"
  shows "f / unit_factor g = f / g * fps_X ^ subdegree g"
  by (simp add: assms fps_divide_shift_denom_conv_times_fps_X_power)

lemma fps_shift_altdef':
  fixes f :: "'a::{semiring_1,inverse,uminus} fps"
  assumes "inverse (1::'a) = 1"
  shows "fps_shift n f = f div fps_X^n"
  using assms
  by (simp add:
            fps_divide_def fps_X_power_subdegree fps_X_power_unit_factor fps_inverse_one'
          )

lemma fps_shift_altdef:
  "fps_shift n f = (f :: 'a :: division_ring fps) div fps_X^n"
  by (rule fps_shift_altdef'[OF inverse_1])

lemma fps_div_fps_X_power_nth':
  fixes f :: "'a::{semiring_1,inverse,uminus} fps"
  assumes "inverse (1::'a) = 1"
  shows "(f div fps_X^n) $ k = f $ (k + n)"
  using assms
  by (simp add: fps_shift_altdef' [symmetric])

lemma fps_div_fps_X_power_nth: "((f :: 'a :: division_ring fps) div fps_X^n) $ k = f $ (k + n)"
  by (rule fps_div_fps_X_power_nth'[OF inverse_1])

lemma fps_div_fps_X_nth':
  fixes f :: "'a::{semiring_1,inverse,uminus} fps"
  assumes "inverse (1::'a) = 1"
  shows "(f div fps_X) $ k = f $ Suc k"
  using assms fps_div_fps_X_power_nth'[of f 1]
  by simp

lemma fps_div_fps_X_nth: "((f :: 'a :: division_ring fps) div fps_X) $ k = f $ Suc k"
  by (rule fps_div_fps_X_nth'[OF inverse_1])

lemma divide_fps_const':
  fixes c :: "'a :: {inverse,comm_monoid_add,uminus,mult_zero}"
  shows "f / fps_const c = f * fps_const (inverse c)"
  by (simp add: fps_divide_def fps_const_inverse)

lemma divide_fps_const [simp]:
  fixes c :: "'a :: {comm_semiring_0,inverse,uminus}"
  shows "f / fps_const c = fps_const (inverse c) * f"
  by (simp add: divide_fps_const' mult.commute)

lemma fps_const_divide: "fps_const (x :: _ :: division_ring) / fps_const y = fps_const (x / y)"
  by (simp add: fps_divide_def fps_const_inverse divide_inverse)

lemma fps_numeral_divide_divide:
  "x / numeral b / numeral c = (x / numeral (b * c) :: 'a :: field fps)"
  by (simp add: fps_divide_by_mult[symmetric])

lemma fps_numeral_mult_divide:
  "numeral b * x / numeral c = (numeral b / numeral c * x :: 'a :: field fps)"
  by (simp add: fps_divide_times2)

lemmas fps_numeral_simps =
  fps_numeral_divide_divide fps_numeral_mult_divide inverse_fps_numeral neg_numeral_fps_const

lemma fps_is_left_unit_iff_zeroth_is_left_unit:
  fixes f :: "'a :: ring_1 fps"
  shows "(∃g. 1 = f * g) ⟷ (∃k. 1 = f$0 * k)"
proof
  assume "∃g. 1 = f * g"
  then obtain g where "1 = f * g" by fast
  hence "1 = f$0 * g$0" using fps_mult_nth_0[of f g] by simp
  thus "∃k. 1 = f$0 * k" by auto
next
  assume "∃k. 1 = f$0 * k"
  then obtain k where "1 = f$0 * k" by fast
  hence "1 = f * fps_right_inverse f k"
    using fps_right_inverse by simp
  thus "∃g. 1 = f * g" by fast
qed

lemma fps_is_right_unit_iff_zeroth_is_right_unit:
  fixes f :: "'a :: ring_1 fps"
  shows "(∃g. 1 = g * f) ⟷ (∃k. 1 = k * f$0)"
proof
  assume "∃g. 1 = g * f"
  then obtain g where "1 = g * f" by fast
  hence "1 = g$0 * f$0" using fps_mult_nth_0[of g f] by simp
  thus "∃k. 1 = k * f$0" by auto
next
  assume "∃k. 1 = k * f$0"
  then obtain k where "1 = k * f$0" by fast
  hence "1 = fps_left_inverse f k * f"
    using fps_left_inverse by simp
  thus "∃g. 1 = g * f" by fast
qed

lemma fps_is_unit_iff [simp]: "(f :: 'a :: field fps) dvd 1 ⟷ f $ 0 ≠ 0"
proof
  assume "f dvd 1"
  then obtain g where "1 = f * g" by (elim dvdE)
  from this[symmetric] have "(f*g) $ 0 = 1" by simp
  thus "f $ 0 ≠ 0" by auto
next
  assume A: "f $ 0 ≠ 0"
  thus "f dvd 1" by (simp add: inverse_mult_eq_1[OF A, symmetric])
qed

lemma subdegree_eq_0_left:
  fixes f :: "'a::{comm_monoid_add,zero_neq_one,mult_zero} fps"
  assumes "∃g. 1 = f * g"
  shows "subdegree f = 0"
proof (intro subdegree_eq_0)
  from assms obtain g where "1 = f * g" by fast
  hence "f$0 * g$0 = 1" using fps_mult_nth_0[of f g] by simp
  thus "f$0 ≠ 0" by auto
qed

lemma subdegree_eq_0_right:
  fixes f :: "'a::{comm_monoid_add,zero_neq_one,mult_zero} fps"
  assumes "∃g. 1 = g * f"
  shows "subdegree f = 0"
proof (intro subdegree_eq_0)
  from assms obtain g where "1 = g * f" by fast
  hence "g$0 * f$0 = 1" using fps_mult_nth_0[of g f] by simp
  thus "f$0 ≠ 0" by auto
qed

lemma subdegree_eq_0' [simp]: "(f :: 'a :: field fps) dvd 1 ==> subdegree f = 0"
  by simp

lemma fps_dvd1_left_trivial_unit_factor:
  fixes f :: "'a::{comm_monoid_add, zero_neq_one, mult_zero} fps"
  assumes "∃g. 1 = f * g"
  shows "unit_factor f = f"
  using assms subdegree_eq_0_left
  by fastforce

lemma fps_dvd1_right_trivial_unit_factor:
  fixes f :: "'a::{comm_monoid_add, zero_neq_one, mult_zero} fps"
  assumes "∃g. 1 = g * f"
  shows "unit_factor f = f"
  using assms subdegree_eq_0_right
  by fastforce

lemma fps_dvd1_trivial_unit_factor:
  "(f :: 'a::comm_semiring_1 fps) dvd 1 ==> unit_factor f = f"
  unfolding dvd_def by (rule fps_dvd1_left_trivial_unit_factor) simp

lemma fps_unit_dvd_left:
  fixes f :: "'a :: division_ring fps"
  assumes "f $ 0 ≠ 0"
  shows "∃g. 1 = f * g"
  using assms fps_is_left_unit_iff_zeroth_is_left_unit right_inverse
  by fastforce

lemma fps_unit_dvd_right:
  fixes f :: "'a :: division_ring fps"
  assumes "f $ 0 ≠ 0"
  shows "∃g. 1 = g * f"
  using assms fps_is_right_unit_iff_zeroth_is_right_unit left_inverse
  by fastforce

lemma fps_unit_dvd [simp]: "(f $ 0 :: 'a :: field) ≠ 0 ==> f dvd g"
  using fps_unit_dvd_left dvd_trans[of f 1] by simp

lemma dvd_left_imp_subdegree_le:
  fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
  assumes "∃k. g = f * k" "g ≠ 0"
  shows "subdegree f ≤ subdegree g"
  using assms fps_mult_subdegree_ge
  by fastforce

lemma dvd_right_imp_subdegree_le:
  fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
  assumes "∃k. g = k * f" "g ≠ 0"
  shows "subdegree f ≤ subdegree g"
  using assms fps_mult_subdegree_ge
  by fastforce

lemma dvd_imp_subdegree_le:
  "f dvd g ==> g ≠ 0 ==> subdegree f ≤ subdegree g"
  using dvd_left_imp_subdegree_le by fast

lemma subdegree_le_imp_dvd_left_ring1:
  fixes f g :: "'a :: ring_1 fps"
  assumes "∃y. f $ subdegree f * y = 1" "subdegree f ≤ subdegree g"
  shows "∃k. g = f * k"
proof-
  define h :: "'a fps" where "h ≡ fps_X ^ (subdegree g - subdegree f)"
  from assms(1) obtain y where "f $ subdegree f * y = 1" by fast
  hence "unit_factor f $ 0 * y = 1" by simp
  from this obtain k where "1 = unit_factor f * k"
    using fps_is_left_unit_iff_zeroth_is_left_unit[of "unit_factor f"] by auto
  hence "fps_X ^ subdegree f = fps_X ^ subdegree f * unit_factor f * k"
    by (simp add: mult.assoc)
  moreover have "fps_X ^ subdegree f * unit_factor f = f"
    by (rule fps_unit_factor_decompose'[symmetric])
  ultimately have
    "fps_X ^ (subdegree f + (subdegree g - subdegree f)) = f * k * h"
    by (simp add: power_add h_def)
  hence "g = f * (k * h * unit_factor g)"
    using fps_unit_factor_decompose'[of g]
    by (simp add: assms(2) mult.assoc)
  thus ?thesis by fast
qed

lemma subdegree_le_imp_dvd_left_divring:
  fixes f g :: "'a :: division_ring fps"
  assumes "f ≠ 0" "subdegree f ≤ subdegree g"
  shows "∃k. g = f * k"
proof (intro subdegree_le_imp_dvd_left_ring1)
  from assms(1) have "f $ subdegree f ≠ 0" by simp
  thus "∃y. f $ subdegree f * y = 1" using right_inverse by blast
qed (rule assms(2))

lemma subdegree_le_imp_dvd_right_ring1:
  fixes f g :: "'a :: ring_1 fps"
  assumes "∃x. x * f $ subdegree f = 1" "subdegree f ≤ subdegree g"
  shows "∃k. g = k * f"
proof-
  define h :: "'a fps" where "h ≡ fps_X ^ (subdegree g - subdegree f)"
  from assms(1) obtain x where "x * f $ subdegree f = 1" by fast
  hence "x * unit_factor f $ 0 = 1" by simp
  from this obtain k where "1 = k * unit_factor f"
    using fps_is_right_unit_iff_zeroth_is_right_unit[of "unit_factor f"] by auto
  hence "fps_X ^ subdegree f = k * (unit_factor f * fps_X ^ subdegree f)"
    by (simp add: mult.assoc[symmetric])
  moreover have "unit_factor f * fps_X ^ subdegree f = f"
    by (rule fps_unit_factor_decompose[symmetric])
  ultimately have "fps_X ^ (subdegree g - subdegree f + subdegree f) = h * k * f"
    by (simp add: power_add h_def mult.assoc)
  hence "g = unit_factor g * h * k * f"
    using fps_unit_factor_decompose[of g]
    by (simp add: assms(2) mult.assoc)
  thus ?thesis by fast
qed

lemma subdegree_le_imp_dvd_right_divring:
  fixes f g :: "'a :: division_ring fps"
  assumes "f ≠ 0" "subdegree f ≤ subdegree g"
  shows "∃k. g = k * f"
proof (intro subdegree_le_imp_dvd_right_ring1)
  from assms(1) have "f $ subdegree f ≠ 0" by simp
  thus "∃x. x * f $ subdegree f = 1" using left_inverse by blast
qed (rule assms(2))

lemma fps_dvd_iff:
  assumes "(f :: 'a :: field fps) ≠ 0" "g ≠ 0"
  shows "f dvd g ⟷ subdegree f ≤ subdegree g"
proof
  assume "subdegree f ≤ subdegree g"
  with assms show "f dvd g"
    using subdegree_le_imp_dvd_left_divring
    by (auto intro: dvdI)
qed (simp add: assms dvd_imp_subdegree_le)

lemma subdegree_div':
  fixes p q :: "'a::division_ring fps"
  assumes "∃k. p = k * q"
  shows "subdegree (p div q) = subdegree p - subdegree q"
proof (cases "p = 0")
  case False
  from assms(1) obtain k where k: "p = k * q" by blast
  with False have "subdegree (p div q) = subdegree k" by (simp add: fps_divide_times_eq)
  moreover have "k $ subdegree k * q $ subdegree q ≠ 0"
  proof
    assume "k $ subdegree k * q $ subdegree q = 0"
    hence "k $ subdegree k * q $ subdegree q * inverse (q $ subdegree q) = 0" by simp
    with False k show False by (simp add: mult.assoc)
  qed
  ultimately show ?thesis by (simp add: k subdegree_mult')
qed simp

lemma subdegree_div:
  fixes p q :: "'a :: field fps"
  assumes "q dvd p"
  shows "subdegree (p div q) = subdegree p - subdegree q"
  using assms
  unfolding dvd_def
  by (auto intro: subdegree_div')

lemma subdegree_div_unit':
  fixes p q :: "'a :: {ab_group_add,mult_zero,inverse} fps"
  assumes "q $ 0 ≠ 0" "p $ subdegree p * inverse (q $ 0) ≠ 0"
  shows "subdegree (p div q) = subdegree p"
  using assms subdegree_mult'[of p "inverse q"]
  by (auto simp add: fps_divide_unit)

lemma subdegree_div_unit'':
  fixes p q :: "'a :: {ring_no_zero_divisors,inverse} fps"
  assumes "q $ 0 ≠ 0" "inverse (q $ 0) ≠ 0"
  shows "subdegree (p div q) = subdegree p"
  by (cases "p = 0") (auto intro: subdegree_div_unit' simp: assms)

lemma subdegree_div_unit:
  fixes p q :: "'a :: division_ring fps"
  assumes "q $ 0 ≠ 0"
  shows "subdegree (p div q) = subdegree p"
  by (intro subdegree_div_unit'') (simp_all add: assms)

instantiation fps :: ("{comm_semiring_1,inverse,uminus}") modulo
begin

definition fps_mod_def:
  "f mod g = (if g = 0 then f else
     let h = unit_factor g in  fps_cutoff (subdegree g) (f * inverse h) * h)"

instance ..

end

lemma fps_mod_zero [simp]:
  "(f::'a::{comm_semiring_1,inverse,uminus} fps) mod 0 = f"
  by (simp add: fps_mod_def)

lemma fps_mod_eq_zero:
  assumes "g ≠ 0" and "subdegree f ≥ subdegree g"
  shows "f mod g = 0"
proof (cases "f * inverse (unit_factor g) = 0")
  case False
  have "fps_cutoff (subdegree g) (f * inverse (unit_factor g)) = 0"
    using False assms(2) fps_mult_subdegree_ge fps_cutoff_zero_iff by force
  with assms(1) show ?thesis by (simp add: fps_mod_def Let_def)
qed (simp add: assms fps_mod_def)

lemma fps_mod_unit [simp]: "g$0 ≠ 0 ==> f mod g = 0"
  by (intro fps_mod_eq_zero) auto

lemma subdegree_mod:
  assumes "subdegree (f::'a::field fps) < subdegree g"
  shows "subdegree (f mod g) = subdegree f"
proof (cases "f = 0")
  case False
  with assms show ?thesis
    by (intro subdegreeI)
        (auto simp: inverse_mult_eq_1 fps_mod_def Let_def fps_cutoff_left_mult_nth mult.assoc)
qed (simp add: fps_mod_def)

instance fps :: (field) idom_modulo
proof

  fix f g :: "'a fps"

  define n where "n = subdegree g"
  define h where "h = f * inverse (unit_factor g)"

  show "f div g * g + f mod g = f"
  proof (cases "g = 0")
    case False
    with n_def h_def have
      "f div g * g + f mod g = (fps_shift n h * fps_X ^ n + fps_cutoff n h) * unit_factor g"
      by (simp add: fps_divide_def fps_mod_def Let_def subdegree_decompose algebra_simps)
    with False show ?thesis
      by (simp add: fps_shift_cutoff h_def inverse_mult_eq_1)
  qed auto

qed (rule fps_divide_times_eq, simp_all add: fps_divide_def)

instantiation fps :: (field) normalization_semidom_multiplicative
begin

definition fps_normalize_def [simp]:
  "normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)"

instance proof
  fix f g :: "'a fps"
  assume "is_unit f"
  thus "unit_factor (f * g) = f * unit_factor g"
    using fps_unit_factor_mult[of f g] by simp
next
  fix f g :: "'a fps"
  show "unit_factor f * normalize f = f"
    by (simp add: fps_shift_times_fps_X_power)
next
  fix f g :: "'a fps"
  show "unit_factor (f * g) = unit_factor f * unit_factor g"
    using fps_unit_factor_mult[of f g] by simp
qed (simp_all add: fps_divide_def Let_def)

end


subsection ‹Computing reciprocals via Hensel lifting›

lemma inverse_fps_hensel_lifting:
  fixes F G :: "'a :: field fps" and n :: nat
  assumes G_eq: "fps_cutoff n G = fps_cutoff n (inverse F)"
  assumes unit: "fps_nth F 0 ≠ 0"
  shows "fps_cutoff (2*n) (inverse F) = fps_cutoff (2*n) (G * (2 - F * G))"
proof -
  define R where "R = inverse F - G"
  have eq: "G = inverse F - R"
    by (simp add: R_def)
  from assms have "fps_cutoff n R = 0"
    by (simp add: R_def fps_cutoff_diff)
  hence R: "R = 0 ∨ subdegree R ≥ n"
    by (simp add: fps_cutoff_zero_iff)

  have "G * (2 - F * G) - inverse F = 
         inverse F + F * inverse F * R * 2 - F * R🪙2 - R * 2 - F * inverse F * inverse F"
    by (simp add: eq algebra_simps power2_eq_square)
  also have "F * inverse F = 1"
    using unit by (simp add: inverse_mult_eq_1')
  also have "inverse F + 1 * R * 2 - F * R🪙2 - R * 2 - 1 * inverse F = -F * R🪙2"
    by (simp add: algebra_simps)
  finally have "fps_cutoff (2*n) (G * (2 - F * G) - inverse F) = fps_cutoff (2*n) (-F * R🪙2)"
    by (rule arg_cong)
  also have "… = 0"
  proof (cases "-F * R🪙2 = 0")
    case False
    have "2 * n ≤ subdegree (-F * R🪙2)"
      using False R unit by simp
    thus ?thesis
      by (simp add: fps_cutoff_zero_iff)
  qed auto
  finally show ?thesis
    by (simp add: fps_cutoff_diff)
qed

lemma inverse_fps_hensel_lifting':
  fixes F G :: "'a :: field fps" and n :: nat
  assumes G_eq: "fps_cutoff n G = fps_cutoff n (inverse F)"
  assumes unit: "fps_nth F 0 ≠ 0"
  defines "P ≡ fps_shift n (F * G - 1)"
  shows "fps_cutoff (2*n) (inverse F) = fps_cutoff (2*n) (G * (1 - fps_X ^ n * P))"
proof -
  define R where "R = inverse F - G"
  have eq: "G = inverse F - R"
    by (simp add: R_def)
  from assms have "fps_cutoff n R = 0"
    by (simp add: R_def fps_cutoff_diff)
  hence R: "R = 0 ∨ subdegree R ≥ n"
    by (simp add: fps_cutoff_zero_iff)

  have FG_eq: "F * G = 1 + fps_X ^ n * P"
  proof (cases "F * G - 1 = 0")
    case False
    have eq: "F * G - 1 = F * (G - inverse F)"
      using unit by (simp add: inverse_mult_eq_1' ring_distribs)
    have "subdegree (F * (G - inverse F)) ≥ n"
    proof -
      have "fps_cutoff n (G - inverse F) = 0"
        using G_eq by (simp add: fps_cutoff_diff)
      hence "n ≤ subdegree (G - inverse F)"
        using False unfolding eq by (simp add: fps_cutoff_zero_iff)
      also have "subdegree (G - inverse F) = subdegree (F * (G - inverse F))"
        by (subst subdegree_mult) (use unit False in ‹auto simp: eq›)
      finally have "n ≤ subdegree (F * (G - inverse F))" .
      thus ?thesis
        by blast
    qed
    hence "F * G - 1 = fps_X ^ n * P"
      unfolding eq P_def by (intro fps_conv_fps_X_power_mult_fps_shift) auto
    thus ?thesis
      by (simp add: algebra_simps)
  qed (auto simp: P_def)

  have "G * (1 - fps_X ^ n * P) - inverse F = G * (2 - F * G) - inverse F"
    by (auto simp: FG_eq)
  also have "G * (2 - F * G) - inverse F = 
         inverse F + F * inverse F * R * 2 - F * R🪙2 - R * 2 - F * inverse F * inverse F"
    by (simp add: eq algebra_simps power2_eq_square)
  also have "F * inverse F = 1"
    using unit by (simp add: inverse_mult_eq_1')
  also have "inverse F + 1 * R * 2 - F * R🪙2 - R * 2 - 1 * inverse F = -F * R🪙2"
    by (simp add: algebra_simps)
  finally have "fps_cutoff (2*n) (G * (1 - fps_X ^ n * P) - inverse F) = fps_cutoff (2*n) (-F * R🪙2)"
    by (rule arg_cong)
  also have "… = 0"
  proof (cases "-F * R🪙2 = 0")
    case False
    have "2 * n ≤ subdegree (-F * R🪙2)"
      using False R unit by simp
    thus ?thesis
      by (simp add: fps_cutoff_zero_iff)
  qed auto
  finally show ?thesis
    by (simp add: fps_cutoff_diff)
qed


subsection ‹Euclidean division›

instantiation fps :: (field) euclidean_ring_cancel
begin

definition fps_euclidean_size_def:
  "euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)"

instance proof
  fix f g :: "'a fps" assume [simp]: "g ≠ 0"
  show "euclidean_size f ≤ euclidean_size (f * g)"
    by (cases "f = 0") (simp_all add: fps_euclidean_size_def)
  show "euclidean_size (f mod g) < euclidean_size g"
  proof (cases "f = 0")
    case True
    then show ?thesis
      by (simp add: fps_euclidean_size_def)
  next
    case False
    then show ?thesis
      using le_less_linear[of "subdegree g" "subdegree f"]
      by (force simp add: fps_mod_eq_zero fps_euclidean_size_def subdegree_mod)
  qed
next
  fix f g h :: "'a fps" assume [simp]: "h ≠ 0"
  show "(h * f) div (h * g) = f div g"
    by (simp add: fps_divide_cancel mult.commute)
  show "(f + g * h) div h = g + f div h"
    by (simp add: fps_divide_add fps_divide_times_eq)
qed (simp add: fps_euclidean_size_def)

end

instance fps :: (field) normalization_euclidean_semiring ..

instantiation fps :: (field) euclidean_ring_gcd
begin
definition fps_gcd_def: "(gcd :: 'a fps ==> _) = Euclidean_Algorithm.gcd"
definition fps_lcm_def: "(lcm :: 'a fps ==> _) = Euclidean_Algorithm.lcm"
definition fps_Gcd_def: "(Gcd :: 'a fps set ==> _) = Euclidean_Algorithm.Gcd"
definition fps_Lcm_def: "(Lcm :: 'a fps set ==> _) = Euclidean_Algorithm.Lcm"
instance by standard (simp_all add: fps_gcd_def fps_lcm_def fps_Gcd_def fps_Lcm_def)
end

lemma fps_gcd:
  assumes [simp]: "f ≠ 0" "g ≠ 0"
  shows "gcd f g = fps_X ^ min (subdegree f) (subdegree g)"
proof -
  let ?m = "min (subdegree f) (subdegree g)"
  show "gcd f g = fps_X ^ ?m"
  proof (rule sym, rule gcdI)
    fix d assume "d dvd f" "d dvd g"
    thus "d dvd fps_X ^ ?m" by (cases "d = 0") (simp_all add: fps_dvd_iff)
  qed (simp_all add: fps_dvd_iff)
qed

lemma fps_gcd_altdef: "gcd f g =
  (if f = 0 ∧ g = 0 then 0 else
   if f = 0 then fps_X ^ subdegree g else
   if g = 0 then fps_X ^ subdegree f else
     fps_X ^ min (subdegree f) (subdegree g))"
  by (simp add: fps_gcd)

lemma fps_lcm:
  assumes [simp]: "f ≠ 0" "g ≠ 0"
  shows "lcm f g = fps_X ^ max (subdegree f) (subdegree g)"
proof -
  let ?m = "max (subdegree f) (subdegree g)"
  show "lcm f g = fps_X ^ ?m"
  proof (rule sym, rule lcmI)
    fix d assume "f dvd d" "g dvd d"
    thus "fps_X ^ ?m dvd d" by (cases "d = 0") (simp_all add: fps_dvd_iff)
  qed (simp_all add: fps_dvd_iff)
qed

lemma fps_lcm_altdef: "lcm f g =
  (if f = 0 ∨ g = 0 then 0 else fps_X ^ max (subdegree f) (subdegree g))"
  by (simp add: fps_lcm)

lemma fps_Gcd:
  assumes "A - {0} ≠ {}"
  shows "Gcd A = fps_X ^ (INF f∈A-{0}. subdegree f)"
proof (rule sym, rule GcdI)
  fix f assume "f ∈ A"
  thus "fps_X ^ (INF f∈A - {0}. subdegree f) dvd f"
    by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cINF_lower)
next
  fix d assume d: "∧f. f ∈ A ==> d dvd f"
  from assms obtain f where "f ∈ A - {0}" by auto
  with d[of f] have [simp]: "d ≠ 0" by auto
  from d assms have "subdegree d ≤ (INF f∈A-{0}. subdegree f)"
    by (intro cINF_greatest) (simp_all add: fps_dvd_iff[symmetric])
  with d assms show "d dvd fps_X ^ (INF f∈A-{0}. subdegree f)" by (simp add: fps_dvd_iff)
qed simp_all

lemma fps_Gcd_altdef: "Gcd A =
  (if A ⊆ {0} then 0 else fps_X ^ (INF f∈A-{0}. subdegree f))"
  using fps_Gcd by auto

lemma fps_Lcm:
  assumes "A ≠ {}" "0 ∉ A" "bdd_above (subdegree`A)"
  shows "Lcm A = fps_X ^ (SUP f∈A. subdegree f)"
proof (rule sym, rule LcmI)
  fix f assume "f ∈ A"
  moreover from assms(3) have "bdd_above (subdegree ` A)" by auto
  ultimately show "f dvd fps_X ^ (SUP f∈A. subdegree f)" using assms(2)
    by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cSUP_upper)
next
  fix d assume d: "∧f. f ∈ A ==> f dvd d"
  from assms obtain f where f: "f ∈ A" "f ≠ 0" by auto
  show "fps_X ^ (SUP f∈A. subdegree f) dvd d"
  proof (cases "d = 0")
    assume "d ≠ 0"
    moreover from d have "∧f. f ∈ A ==> f ≠ 0 ==> f dvd d" by blast
    ultimately have "subdegree d ≥ (SUP f∈A. subdegree f)" using assms
      by (intro cSUP_least) (auto simp: fps_dvd_iff)
    with ‹d ≠ 0› show ?thesis by (simp add: fps_dvd_iff)
  qed simp_all
qed simp_all

lemma fps_Lcm_altdef:
  "Lcm A =
     (if 0 ∈ A ∨ ¬bdd_above (subdegree`A) then 0 else
      if A = {} then 1 else fps_X ^ (SUP f∈A. subdegree f))"
proof (cases "bdd_above (subdegree`A)")
  assume unbounded: "¬bdd_above (subdegree`A)"
  have "Lcm A = 0"
  proof (rule ccontr)
    assume "Lcm A ≠ 0"
    from unbounded obtain f where f: "f ∈ A" "subdegree (Lcm A) < subdegree f"
      unfolding bdd_above_def by (auto simp: not_le)
    moreover from f and ‹Lcm A ≠ 0› have "subdegree f ≤ subdegree (Lcm A)"
      by (intro dvd_imp_subdegree_le dvd_Lcm) simp_all
    ultimately show False by simp
  qed
  with unbounded show ?thesis by simp
qed (simp_all add: fps_Lcm Lcm_eq_0_I)



subsection ‹Formal Derivatives›

definition "fps_deriv f = Abs_fps (λn. 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_0th_higher_deriv:
  "(fps_deriv ^^ n) f $ 0 = fact n * f $ n"
  by (induction n arbitrary: f)
      (simp_all add: funpow_Suc_right mult_of_nat_commute algebra_simps del: funpow.simps)

lemma fps_deriv_mult[simp]:
  "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
proof (intro fps_ext)
  fix n
  have LHS: "fps_deriv (f * g) $ n = (∑i=0..Suc n. of_nat (n+1) * f$i * g$(Suc n - i))"
    by (simp add: fps_mult_nth sum_distrib_left algebra_simps)

  have "∀i∈{1..n}. n - (i - 1) = n - i + 1" by auto
  moreover have
    "(∑i=0..n. of_nat (i+1) * f$(i+1) * g$(n - i)) =
      (∑i=1..Suc n. of_nat i * f$i * g$(n - (i - 1)))"
    by (intro sum.reindex_bij_witness[where i="λx. x-1" and j="λx. x+1"]) auto
  ultimately have
    "(f * fps_deriv g + fps_deriv f * g) $ n =
      of_nat (Suc n) * f$0 * g$(Suc n) +
      (∑i=1..n. (of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1)) +
          of_nat (Suc n) * f$(Suc n) * g$0"
    by (simp add: fps_mult_nth algebra_simps mult_of_nat_commute sum.atLeast_Suc_atMost sum.distrib)
  moreover have
    "∀i∈{1..n}.
      (of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1) =
      of_nat (n + 1) * f $ i * g $ (Suc n - i)"
  proof
    fix i assume i: "i ∈ {1..n}"
    from i have "of_nat (n - i + 1) + (of_nat i :: 'a) = of_nat (n + 1)"
      using of_nat_add[of "n-i+1" i,symmetric] by simp
    moreover from i have "Suc n - i = n - i + 1" by auto
    ultimately show "(of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1) =
            of_nat (n + 1) * f $ i * g $ (Suc n - i)"
      by simp
  qed
  ultimately have
    "(f * fps_deriv g + fps_deriv f * g) $ n =
      (∑i=0..Suc n. of_nat (Suc n) * f $ i * g $ (Suc n - i))"
    by (simp add: sum.atLeast_Suc_atMost)
  with LHS show "fps_deriv (f * g) $ n = (f * fps_deriv g + fps_deriv f * g) $ n"
    by simp
qed

lemma fps_deriv_fps_X[simp]: "fps_deriv fps_X = 1"
  by (simp add: fps_deriv_def fps_X_def fps_eq_iff)

lemma fps_deriv_neg[simp]:
  "fps_deriv (- (f:: 'a::ring_1 fps)) = - (fps_deriv f)"
  by (simp add: fps_eq_iff fps_deriv_def)

lemma fps_deriv_add[simp]: "fps_deriv (f + g) = fps_deriv f + fps_deriv g"
  by (auto intro: fps_ext simp: algebra_simps)

lemma fps_deriv_sub[simp]:
  "fps_deriv ((f:: 'a::ring_1 fps) - g) = fps_deriv f - fps_deriv g"
  using fps_deriv_add [of f "- g"] by simp

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

lemma fps_deriv_of_nat [simp]: "fps_deriv (of_nat n) = 0"
  by (simp add: fps_of_nat [symmetric])

lemma fps_deriv_of_int [simp]: "fps_deriv (of_int n) = 0"
  by (simp add: fps_of_int [symmetric])

lemma fps_deriv_numeral [simp]: "fps_deriv (numeral n) = 0"
  by (simp add: numeral_fps_const)

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

lemma fps_deriv_linear[simp]:
  "fps_deriv (fps_const a * f + fps_const b * g) =
    fps_const a * fps_deriv f + fps_const b * fps_deriv g"
  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) = fps_deriv f * fps_const c"
  by simp

lemma fps_deriv_sum:
  "fps_deriv (sum f S) = sum (λi. fps_deriv (f i)) S"
proof (cases "finite S")
  case False
  then show ?thesis by simp
next
  case True
  show ?thesis by (induct rule: finite_induct [OF True]) simp_all
qed

lemma fps_deriv_eq_0_iff [simp]:
  "fps_deriv f = 0 ⟷ f = fps_const (f$0 :: 'a::{semiring_no_zero_divisors,semiring_char_0})"
proof
  assume f: "fps_deriv f = 0"
  show "f = fps_const (f$0)"
  proof (intro fps_ext)
    fix n show "f $ n = fps_const (f$0) $ n"
    proof (cases n)
      case (Suc m)
      have "(of_nat (Suc m) :: 'a) ≠ 0" by (rule of_nat_neq_0)
      with f Suc show ?thesis using fps_deriv_nth[of f] by auto
    qed simp
  qed
next
  show "f = fps_const (f$0) ==> fps_deriv f = 0" using fps_deriv_const[of "f$0"] by simp
qed

lemma fps_deriv_eq_iff:
  fixes f g :: "'a::{ring_1_no_zero_divisors,semiring_char_0} fps"
  shows "fps_deriv f = fps_deriv g ⟷ (f = fps_const(f$0 - g$0) + g)"
proof -
  have "fps_deriv f = fps_deriv g ⟷ fps_deriv (f - g) = 0"
    using fps_deriv_sub[of f g]
    by simp
  also have "… ⟷ f - g = fps_const ((f - g) $ 0)"
    unfolding fps_deriv_eq_0_iff ..
  finally show ?thesis
    by (simp add: field_simps)
qed

lemma fps_deriv_eq_iff_ex:
  fixes f g :: "'a::{ring_1_no_zero_divisors,semiring_char_0} fps"
  shows "(fps_deriv f = fps_deriv g) ⟷ (∃c. f = fps_const c + g)"
  by (auto simp: fps_deriv_eq_iff)


fun fps_nth_deriv :: "nat ==> 'a::semiring_1 fps ==> '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 * 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

lemma fps_nth_deriv_neg[simp]:
  "fps_nth_deriv n (- (f :: 'a::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::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::ring_1 fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g"
  using fps_nth_deriv_add [of n f "- g"] 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 * 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) = fps_nth_deriv n f * fps_const c"
  by (induct n arbitrary: f) auto

lemma fps_nth_deriv_sum:
  "fps_nth_deriv n (sum f S) = sum (λi. fps_nth_deriv n (f i :: 'a::ring_1 fps)) S"
proof (cases "finite S")
  case True
  show ?thesis by (induct rule: finite_induct [OF True]) simp_all
next
  case False
  then show ?thesis by simp
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) (simp_all add: field_simps)

lemma fps_deriv_lr_inverse:
  fixes x y :: "'a::ring_1"
  assumes "x * f$0 = 1" "f$0 * y = 1"
  🍋 ‹These assumptions imply $x$ equals $y$, but no need to assume that.›
  shows "fps_deriv (fps_left_inverse f x) =
            - fps_left_inverse f x * fps_deriv f * fps_left_inverse f x"
  and "fps_deriv (fps_right_inverse f y) =
            - fps_right_inverse f y * fps_deriv f * fps_right_inverse f y"
proof-

  define L where "L ≡ fps_left_inverse f x"
  hence "fps_deriv (L * f) = 0" using fps_left_inverse[OF assms(1)] by simp
  with assms show "fps_deriv L = - L * fps_deriv f * L"
    using fps_right_inverse'[OF assms]
    by (simp add: minus_unique mult.assoc L_def)

  define R where "R ≡ fps_right_inverse f y"
  hence "fps_deriv (f * R) = 0" using fps_right_inverse[OF assms(2)] by simp
  hence 1: "f * fps_deriv R + fps_deriv f * R = 0" by simp
  have "R * f * fps_deriv R = - R * fps_deriv f * R"
    using iffD2[OF eq_neg_iff_add_eq_0, OF 1] by (simp add: mult.assoc)
  thus "fps_deriv R = - R * fps_deriv f * R"
    using fps_left_inverse'[OF assms] by (simp add: R_def)

qed

lemma fps_deriv_lr_inverse_comm:
  fixes x :: "'a::comm_ring_1"
  assumes "x * f$0 = 1"
  shows "fps_deriv (fps_left_inverse f x) = - fps_deriv f * (fps_left_inverse f x)🪙2"
  and "fps_deriv (fps_right_inverse f x) = - fps_deriv f * (fps_right_inverse f x)🪙2"
  using assms fps_deriv_lr_inverse[of x f x]
  by (simp_all add: mult.commute power2_eq_square)

lemma fps_inverse_deriv_divring:
  fixes a :: "'a::division_ring fps"
  assumes "a$0 ≠ 0"
  shows "fps_deriv (inverse a) = - inverse a * fps_deriv a * inverse a"
  using assms fps_deriv_lr_inverse(2)[of "inverse (a$0)" a "inverse (a$0)"]
  by (simp add: fps_inverse_def)

lemma fps_inverse_deriv:
  fixes a :: "'a::field fps"
  assumes "a$0 ≠ 0"
  shows "fps_deriv (inverse a) = - fps_deriv a * (inverse a)🪙2"
  using assms fps_deriv_lr_inverse_comm(2)[of "inverse (a$0)" a]
  by (simp add: fps_inverse_def)

lemma fps_inverse_deriv':
  fixes a :: "'a::field fps"
  assumes a0: "a $ 0 ≠ 0"
  shows "fps_deriv (inverse a) = - fps_deriv a / a🪙2"
  using fps_inverse_deriv[OF a0] a0
  by (simp add: fps_divide_unit power2_eq_square fps_inverse_mult)

(* FIXME: The last part of this proof should go through by simp once we have a proper
   theorem collection for simplifying division on rings *)
lemma fps_divide_deriv:
  assumes "b dvd (a :: 'a :: field fps)"
  shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b^2"
proof -
  have eq_divide_imp: "c ≠ 0 ==> a * c = b ==> a = b div c" for a b c :: "'a :: field fps"
    by (drule sym) (simp add: mult.assoc)
  from assms have "a = a / b * b" by simp
  also have "fps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b" by simp
  finally have "fps_deriv (a / b) * b^2 = fps_deriv a * b - a * fps_deriv b" using assms
    by (simp add: power2_eq_square algebra_simps)
  thus ?thesis by (cases "b = 0") (simp_all add: eq_divide_imp)
qed

lemma fps_nth_deriv_fps_X[simp]: "fps_nth_deriv n fps_X = (if n = 0 then fps_X else if n=1 then 1 else 0)"
  by (cases n) simp_all


subsection ‹Powers›

lemma fps_power_zeroth: "(a^n) $ 0 = (a$0)^n"
  by (induct n) auto

lemma fps_power_zeroth_eq_one: "a$0 = 1 ==> a^n $ 0 = 1"
  by (simp add: fps_power_zeroth)

lemma fps_power_first:
  fixes a :: "'a::comm_semiring_1 fps"
  shows "(a^n) $ 1 = of_nat n * (a$0)^(n-1) * a$1"
proof (cases n)
  case (Suc m)
  have "(a ^ Suc m) $ 1 = of_nat (Suc m) * (a$0)^(Suc m - 1) * a$1"
  proof (induct m)
    case (Suc k)
    hence "(a ^ Suc (Suc k)) $ 1 =
            a$0 * of_nat (Suc k) * (a $ 0)^k * a$1 + a$1 * ((a$0)^(Suc k))"
      using fps_mult_nth_1[of a] by (simp add: fps_power_zeroth[symmetric] mult.assoc)
    thus ?case by (simp add: algebra_simps)
  qed simp
  with Suc show ?thesis by simp
qed simp

lemma fps_power_first_eq: "a $ 0 = 1 ==> a^n $ 1 = of_nat n * a$1"
proof (induct n)
  case (Suc n)
  show ?case unfolding power_Suc fps_mult_nth
    using Suc.hyps[OF ‹a$0 = 1›] ‹a$0 = 1› fps_power_zeroth_eq_one[OF ‹a$0=1›]
    by (simp add: algebra_simps)
qed simp

lemma fps_power_first_eq':
  assumes "a $ 1 = 1"
  shows "a^n $ 1 = of_nat n * (a$0)^(n-1)"
proof (cases n)
  case (Suc m)
  from assms have "(a ^ Suc m) $ 1 = of_nat (Suc m) * (a$0)^(Suc m - 1)"
    using fps_mult_nth_1[of a]
    by (induct m)
          (simp_all add: algebra_simps mult_of_nat_commute fps_power_zeroth)
  with Suc show ?thesis by simp
qed simp

lemmas startsby_one_power = fps_power_zeroth_eq_one

lemma startsby_zero_power: "a $ 0 = 0 ==> n > 0 ==> a^n $0 = 0"
  by (simp add: fps_power_zeroth zero_power)

lemma startsby_power: "a $0 = v ==> a^n $0 = v^n"
  by (simp add: fps_power_zeroth)

lemma startsby_nonzero_power:
  fixes a :: "'a::semiring_1_no_zero_divisors fps"
  shows "a $ 0 ≠ 0 ==> a^n $ 0 ≠ 0"
  by (simp add: startsby_power)

lemma startsby_zero_power_iff[simp]:
  "a^n $0 = (0::'a::semiring_1_no_zero_divisors) ⟷ n ≠ 0 ∧ a$0 = 0"
proof
  show "a ^ n $ 0 = 0 ==> n ≠ 0 ∧ a $ 0 = 0"
  proof
    assume a: "a^n $ 0 = 0"
    thus "a $ 0 = 0" using startsby_nonzero_power by auto
    have "n = 0 ==> a^n $ 0 = 1" by simp
    with a show "n ≠ 0" by fastforce
  qed
  show "n ≠ 0 ∧ a $ 0 = 0 ==> a ^ n $ 0 = 0"
    by (cases n) auto
qed

lemma startsby_zero_power_prefix:
  assumes a0: "a $ 0 = 0"
  shows "∀n < k. a ^ k $ n = 0"
proof (induct k rule: nat_less_induct, clarify)
  case (1 k)
  fix j :: nat assume j: "j < k"
  show "a ^ k $ j = 0"
  proof (cases k)
    case 0 with j show ?thesis by simp
  next
    case (Suc i)
    with 1 j have "∀m∈{0<..j}. a ^ i $ (j - m) = 0" by auto
    with Suc a0 show ?thesis by (simp add: fps_mult_nth sum.atLeast_Suc_atMost)
  qed
qed

lemma startsby_zero_sum_depends:
  assumes a0: "a $0 = 0"
    and kn: "n ≥ k"
  shows "sum (λi. (a ^ i)$k) {0 .. n} = sum (λi. (a ^ i)$k) {0 .. k}"
proof (intro strip sum.mono_neutral_right)
  show "∧i. i ∈ {0..n} - {0..k} ==> a ^ i $ k = 0"
    by (simp add: a0 startsby_zero_power_prefix)
qed (use kn in auto)

lemma startsby_zero_power_nth_same:
  assumes a0: "a$0 = 0"
  shows "a^n $ n = (a$1) ^ n"
proof (induct n)
  case (Suc n)
  have "∀i∈{Suc 1..Suc n}. a ^ n $ (Suc n - i) = 0"
    using a0 startsby_zero_power_prefix[of a n] by auto
  thus ?case
    using a0 Suc sum.atLeast_Suc_atMost[of 0 "Suc n" "λi. a $ i * a ^ n $ (Suc n - i)"]
          sum.atLeast_Suc_atMost[of 1 "Suc n" "λi. a $ i * a ^ n $ (Suc n - i)"]
    by (simp add: fps_mult_nth)
qed simp

lemma fps_lr_inverse_power:
  fixes a :: "'a::ring_1 fps"
  assumes "x * a$0 = 1" "a$0 * x = 1"
  shows "fps_left_inverse (a^n) (x^n) = fps_left_inverse a x ^ n"
  and "fps_right_inverse (a^n) (x^n) = fps_right_inverse a x ^ n"
proof-

  from assms have xn: "∧n. x^n * (a^n $ 0) = 1" "∧n. (a^n $ 0) * x^n = 1"
    by (simp_all add: left_right_inverse_power fps_power_zeroth)

  show "fps_left_inverse (a^n) (x^n) = fps_left_inverse a x ^ n"
  proof (induct n)
    case 0
    then show ?case by (simp add: fps_lr_inverse_one_one(1))
  next
    case (Suc n)
    with assms show ?case
      using xn fps_lr_inverse_mult_ring1(1)[of x a "x^n" "a^n"]
      by (simp add: power_Suc2[symmetric])
  qed

  moreover have "fps_right_inverse (a^n) (x^n) = fps_left_inverse (a^n) (x^n)"
    using xn by (intro fps_left_inverse_eq_fps_right_inverse[symmetric])
  moreover have "fps_right_inverse a x = fps_left_inverse a x"
    using assms by (intro fps_left_inverse_eq_fps_right_inverse[symmetric])
  ultimately show "fps_right_inverse (a^n) (x^n) = fps_right_inverse a x ^ n"
    by simp

qed

lemma fps_inverse_power:
  fixes a :: "'a::division_ring fps"
  shows "inverse (a^n) = inverse a ^ n"
proof (cases "n=0" "a$0 = 0" rule: case_split[case_product case_split])
  case False_True
  hence LHS: "inverse (a^n) = 0" and RHS: "inverse a ^ n = 0"
    by (simp_all add: startsby_zero_power)
  show ?thesis using trans_sym[OF LHS RHS] by fast
next
  case False_False
  from False_False(2) show ?thesis
    by (simp add:
          fps_inverse_def fps_power_zeroth power_inverse fps_lr_inverse_power(2)[symmetric]
        )
qed auto

lemma fps_deriv_power':
  fixes a :: "'a::comm_semiring_1 fps"
  shows "fps_deriv (a ^ n) = (of_nat n) * fps_deriv a * a ^ (n - 1)"
proof (cases n)
  case (Suc m)
  moreover have "fps_deriv (a^Suc m) = of_nat (Suc m) * fps_deriv a * a^m"
    by (induct m) (simp_all add: algebra_simps)
  ultimately show ?thesis by simp
qed simp

lemma fps_deriv_power:
  fixes a :: "'a::comm_semiring_1 fps"
  shows "fps_deriv (a ^ n) = fps_const (of_nat n) * fps_deriv a * a ^ (n - 1)"
  by (simp add: fps_deriv_power' fps_of_nat)


subsection ‹Finite and infinite products›

lemma fps_prod_nth':
  assumes "finite A"
  shows "fps_nth (∏x∈A. f x) n = (∑X∈multisets_of_size A n. ∏x∈A. fps_nth (f x) (count X x))"
  using assms
proof (induction A arbitrary: n rule: finite_induct)
  case (insert a A n)
  note [simp] = ‹a ∉ A›
  note [intro, simp] = ‹finite A›
  have "(∑X∈multisets_of_size (insert a A) n. ∏x∈insert a A. fps_nth (f x) (count X x)) =
        (∑(m,X)∈(SIGMA m:{0..n}. multisets_of_size A (n-m)). 
           ∏x∈insert a A. fps_nth (f x) (count (X + replicate_mset m a) x))"
    by (subst sum.reindex_bij_betw[OF bij_betw_multisets_of_size_insert, symmetric])
       (simp_all add: case_prod_unfold)
  also have "… = (∑m=0..n. ∑X∈multisets_of_size A (n-m). 
                    ∏x∈insert a A. fps_nth (f x) (count (X + replicate_mset m a) x))"
    by (rule sum.Sigma [symmetric]) auto
  also have "… = (∑m=0..n. fps_nth (f a) m * fps_nth (∏x∈A. f x) (n - m))"
  proof (rule sum.cong)
    fix m
    assume m: "m ∈ {0..n}"
    have "(∑X∈multisets_of_size A (n-m). 
              ∏x∈insert a A. fps_nth (f x) (count (X + replicate_mset m a) x)) =
          (∑X∈multisets_of_size A (n-m). fps_nth (f a) (count X a + m) *
              (∏x∈A. fps_nth (f x) (count (X + replicate_mset m a) x)))"
      by simp
    also have "… = (∑X∈multisets_of_size A (n-m). fps_nth (f a) m *
                      (∏x∈A. fps_nth (f x) (count (X + replicate_mset m a) x)))"
      by (intro sum.cong arg_cong2[of _ _ _ _ "(*)"] arg_cong2[of _ _ _ _ fps_nth] refl)
         (auto simp: multisets_of_size_def simp flip: not_in_iff)
    also have "… = fps_nth (f a) m * (∑X∈multisets_of_size A (n-m).
                      (∏x∈A. fps_nth (f x) (count (X + replicate_mset m a) x)))"
      by (simp add: sum_distrib_left)
    also have "(∑X∈multisets_of_size A (n-m). ∏x∈A. fps_nth (f x) (count (X + replicate_mset m a) x)) =
               (∑X∈multisets_of_size A (n-m). ∏x∈A. fps_nth (f x) (count X x))"
      by (intro sum.cong prod.cong) auto
    also have "… = fps_nth (∏x∈A. f x) (n - m)"
      by (rule insert.IH [symmetric])
    finally show "(∑X∈multisets_of_size A (n-m). ∏x∈insert a A. fps_nth (f x) (count (X + replicate_mset m a) x)) =
                    fps_nth (f a) m * fps_nth (∏x∈A. f x) (n - m)" .
  qed auto
  also have "… = fps_nth (∏x∈insert a A. f x) n"
    by (simp add: fps_mult_nth)
  finally show ?case ..
qed auto

theorem tendsto_prod_fps:
  fixes f :: "nat ==> 'a :: {idom, t2_space} fps"
  assumes [simp]: "∧k. f k ≠ 0"
  assumes g: "∧n k. k > g n ==> subdegree (f k - 1) > n"
  defines "P ≡ Abs_fps (λn. (∑X∈multisets_of_size {..g n} n. ∏i≤g n. fps_nth (f i) (count X i)))"
  shows   "(λn. ∏k≤n. f k) <---- P"
proof (rule tendsto_fpsI)
  fix n :: nat
  show "eventually (λN. fps_nth (prod f {..N}) n = fps_nth P n) at_top"
    using eventually_ge_at_top[of "g n"]
  proof eventually_elim
    case (elim N)
    have "fps_nth (prod f {..N}) n = (∑X∈multisets_of_size {..N} n. ∏x≤N. fps_nth (f x) (count X x))"
      by (subst fps_prod_nth') auto
    also have "… = (∑X | X ∈ multisets_of_size {..N} n ∧ (∀x≤N. fps_nth (f x) (count X x) ≠ 0).
                      ∏x≤N. fps_nth (f x) (count X x))"
      by (intro sum.mono_neutral_right) auto

    also have "{X. X ∈ multisets_of_size {..N} n ∧ (∀x≤N. fps_nth (f x) (count X x) ≠ 0)} =
               {X. X ∈ multisets_of_size {..g n} n ∧ (∀x≤N. fps_nth (f x) (count X x) ≠ 0)}" 
      (is "?lhs = ?rhs")
    proof (intro equalityI subsetI)
      fix X assume "X ∈ ?rhs"
      thus "X ∈ ?lhs" using elim multisets_of_size_mono[of "{..g n}" "{..N}"] by auto
    next
      fix X assume "X ∈ ?lhs"
      hence X: "set_mset X ⊆ {..N}" "size X = n"  "∧x. x ≤ N ==> fps_nth (f x) (count X x) ≠ 0"
        by (auto simp: multisets_of_size_def)
      have "set_mset X ⊆ {..g n}"
      proof
        fix x assume *: "x ∈ set_mset X"
        show "x ∈ {..g n}"
        proof (rule ccontr)
          assume "x ∉ {..g n}"
          hence x: "x > g n" "x ≤ N"
            using X(1) * by auto
          have "count X x ≤ n"
            using X x count_le_size[of X x] by (auto simp: Pi_def)
          also have "n < subdegree (f x - 1)"
            by (rule g) (use x in auto)
          finally have "fps_nth (f x - 1) (count X x) = 0"
            by blast
          hence "fps_nth (f x) (count X x) = 0"
            using * by simp
          moreover have "fps_nth (f x) (count X x) ≠ 0"
            by (intro X(3)) (use x in auto)
          ultimately show False by contradiction
        qed
      qed
      thus "X ∈ ?rhs" using X
        by (auto simp: multisets_of_size_def)
    qed

    also have "(∑X | X ∈ multisets_of_size {..g n} n ∧ (∀x≤N. fps_nth (f x) (count X x) ≠ 0).
                      ∏x≤N. fps_nth (f x) (count X x)) =
               (∑X | X ∈ multisets_of_size {..g n} n ∧ (∀x≤N. fps_nth (f x) (count X x) ≠ 0).
                      ∏i≤g n. fps_nth (f i) (count X i))"
    proof (intro sum.cong prod.mono_neutral_right ballI)
      fix X i 
      assume i: "i ∈ {..N} - {..g n}"
      assume "X ∈ {X. X ∈ multisets_of_size {..g n} n ∧ (∀x≤N. fps_nth (f x) (count X x) ≠ 0)}"
      hence h: "X ∈ multisets_of_size {..g n} n" "∧x. x ≤ N ==> fps_nth (f x) (count X x) ≠ 0"
        by blast+
      have "i ∉# X"
        using i h unfolding multisets_of_size_def by auto
      have "n < subdegree (f i - 1)"
        by (intro g) (use i in auto)
      moreover have "count X i ≤ n"
        using ‹i ∉# X› by (simp add: not_in_iff)
      ultimately have "fps_nth (f i - 1) (count X i) = 0"
        by (intro nth_less_subdegree_zero) auto
      thus "fps_nth (f i) (count X i) = 1"
        using h(2) i ‹i ∉# X› by (auto split: if_splits)
    qed (use elim in auto)

    also have "(∑X | X ∈ multisets_of_size {..g n} n ∧ (∀x≤N. fps_nth (f x) (count X x) ≠ 0).
                   ∏i≤g n. fps_nth (f i) (count X i)) =
               (∑X ∈ multisets_of_size {..g n} n. ∏i≤g n. fps_nth (f i) (count X i))"
    proof (intro sum.mono_neutral_left ballI)
      fix X assume "X ∈ multisets_of_size {..g n} n -
                      {X∈multisets_of_size {..g n} n. ∀x≤N. fps_nth (f x) (count X x) ≠ 0}"
      then obtain i
        where h: "X ∈ multisets_of_size {..g n} n" 
        and i: "i ≤ N" "fps_nth (f i) (count X i) = 0"
        by blast
      have "¬(i > g n)"
      proof
        assume i': "i > g n"
        hence "count X i = 0"
          using h by (auto simp: multisets_of_size_def simp flip: not_in_iff)
        have "subdegree (f i - 1) > n"
          by (intro g) (use i' in auto)
        hence "subdegree (f i - 1) > 0"
          by simp
        hence "fps_nth (f i - 1) 0 = 0"
          by blast
        hence "fps_nth (f i) (count X i) = 1"
          using ‹count X i = 0› by simp
        thus False using i by simp
      qed
      thus " (∏x≤g n. fps_nth (f x) (count X x)) = 0"
        using i by auto
    qed auto

    also have "… = fps_nth P n"
      by (simp add: P_def)
    finally show "fps_nth (∏k≤N. f k) n = fps_nth P n" .
  qed
qed



subsection ‹Integration›

definition fps_integral :: "'a::{semiring_1,inverse} fps ==> 'a ==> 'a fps"
  where "fps_integral a a0 =
          Abs_fps (λn. if n=0 then a0 else inverse (of_nat n) * a$(n - 1))"

abbreviation "fps_integral0 a ≡ fps_integral a 0"

lemma fps_integral_nth_0_Suc [simp]:
  fixes a :: "'a::{semiring_1,inverse} fps"
  shows "fps_integral a a0 $ 0 = a0"
  and   "fps_integral a a0 $ Suc n = inverse (of_nat (Suc n)) * a $ n"
  by    (auto simp: fps_integral_def)

lemma fps_integral_conv_plus_const:
  "fps_integral a a0 = fps_integral a 0 + fps_const a0"
  unfolding fps_integral_def by (intro fps_ext) simp

lemma fps_deriv_fps_integral:
  fixes a :: "'a::{division_ring,ring_char_0} fps"
  shows "fps_deriv (fps_integral a a0) = a"
proof (intro fps_ext)
  fix n
  have "(of_nat (Suc n) :: 'a) ≠ 0" by (rule of_nat_neq_0)
  hence "of_nat (Suc n) * inverse (of_nat (Suc n) :: 'a) = 1" by simp
  moreover have
    "fps_deriv (fps_integral a a0) $ n = of_nat (Suc n) * inverse (of_nat (Suc n)) * a $ n"
    by (simp add: mult.assoc)
  ultimately show "fps_deriv (fps_integral a a0) $ n = a $ n" by simp
qed

lemma fps_integral0_deriv:
  fixes a :: "'a::{division_ring,ring_char_0} fps"
  shows "fps_integral0 (fps_deriv a) = a - fps_const (a$0)"
proof (intro fps_ext)
  fix n
  show "fps_integral0 (fps_deriv a) $ n = (a - fps_const (a$0)) $ n"
  proof (cases n)
    case (Suc m)
    have "(of_nat (Suc m) :: 'a) ≠ 0" by (rule of_nat_neq_0)
    hence "inverse (of_nat (Suc m) :: 'a) * of_nat (Suc m) = 1" by simp
    moreover have
      "fps_integral0 (fps_deriv a) $ Suc m =
        inverse (of_nat (Suc m)) * of_nat (Suc m) * a $ (Suc m)"
      by (simp add: mult.assoc)
    ultimately show ?thesis using Suc by simp
  qed simp
qed

lemma fps_integral_deriv:
  fixes a :: "'a::{division_ring,ring_char_0} fps"
  shows "fps_integral (fps_deriv a) (a$0) = a"
  using fps_integral_conv_plus_const[of "fps_deriv a" "a$0"]
  by    (simp add: fps_integral0_deriv)

lemma fps_integral0_zero:
  "fps_integral0 (0::'a::{semiring_1,inverse} fps) = 0"
  by (intro fps_ext) (simp add: fps_integral_def)

lemma fps_integral0_fps_const':
  fixes   c :: "'a::{semiring_1,inverse}"
  assumes "inverse (1::'a) = 1"
  shows   "fps_integral0 (fps_const c) = fps_const c * fps_X"
proof (intro fps_ext)
  fix n
  show "fps_integral0 (fps_const c) $ n = (fps_const c * fps_X) $ n"
    by (cases n) (simp_all add: assms mult_delta_right)
qed

lemma fps_integral0_fps_const:
  fixes c :: "'a::division_ring"
  shows "fps_integral0 (fps_const c) = fps_const c * fps_X"
  by    (rule fps_integral0_fps_const'[OF inverse_1])

lemma fps_integral0_one':
  assumes "inverse (1::'a::{semiring_1,inverse}) = 1"
  shows   "fps_integral0 (1::'a fps) = fps_X"
  using   assms fps_integral0_fps_const'[of "1::'a"]
  by      simp

lemma fps_integral0_one:
  "fps_integral0 (1::'a::division_ring fps) = fps_X"
  by (rule fps_integral0_one'[OF inverse_1])

lemma fps_integral0_fps_const_mult_left:
  fixes a :: "'a::division_ring fps"
  shows "fps_integral0 (fps_const c * a) = fps_const c * fps_integral0 a"
proof (intro fps_ext)
  fix n
  show "fps_integral0 (fps_const c * a) $ n = (fps_const c * fps_integral0 a) $ n"
    using mult_inverse_of_nat_commute[of n c, symmetric]
          mult.assoc[of "inverse (of_nat n)" c "a$(n-1)"]
          mult.assoc[of c "inverse (of_nat n)" "a$(n-1)"]
    by    (simp add: fps_integral_def)
qed

lemma fps_integral0_fps_const_mult_right:
  fixes a :: "'a::{semiring_1,inverse} fps"
  shows "fps_integral0 (a * fps_const c) = fps_integral0 a * fps_const c"
  by    (intro fps_ext) (simp add: fps_integral_def algebra_simps)

lemma fps_integral0_neg:
  fixes a :: "'a::{ring_1,inverse} fps"
  shows "fps_integral0 (-a) = - fps_integral0 a"
  using fps_integral0_fps_const_mult_right[of a "-1"]
  by    (simp add: fps_const_neg[symmetric])

lemma fps_integral0_add:
  "fps_integral0 (a+b) = fps_integral0 a + fps_integral0 b"
  by (intro fps_ext) (simp add: fps_integral_def algebra_simps)

lemma fps_integral0_linear:
  fixes a b :: "'a::division_ring"
  shows "fps_integral0 (fps_const a * f + fps_const b * g) =
          fps_const a * fps_integral0 f + fps_const b * fps_integral0 g"
  by    (simp add: fps_integral0_add fps_integral0_fps_const_mult_left)
  
lemma fps_integral0_linear2:
  "fps_integral0 (f * fps_const a + g * fps_const b) =
    fps_integral0 f * fps_const a + fps_integral0 g * fps_const b"
  by (simp add: fps_integral0_add fps_integral0_fps_const_mult_right)

lemma fps_integral_linear:
  fixes a b a0 b0 :: "'a::division_ring"
  shows
  "fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) =
    fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0"
  using fps_integral_conv_plus_const[of
          "fps_const a * f + fps_const b * g"
          "a*a0 + b*b0"
        ]
        fps_integral_conv_plus_const[of f a0] fps_integral_conv_plus_const[of g b0]
  by    (simp add: fps_integral0_linear algebra_simps)

lemma fps_integral0_sub:
  fixes a b :: "'a::{ring_1,inverse} fps"
  shows "fps_integral0 (a-b) = fps_integral0 a - fps_integral0 b"
  using fps_integral0_linear2[of a 1 b "-1"]
  by    (simp add: fps_const_neg[symmetric])

lemma fps_integral0_of_nat:
  "fps_integral0 (of_nat n :: 'a::division_ring fps) = of_nat n * fps_X"
  using fps_integral0_fps_const[of "of_nat n :: 'a"] by (simp add: fps_of_nat)

lemma fps_integral0_sum:
  "fps_integral0 (sum f S) = sum (λi. fps_integral0 (f i)) S"
proof (cases "finite S")
  case True show ?thesis
    by  (induct rule: finite_induct [OF True])
        (simp_all add: fps_integral0_zero fps_integral0_add)
qed (simp add: fps_integral0_zero)

lemma fps_integral0_by_parts:
  fixes a b :: "'a::{division_ring,ring_char_0} fps"
  shows
    "fps_integral0 (a * b) =
      a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)"
proof-
  have "fps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b"
    using fps_integral0_deriv[of "(a * fps_integral0 b)"] by simp
  moreover have
    "fps_integral0 (a * b) =
      fps_integral0 (fps_deriv (a * fps_integral0 b)) -
      fps_integral0 (fps_deriv a * fps_integral0 b)"
    by (auto simp: fps_deriv_fps_integral fps_integral0_sub[symmetric])
  ultimately show ?thesis by simp
qed

lemma fps_integral0_fps_X:
  "fps_integral0 (fps_X::'a::{semiring_1,inverse} fps) =
    fps_const (inverse (of_nat 2)) * fps_X🪙2"
  by (intro fps_ext) (auto simp: fps_integral_def)

lemma fps_integral0_fps_X_power:
  "fps_integral0 ((fps_X::'a::{semiring_1,inverse} fps) ^ n) =
            fps_const (inverse (of_nat (Suc n))) * fps_X ^ Suc n"
proof (intro fps_ext)
  fix k show
    "fps_integral0 ((fps_X::'a fps) ^ n) $ k =
      (fps_const (inverse (of_nat (Suc n))) * fps_X ^ Suc n) $ k"
    by (cases k) simp_all
qed


subsection ‹Composition›

definition fps_compose :: "'a::semiring_1 fps ==> 'a fps ==> 'a fps"  (infixl ‹oo› 55)
  where "a oo b = Abs_fps (λn. sum (λi. a$i * (b^i$n)) {0..n})"

lemma fps_compose_nth: "(a oo b)$n = sum (λi. a$i * (b^i$n)) {0..n}"
  by (simp add: fps_compose_def)

lemma fps_compose_nth_0 [simp]: "(f oo g) $ 0 = f $ 0"
  by (simp add: fps_compose_nth)

lemma fps_compose_fps_X[simp]: "a oo fps_X = (a :: 'a::comm_ring_1 fps)"
  by (simp add: fps_ext fps_compose_def mult_delta_right)

lemma fps_const_compose[simp]: "fps_const (a::'a::comm_ring_1) oo b = fps_const a"
  by (simp add: fps_eq_iff fps_compose_nth mult_delta_left)

lemma numeral_compose[simp]: "(numeral k :: 'a::comm_ring_1 fps) oo b = numeral k"
  unfolding numeral_fps_const by simp

lemma neg_numeral_compose[simp]: "(- numeral k :: 'a::comm_ring_1 fps) oo b = - numeral k"
  unfolding neg_numeral_fps_const by simp

lemma fps_X_fps_compose_startby0[simp]: "a$0 = 0 ==> fps_X oo a = (a :: 'a::comm_ring_1 fps)"
  by (simp add: fps_eq_iff fps_compose_def mult_delta_left not_le)


subsection ‹Rules from Herbert Wilf's Generatingfunctionology›

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

lemma fps_power_mult_eq_shift:
  "fps_X^Suc k * Abs_fps (λn. a (n + Suc k)) =
    Abs_fps a - sum (λi. fps_const (a i :: 'a::comm_ring_1) * fps_X^i) {0 .. k}"
  (is "?lhs = ?rhs")
proof -
  have "?lhs $ n = ?rhs $ n" for n :: nat
  proof -
    have "?lhs $ n = (if n < Suc k then 0 else a n)"
      unfolding fps_X_power_mult_nth by auto
    also have "… = ?rhs $ n"
    proof (induct k)
      case 0
      then show ?case
        by (simp add: fps_sum_nth)
    next
      case (Suc k)
      have "(Abs_fps a - sum (λi. fps_const (a i :: 'a) * fps_X^i) {0 .. Suc k})$n =
        (Abs_fps a - sum (λi. fps_const (a i :: 'a) * fps_X^i) {0 .. k} -
          fps_const (a (Suc k)) * fps_X^ Suc k) $ n"
        by (simp add: field_simps)
      also have "… = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * fps_X^ Suc k)$n"
        using Suc.hyps[symmetric] unfolding fps_sub_nth by simp
      also have "… = (if n < Suc (Suc k) then 0 else a n)"
        unfolding fps_X_power_mult_right_nth
        by (simp add: not_less le_less_Suc_eq)
      finally show ?case
        by simp
    qed
    finally show ?thesis .
  qed
  then show ?thesis
    by (simp add: fps_eq_iff)
qed


subsubsection ‹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 "fps_XD = (*) fps_X ∘ fps_deriv"

lemma fps_XD_add[simp]:"fps_XD (a + b) = fps_XD a + fps_XD (b :: 'a::comm_ring_1 fps)"
  by (simp add: fps_XD_def field_simps)

lemma fps_XD_mult_const[simp]:"fps_XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * fps_XD a"
  by (simp add: fps_XD_def field_simps)

lemma fps_XD_linear[simp]: "fps_XD (fps_const c * a + fps_const d * b) =
    fps_const c * fps_XD a + fps_const d * fps_XD (b :: 'a::comm_ring_1 fps)"
  by simp

lemma fps_XDN_linear:
  "(fps_XD ^^ n) (fps_const c * a + fps_const d * b) =
    fps_const c * (fps_XD ^^ n) a + fps_const d * (fps_XD ^^ n) (b :: 'a::comm_ring_1 fps)"
  by (induct n) simp_all

lemma fps_mult_fps_X_deriv_shift: "fps_X* fps_deriv a = Abs_fps (λn. of_nat n* a$n)"
  by (simp add: fps_eq_iff)

lemma fps_mult_fps_XD_shift:
  "(fps_XD ^^ k) (a :: 'a::comm_ring_1 fps) = Abs_fps (λn. (of_nat n ^ k) * a$n)"
  by (induct k arbitrary: a) (simp_all add: fps_XD_def fps_eq_iff field_simps del: One_nat_def)


subsubsection ‹Rule 3›

text ‹Rule 3 is trivial and is given by \texttt{fps\_times\_def}.›


subsubsection ‹Rule 5 --- summation and ``division'' by $1 - X$›

lemma fps_divide_fps_X_minus1_sum_lemma:
  "a = ((1::'a::ring_1 fps) - fps_X) * Abs_fps (λn. sum (λi. a $ i) {0..n})"
proof (rule fps_ext)
  define f g :: "'a fps"
    where "f ≡ 1 - fps_X"
    and   "g ≡ Abs_fps (λn. sum (λi. a $ i) {0..n})"
  fix n show "a $ n= (f * g) $ n"
  proof (cases n)
    case (Suc m)
    hence "(f * g) $ n = g $ Suc m - g $ m"
      using fps_mult_nth[of f g "Suc m"]
            sum.atLeast_Suc_atMost[of 0 "Suc m" "λi. f $ i * g $ (Suc m - i)"]
            sum.atLeast_Suc_atMost[of 1 "Suc m" "λi. f $ i * g $ (Suc m - i)"]
      by    (simp add: f_def)
    with Suc show ?thesis by (simp add: g_def)
  qed (simp add: f_def g_def)
qed

lemma fps_divide_fps_X_minus1_sum_ring1:
  assumes "inverse 1 = (1::'a::{ring_1,inverse})"
  shows   "a /((1::'a fps) - fps_X) = Abs_fps (λn. sum (λi. a $ i) {0..n})"
proof-
  from assms have "a /((1::'a fps) - fps_X) = a * Abs_fps (λn. 1)"
    by (simp add: fps_divide_def fps_inverse_def fps_lr_inverse_one_minus_fps_X(2))
  thus ?thesis by (auto intro: fps_ext simp: fps_mult_nth)
qed

lemma fps_divide_fps_X_minus1_sum:
  "a /((1::'a::division_ring fps) - fps_X) = Abs_fps (λn. sum (λi. a $ i) {0..n})"
  using fps_divide_fps_X_minus1_sum_ring1[of a] by simp


subsubsection ‹Rule 4 in its more general form›

text ‹This generalizes Rule 3 for an arbitrary
  finite product of FPS, also the relevant instance of powers of a FPS.›

definition "natpermute n k = {l :: nat list. length l = k ∧ sum_list l = n}"

lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
proof -
  have "[length xs = 1; n = sum_list xs] ==> xs = [sum_list xs]" for xs
    by (cases xs) auto
  then show ?thesis
    by (auto simp add: natpermute_def)
qed

lemma natlist_trivial_Suc0 [simp]: "natpermute n (Suc 0) = {[n]}"
  using natlist_trivial_1 by force

lemma append_natpermute_less_eq:
  assumes "xs @ ys ∈ natpermute n k"
  shows "sum_list xs ≤ n"
    and "sum_list ys ≤ n"
proof -
  from assms have "sum_list (xs @ ys) = n"
    by (simp add: natpermute_def)
  then have "sum_list xs + sum_list ys = n"
    by simp
  then show "sum_list xs ≤ n" and "sum_list ys ≤ n"
    by simp_all
qed

lemma natpermute_split:
  assumes "h ≤ k"
  shows "natpermute n k =
    (∪m ∈{0..n}. {l1 @ l2 |l1 l2. l1 ∈ natpermute m h ∧ l2 ∈ natpermute (n - m) (k - h)})"
  (is "?L = ?R" is "_ = (∪m ∈{0..n}. ?S m)")
proof
  show "?R ⊆ ?L"
  proof
    fix l
    assume l: "l ∈ ?R"
    from l obtain m xs ys where h: "m ∈ {0..n}"
      and xs: "xs ∈ natpermute m h"
      and ys: "ys ∈ natpermute (n - m) (k - h)"
      and leq: "l = xs@ys" by blast
    from xs have xs': "sum_list xs = m"
      by (simp add: natpermute_def)
    from ys have ys': "sum_list ys = n - m"
      by (simp add: natpermute_def)
    show "l ∈ ?L" using leq xs ys h
      using assms by (force simp add: natpermute_def)
  qed
  show "?L ⊆ ?R"
  proof
    fix l
    assume l: "l ∈ natpermute n k"
    let ?xs = "take h l"
    let ?ys = "drop h l"
    let ?m = "sum_list ?xs"
    from l have ls: "sum_list (?xs @ ?ys) = n"
      by (simp add: natpermute_def)
    have xs: "?xs ∈ natpermute ?m h" using l assms
      by (simp add: natpermute_def)
    have l_take_drop: "sum_list l = sum_list (take h l @ drop h l)"
      by simp
    then have ys: "?ys ∈ natpermute (n - ?m) (k - h)"
      using l assms ls by (auto simp add: natpermute_def simp del: append_take_drop_id)
    from ls have m: "?m ∈ {0..n}"
      by (simp add: l_take_drop del: append_take_drop_id)
    have "sum_list (take h l) ≤ sum_list l"
      using l_take_drop ls m by presburger
    with xs ys ls l show "l ∈ ?R"
      by simp (metis append_take_drop_id m)
  qed
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})"
  by (auto simp add: set_replicate_conv_if natpermute_def replicate_length_same)

lemma natpermute_finite: "finite (natpermute n k)"
proof (induct k arbitrary: n)
  case 0
  then show ?case
    by (simp add: natpermute_0)
next
  case (Suc k)
  then show ?case
    using natpermute_split [of k "Suc k"] finite_UN_I by simp
qed

lemma natpermute_contain_maximal:
  "{xs ∈ natpermute n (k + 1). n ∈ set xs} = (∪i∈{0 .. k}. {(replicate (k + 1) 0) [i:=n]})"
  (is "?A = ?B")
proof
  show "?A ⊆ ?B"
  proof
    fix xs
    assume "xs ∈ ?A"
    then have H: "xs ∈ natpermute n (k + 1)" and n: "n ∈ set xs"
      by blast+
    then obtain i where i: "i ∈ {0.. k}" "xs!i = n"
      unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
    have eqs: "({0..k} - {i}) ∪ {i} = {0..k}"
      using i by auto
    have f: "finite({0..k} - {i})" "finite {i}"
      by auto
    have d: "({0..k} - {i}) ∩ {i} = {}"
      using i by auto
    from H have "n = sum (nth xs) {0..k}"
      by (auto simp add: natpermute_def atLeastLessThanSuc_atLeastAtMost sum_list_sum_nth)
    also have "… = n + sum (nth xs) ({0..k} - {i})"
      unfolding sum.union_disjoint[OF f d, unfolded eqs] using i by simp
    finally have zxs: "∀ j∈ {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 presburger+
    have "xs = (replicate (k+1) 0) [i := n]"
    proof (rule nth_equalityI)
      show "length xs = length ((replicate (k + 1) 0)[i := n])"
        by (metis length_list_update length_replicate xsl)
      show "xs ! j = (replicate (k + 1) 0)[i := n] ! j" if "j < length xs" for j
      proof (cases "j = i")
        case True
        then show ?thesis
          by (metis i'(1) i(2) nth_list_update)
      next
        case False
        with that show ?thesis
          by (simp add: xsl zxs del: replicate.simps split: nat.split)
      qed
    qed
    then show "xs ∈ ?B" using i by blast
  qed
  show "?B ⊆ ?A"
  proof
    fix xs
    assume "xs ∈ ?B"
    then obtain i where i: "i ∈ {0..k}" and xs: "xs = (replicate (k + 1) 0) [i:=n]"
      by auto
    have nxs: "n ∈ set xs"
      unfolding xs using set_update_memI i 
      by (metis Suc_eq_plus1 atLeast0AtMost atMost_iff le_simps(2) length_replicate)
    have xsl: "length xs = k + 1"
      by (simp only: xs length_replicate length_list_update)
    have "sum_list xs = sum (nth xs) {0..
      unfolding sum_list_sum_nth xsl ..
    also have "… = sum (λj. if j = i then n else 0) {0..< k+1}"
      by (rule sum.cong) (simp_all add: xs del: replicate.simps)
    also have "… = n" using i by simp
    finally have "xs ∈ natpermute n (k + 1)"
      using xsl unfolding natpermute_def mem_Collect_eq by blast
    then show "xs ∈ ?A"
      using nxs by blast
  qed
qed

text ‹The general form.›
lemma fps_prod_nth:
  fixes m :: nat
    and a :: "nat ==> 'a::comm_ring_1 fps"
  shows "(prod a {0 .. m}) $ n =
    sum (λv. prod (λj. (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: "∀m' < m. ∀n. ?P m' n"
  show "?P m n"
  proof (cases m)
    case 0
    then show ?thesis
      by simp
  next
    case (Suc k)
    then have km: "k < m" by arith
    have u0: "{0 .. k} ∪ {m} = {0..m}"
      using Suc by (simp add: set_eq_iff) presburger
    have f0: "finite {0 .. k}" "finite {m}" by auto
    have d0: "{0 .. k} ∩ {m} = {}" using Suc by auto
    have "(prod a {0 .. m}) $ n = (prod a {0 .. k} * a m) $ n"
      unfolding prod.union_disjoint[OF f0 d0, unfolded u0] by simp
    also have "… = (∑i = 0..n. (∑v∈natpermute i (k + 1).
            (∏j = 0..k. a j $ v ! j) * a m $ (n - i)))"
      unfolding fps_mult_nth H[rule_format, OF km] sum_distrib_right ..
    also have "... = (∑i = 0..n.
                       ∑v∈(λl1. l1 @ [n - i]) ` natpermute i (Suc k).
                        (∏j = 0..k. a j $ v ! j) * a (Suc k) $ v ! Suc k)"
      by (intro sum.cong [OF refl sym] sum.reindex_cong) (auto simp: inj_on_def natpermute_def nth_append Suc)
    also have "... = (∑v∈(∪x∈{0..n}. {l1 @ [n - x] |l1. l1 ∈ natpermute x (Suc k)}).
                      (∏j = 0..k. a j $ v ! j) * a (Suc k) $ v ! Suc k)"
      by (subst sum.UNION_disjoint) (auto simp add: natpermute_finite setcompr_eq_image)
    also have "… = (∑v∈natpermute n (m + 1). ∏j∈{0..m}. a j $ v ! j)"
      using natpermute_split[of m "m + 1"] by (simp add: Suc)
    finally show ?thesis .
  qed
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 = sum (λv. prod (λj. a $ (v!j)) {0..m}) (natpermute n (m+1))"
proof -
  have th0: "a^Suc m = prod (λi. a) {0..m}"
    by (simp add: prod_constant)
  show ?thesis unfolding th0 fps_prod_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 sum (λv. prod (λj. a $ (v!j)) {0..m - 1}) (natpermute n m))"
  by (cases m) (simp_all add: fps_power_nth_Suc del: power_Suc)

lemmas fps_nth_power_0 = fps_power_zeroth

lemma natpermute_max_card:
  assumes n0: "n ≠ 0"
  shows "card {xs ∈ natpermute n (k + 1). n ∈ set xs} = k + 1"
  unfolding natpermute_contain_maximal
proof -
  let ?A = "λi. {(replicate (k + 1) 0)[i := n]}"
  let ?K = "{0 ..k}"
  have fK: "finite ?K"
    by simp
  have fAK: "∀i∈?K. finite (?A i)"
    by auto
  have d: "∀i∈ ?K. ∀j∈ ?K. i ≠ j ⟶
    {(replicate (k + 1) 0)[i := n]} ∩ {(replicate (k + 1) 0)[j := n]} = {}"
  proof clarify
    fix i j
    assume i: "i ∈ ?K" and j: "j ∈ ?K" and ij: "i ≠ j"
    have False if eq: "(replicate (k+1) 0)[i:=n] = (replicate (k+1) 0)[j:= n]"
    proof -
      have "(replicate (k+1) 0) [i:=n] ! i = n"
        using i by (simp del: replicate.simps)
      moreover
      have "(replicate (k+1) 0) [j:=n] ! i = 0"
        using i ij by (simp del: replicate.simps)
      ultimately show ?thesis
        using eq n0 by (simp del: replicate.simps)
    qed
    then show "{(replicate (k + 1) 0)[i := n]} ∩ {(replicate (k + 1) 0)[j := n]} = {}"
      by auto
  qed
  from card_UN_disjoint[OF fK fAK d]
  show "card (∪i∈{0..k}. {(replicate (k + 1) 0)[i := n]}) = k + 1"
    by simp
qed

lemma fps_power_Suc_nth:
  fixes f :: "'a :: comm_ring_1 fps"
  assumes k: "k > 0"
  shows "(f ^ Suc m) $ k =
           of_nat (Suc m) * (f $ k * (f $ 0) ^ m) +
           (∑v∈{v∈natpermute k (m+1). k ∉ set v}. ∏j = 0..m. f $ v ! j)"
proof -
  define A B 
    where "A = {v∈natpermute k (m+1). k ∈ set v}" 
      and  "B = {v∈natpermute k (m+1). k ∉ set v}"
  have [simp]: "finite A" "finite B" "A ∩ B = {}" by (auto simp: A_def B_def natpermute_finite)

  from natpermute_max_card[of k m] k have card_A: "card A = m + 1" by (simp add: A_def)
  {
    fix v assume v: "v ∈ A"
    from v have [simp]: "length v = Suc m" by (simp add: A_def natpermute_def)
    from v have "∃j. j ≤ m ∧ v ! j = k" 
      by (auto simp: set_conv_nth A_def natpermute_def less_Suc_eq_le)
    then obtain j where j: "j ≤ m" "v ! j = k" by auto
    
    from v have "k = sum_list v" by (simp add: A_def natpermute_def)
    also have "… = (∑i=0..m. v ! i)"
      by (simp add: sum_list_sum_nth atLeastLessThanSuc_atLeastAtMost del: sum.op_ivl_Suc)
    also from j have "{0..m} = insert j ({0..m}-{j})" by auto
    also from j have "(∑i∈…. v ! i) = k + (∑i∈{0..m}-{j}. v ! i)"
      by (subst sum.insert) simp_all
    finally have "(∑i∈{0..m}-{j}. v ! i) = 0" by simp
    hence zero: "v ! i = 0" if "i ∈ {0..m}-{j}" for i using that
      by (subst (asm) sum_eq_0_iff) auto
      
    from j have "{0..m} = insert j ({0..m} - {j})" by auto
    also from j have "(∏i∈…. f $ (v ! i)) = f $ k * (∏i∈{0..m} - {j}. f $ (v ! i))"
      by (subst prod.insert) auto
    also have "(∏i∈{0..m} - {j}. f $ (v ! i)) = (∏i∈{0..m} - {j}. f $ 0)"
      by (intro prod.cong) (simp_all add: zero)
    also from j have "… = (f $ 0) ^ m" by (subst prod_constant) simp_all
    finally have "(∏j = 0..m. f $ (v ! j)) = f $ k * (f $ 0) ^ m" .
  } note A = this
  
  have "(f ^ Suc m) $ k = (∑v∈natpermute k (m + 1). ∏j = 0..m. f $ v ! j)"
    by (rule fps_power_nth_Suc)
  also have "natpermute k (m+1) = A ∪ B" unfolding A_def B_def by blast
  also have "(∑v∈…. ∏j = 0..m. f $ (v ! j)) =
               (∑v∈A. ∏j = 0..m. f $ (v ! j)) + (∑v∈B. ∏j = 0..m. f $ (v ! j))"
    by (intro sum.union_disjoint) simp_all   
  also have "(∑v∈A. ∏j = 0..m. f $ (v ! j)) = of_nat (Suc m) * (f $ k * (f $ 0) ^ m)"
    by (simp add: A card_A)
  finally show ?thesis by (simp add: B_def)
qed 
  
lemma fps_power_Suc_eqD:
  fixes f g :: "'a :: {idom,semiring_char_0} fps"
  assumes "f ^ Suc m = g ^ Suc m" "f $ 0 = g $ 0" "f $ 0 ≠ 0"
  shows   "f = g"
proof (rule fps_ext)
  fix k :: nat
  show "f $ k = g $ k"
  proof (induction k rule: less_induct)
    case (less k)
    show ?case
    proof (cases "k = 0")
      case False
      let ?h = "λf. (∑v | v ∈ natpermute k (m + 1) ∧ k ∉ set v. ∏j = 0..m. f $ v ! j)"
      from False fps_power_Suc_nth[of k f m] fps_power_Suc_nth[of k g m]
        have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h f =
                g $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h g" using assms 
        by (simp add: mult_ac del: power_Suc of_nat_Suc)
      also have "v ! i < k" if "v ∈ {v∈natpermute k (m+1). k ∉ set v}" "i ≤ m" for v i
        using that elem_le_sum_list[of i v] unfolding natpermute_def
        by (auto simp: set_conv_nth dest!: spec[of _ i])
      hence "?h f = ?h g"
        by (intro sum.cong refl prod.cong less lessI) (simp add: natpermute_def)
      finally have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) = g $ k * (of_nat (Suc m) * (f $ 0) ^ m)"
        by simp
      with assms show "f $ k = g $ k" 
        by (subst (asm) mult_right_cancel) (auto simp del: of_nat_Suc)
    qed (simp_all add: assms)
  qed
qed

lemma fps_power_Suc_eqD':
  fixes f g :: "'a :: {idom,semiring_char_0} fps"
  assumes "f ^ Suc m = g ^ Suc m" "f $ subdegree f = g $ subdegree g"
  shows   "f = g"
proof (cases "f = 0")
  case False
  have "Suc m * subdegree f = subdegree (f ^ Suc m)"
    by (rule subdegree_power [symmetric])
  also have "f ^ Suc m = g ^ Suc m" by fact
  also have "subdegree … = Suc m * subdegree g" by (rule subdegree_power)
  finally have [simp]: "subdegree f = subdegree g"
    by (subst (asm) Suc_mult_cancel1)
  have "fps_shift (subdegree f) f * fps_X ^ subdegree f = f"
    by (rule subdegree_decompose [symmetric])
  also have "… ^ Suc m = g ^ Suc m" by fact
  also have "g = fps_shift (subdegree g) g * fps_X ^ subdegree g"
    by (rule subdegree_decompose)
  also have "subdegree f = subdegree g" by fact
  finally have "fps_shift (subdegree g) f ^ Suc m = fps_shift (subdegree g) g ^ Suc m"
    by (simp add: algebra_simps power_mult_distrib del: power_Suc)
  hence "fps_shift (subdegree g) f = fps_shift (subdegree g) g"
    by (rule fps_power_Suc_eqD) (insert assms False, auto)
  with subdegree_decompose[of f] subdegree_decompose[of g] show ?thesis by simp
qed (insert assms, simp_all)

lemma fps_power_eqD':
  fixes f g :: "'a :: {idom,semiring_char_0} fps"
  assumes "f ^ m = g ^ m" "f $ subdegree f = g $ subdegree g" "m > 0"
  shows   "f = g"
  using fps_power_Suc_eqD'[of f "m-1" g] assms by simp

lemma fps_power_eqD:
  fixes f g :: "'a :: {idom,semiring_char_0} fps"
  assumes "f ^ m = g ^ m" "f $ 0 = g $ 0" "f $ 0 ≠ 0" "m > 0"
  shows   "f = g"
  by (rule fps_power_eqD'[of f m g]) (insert assms, simp_all)

lemma fps_compose_inj_right:
  assumes a0: "a$0 = (0::'a::idom)"
    and a1: "a$1 ≠ 0"
  shows "(b oo a = c oo a) ⟷ b = c"
  (is "?lhs ⟷?rhs")
proof
  show ?lhs if ?rhs using that by simp
  show ?rhs if ?lhs
  proof -
    have "b$n = c$n" for n
    proof (induct n rule: nat_less_induct)
      fix n
      assume H: "∀m
      show "b$n = c$n"
      proof (cases n)
        case 0
        from ‹?lhs› have "(b oo a)$n = (c oo a)$n"
          by simp
        then show ?thesis
          using 0 by (simp add: fps_compose_nth)
      next
        case (Suc n1)
        have f: "finite {0 .. n1}" "finite {n}" by simp_all
        have eq: "{0 .. n1} ∪ {n} = {0 .. n}" using Suc by auto
        have d: "{0 .. n1} ∩ {n} = {}" using Suc by auto
        have seq: "(∑i = 0..n1. b $ i * a ^ i $ n) = (∑i = 0..n1. c $ i * a ^ i $ n)"
          using H Suc by auto
        have th0: "(b oo a) $n = (∑i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n"
          unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq] seq
          using startsby_zero_power_nth_same[OF a0]
          by simp
        have th1: "(c oo a) $n = (∑i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n"
          unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq]
          using startsby_zero_power_nth_same[OF a0]
          by simp
        from ‹?lhs›[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
        show ?thesis by auto
      qed
    qed
    then show ?rhs by (simp add: fps_eq_iff)
  qed
qed


subsection ‹Radicals›

declare prod.cong [fundef_cong]

function radical :: "(nat ==> 'a ==> 'a) ==> nat ==> 'a::field fps ==> nat ==> 'a"
where
  "radical r 0 a 0 = 1"
| "radical r 0 a (Suc n) = 0"
| "radical r (Suc k) a 0 = r (Suc k) (a$0)"
| "radical r (Suc k) a (Suc n) =
    (a$ Suc n - sum (λxs. prod (λj. radical r (Suc k) a (xs ! j)) {0..k})
      {xs. xs ∈ natpermute (Suc n) (Suc k) ∧ Suc n ∉ set xs}) /
    (of_nat (Suc k) * (radical r (Suc k) a 0)^k)"
  by pat_completeness auto

termination radical
proof
  let ?R = "measure (λ(r, k, a, n). n)"
  {
    show "wf ?R" by auto
  next
    fix r :: "nat ==> 'a ==> 'a"
    and a :: "'a fps"
    and k n xs i
    assume xs: "xs ∈ {xs ∈ natpermute (Suc n) (Suc k). Suc n ∉ set xs}" and i: "i ∈ {0..k}"
    have False if c: "Suc n ≤ xs ! i"
    proof -
      from xs i have "xs !i ≠ Suc n"
        by (simp add: in_set_conv_nth natpermute_def)
      with c have c': "Suc n < xs!i" by arith
      have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..
        by simp_all
      have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}"
        by auto
      have eqs: "{0..∪ ({i} ∪ {i+1 ..< Suc k})"
        using i by auto
      from xs have "Suc n = sum_list xs"
        by (simp add: natpermute_def)
      also have "… = sum (nth xs) {0.. using xs
        by (simp add: natpermute_def sum_list_sum_nth)
      also have "… = xs!i + sum (nth xs) {0..
        unfolding eqs  sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
        unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)]
        by simp
      finally show ?thesis using c' by simp
    qed
    then show "((r, Suc k, a, xs!i), r, Suc k, a, Suc n) ∈ ?R"
      using not_less by auto
  next
    fix r :: "nat ==> 'a ==> 'a"
    and a :: "'a fps"
    and k n
    show "((r, Suc k, a, 0), r, Suc k, a, Suc n) ∈ ?R" by simp
  }
qed

definition "fps_radical r n a = Abs_fps (radical r n a)"

lemma radical_0 [simp]: "∧n. 0 < n ==> radical r 0 a n = 0"
  using radical.elims by blast

lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
  by (auto simp add: fps_eq_iff fps_radical_def)

lemma fps_radical_nth_0[simp]: "fps_radical r n a $ 0 = (if n = 0 then 1 else r n (a$0))"
  by (cases n) (simp_all add: fps_radical_def)

lemma fps_radical_power_nth[simp]:
  assumes r: "(r k (a$0)) ^ k = a$0"
  shows "fps_radical r k a ^ k $ 0 = (if k = 0 then 1 else a$0)"
proof (cases k)
  case 0
  then show ?thesis by simp
next
  case (Suc h)
  have eq1: "fps_radical r k a ^ k $ 0 = (∏j∈{0..h}. fps_radical r k a $ (replicate k 0) ! j)"
    unfolding fps_power_nth Suc by simp
  also have "… = (∏j∈{0..h}. r k (a$0))"
  proof (rule prod.cong [OF refl])
    show "fps_radical r k a $ replicate k 0 ! j = r k (a $ 0)" if "j ∈ {0..h}" for j
    proof -
      have "j < Suc h"
        using that by presburger
      then show ?thesis
        by (metis Suc fps_radical_nth_0 nth_replicate old.nat.distinct(2))
    qed
  qed
  also have "… = a$0"
    using r Suc by simp
  finally show ?thesis
    using Suc by simp
qed

lemma power_radical:
  fixes a:: "'a::field_char_0 fps"
  assumes a0: "a$0 ≠ 0"
  shows "(r (Suc k) (a$0)) ^ Suc k = a$0 ⟷ (fps_radical r (Suc k) a) ^ (Suc k) = a"
    (is "?lhs ⟷ ?rhs")
proof
  let ?r = "fps_radical r (Suc k) a"
  show ?rhs if r0: ?lhs
  proof -
    from a0 r0 have r00: "r (Suc k) (a$0) ≠ 0" by auto
    have "?r ^ Suc k $ z = a$z" for z
    proof (induct z rule: nat_less_induct)
      fix n
      assume H: "∀m
      show "?r ^ Suc k $ n = a $n"
      proof (cases n)
        case 0
        then show ?thesis
          using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp
      next
        case (Suc n1)
        then have "n ≠ 0" by simp
        let ?Pnk = "natpermute n (k + 1)"
        let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}"
        let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}"
        have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast
        have d: "?Pnkn ∩ ?Pnknn = {}" by blast
        have f: "finite ?Pnkn" "finite ?Pnknn"
          using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
          by (metis natpermute_finite)+
        let ?f = "λv. ∏j∈{0..k}. ?r $ v ! j"
        have "sum ?f ?Pnkn = sum (λv. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
        proof (rule sum.cong)
          fix v assume v: "v ∈ {xs ∈ natpermute n (k + 1). n ∈ set xs}"
          let ?ths = "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) =
            fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k"
          from v obtain i where i: "i ∈ {0..k}" "v = (replicate (k+1) 0) [i:= n]"
            unfolding natpermute_contain_maximal by auto
          have "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) =
                (∏j∈{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
              using i r0 by (auto simp del: replicate.simps intro: prod.cong)
          also have "… = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
            using i r0 by (simp add: prod_gen_delta)
          finally show ?ths .
        qed rule
        then have "sum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"
          by (simp add: natpermute_max_card[OF ‹n ≠ 0›, simplified])
        also have "… = a$n - sum ?f ?Pnknn"
          unfolding Suc using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
        finally have fn: "sum ?f ?Pnkn = a$n - sum ?f ?Pnknn" .
        have "(?r ^ Suc k)$n = sum ?f ?Pnkn + sum ?f ?Pnknn"
          unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] ..
        also have "… = a$n" unfolding fn by simp
        finally show ?thesis .
      qed
    qed
    then show ?thesis using r0 by (simp add: fps_eq_iff)
  qed
  show ?lhs if ?rhs
  proof -
    from that have "((fps_radical r (Suc k) a) ^ (Suc k))$0 = a$0"
      by simp
    then show ?thesis
      unfolding fps_power_nth_Suc
      by (simp add: prod_constant del: replicate.simps)
  qed
qed

lemma radical_unique:
  assumes r0: "(r (Suc k) (b$0)) ^ Suc k = b$0"
    and a0: "r (Suc k) (b$0 ::'a::field_char_0) = a$0"
    and b0: "b$0 ≠ 0"
  shows "a^(Suc k) = b ⟷ a = fps_radical r (Suc k) b"
    (is "?lhs ⟷ ?rhs" is "_ ⟷ a = ?r")
proof
  show ?lhs if ?rhs
    using that using power_radical[OF b0, of r k, unfolded r0] by simp
  show ?rhs if ?lhs
  proof -
    have r00: "r (Suc k) (b$0) ≠ 0" using b0 r0 by auto
    have ceq: "card {0..k} = Suc k" by simp
    from a0 have a0r0: "a$0 = ?r$0" by simp
    have "a $ n = ?r $ n" for n
    proof (induct n rule: nat_less_induct)
      fix n
      assume h: "∀m
      show "a$n = ?r $ n"
      proof (cases n)
        case 0
        then show ?thesis using a0 by simp
      next
        case (Suc n1)
        have fK: "finite {0..k}" by simp
        have nz: "n ≠ 0" using Suc by simp
        let ?Pnk = "natpermute n (Suc k)"
        let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}"
        let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}"
        have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast
        have d: "?Pnkn ∩ ?Pnknn = {}" by blast
        have f: "finite ?Pnkn" "finite ?Pnknn"
          using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
          by (metis natpermute_finite)+
        let ?f = "λv. ∏j∈{0..k}. ?r $ v ! j"
        let ?g = "λv. ∏j∈{0..k}. a $ v ! j"
        have "sum ?g ?Pnkn = sum (λv. a $ n * (?r$0)^k) ?Pnkn"
        proof (rule sum.cong)
          fix v
          assume v: "v ∈ {xs ∈ natpermute n (Suc k). n ∈ set xs}"
          let ?ths = "(∏j∈{0..k}. a $ v ! j) = a $ n * (?r$0)^k"
          from v obtain i where i: "i ∈ {0..k}" "v = (replicate (k+1) 0) [i:= n]"
            unfolding Suc_eq_plus1 natpermute_contain_maximal
            by (auto simp del: replicate.simps)
          have "(∏j∈{0..k}. a $ v ! j) = (∏j∈{0..k}. if j = i then a $ n else r (Suc k) (b$0))"
            using i a0 by (auto simp del: replicate.simps intro: prod.cong)
          also have "… = a $ n * (?r $ 0)^k"
            using i by (simp add: prod_gen_delta)
          finally show ?ths .
        qed rule
        then have th0: "sum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k"
          by (simp add: natpermute_max_card[OF nz, simplified])
        have th1: "sum ?g ?Pnknn = sum ?f ?Pnknn"
        proof (rule sum.cong, rule refl, rule prod.cong, simp)
          fix xs i
          assume xs: "xs ∈ ?Pnknn" and i: "i ∈ {0..k}"
          have False if c: "n ≤ xs ! i"
          proof -
            from xs i have "xs ! i ≠ n"
              by (simp add: in_set_conv_nth natpermute_def)
            with c have c': "n < xs!i" by arith
            have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..
              by simp_all
            have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}"
              by auto
            have eqs: "{0..∪ ({i} ∪ {i+1 ..< Suc k})"
              using i by auto
            from xs have "n = sum_list xs"
              by (simp add: natpermute_def)
            also have "… = sum (nth xs) {0..
              using xs by (simp add: natpermute_def sum_list_sum_nth)
            also have "… = xs!i + sum (nth xs) {0..
              unfolding eqs  sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
              unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)]
              by simp
            finally show ?thesis using c' by simp
          qed
          then have thn: "xs!i < n" by presburger
          from h[rule_format, OF thn] show "a$(xs !i) = ?r$(xs!i)" .
        qed
        have th00: "∧x::'a. of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
          by (simp add: field_simps del: of_nat_Suc)
        from ‹?lhs› have "b$n = a^Suc k $ n"
          by (simp add: fps_eq_iff)
        also have "a ^ Suc k$n = sum ?g ?Pnkn + sum ?g ?Pnknn"
          unfolding fps_power_nth_Suc
          using sum.union_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric],
            unfolded eq, of ?g] by simp
        also have "… = of_nat (k+1) * a $ n * (?r $ 0)^k + sum ?f ?Pnknn"
          unfolding th0 th1 ..
        finally have 🍋: "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - sum ?f ?Pnknn"
          by simp
        have "a$n = (b$n - sum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)"
          apply (rule eq_divide_imp)
          using r00 🍋 by (simp_all add: ac_simps del: of_nat_Suc)
        then show ?thesis
          unfolding fps_radical_def Suc
          by (simp del: of_nat_Suc)
      qed
    qed
    then show ?rhs by (simp add: fps_eq_iff)
  qed
qed


lemma radical_power:
  assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0"
    and a0: "(a$0 :: 'a::field_char_0) ≠ 0"
  shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
proof -
  let ?ak = "a^ Suc k"
  have ak0: "?ak $ 0 = (a$0) ^ Suc k"
    by (simp add: fps_nth_power_0 del: power_Suc)
  from r0 have th0: "r (Suc k) (a ^ Suc k $ 0) ^ Suc k = a ^ Suc k $ 0"
    using ak0 by auto
  from r0 ak0 have th1: "r (Suc k) (a ^ Suc k $ 0) = a $ 0"
    by auto
  from ak0 a0 have ak00: "?ak $ 0 ≠0 "
    by auto
  from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis
    by metis
qed

lemma fps_deriv_radical':
  fixes a :: "'a::field_char_0 fps"
  assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0"
    and a0: "a$0 ≠ 0"
  shows "fps_deriv (fps_radical r (Suc k) a) =
    fps_deriv a / ((of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
proof -
  let ?r = "fps_radical r (Suc k) a"
  let ?w = "(of_nat (Suc k)) * ?r ^ k"
  from a0 r0 have r0': "r (Suc k) (a$0) ≠ 0"
    by auto
  from r0' have w0: "?w $ 0 ≠ 0"
    by (simp del: of_nat_Suc)
  note th0 = inverse_mult_eq_1[OF w0]
  let ?iw = "inverse ?w"
  from iffD1[OF power_radical[of a r], OF a0 r0]
  have "fps_deriv (?r ^ Suc k) = fps_deriv a"
    by simp
  then have "fps_deriv ?r * ?w = fps_deriv a"
    by (simp add: fps_deriv_power' ac_simps del: power_Suc)
  then have "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a"
    by simp
  with a0 r0 have "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
    by (subst fps_divide_unit) (auto simp del: of_nat_Suc)
  then show ?thesis unfolding th0 by simp
qed

lemma fps_deriv_radical:
  fixes a :: "'a::field_char_0 fps"
  assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0"
    and a0: "a$0 ≠ 0"
  shows "fps_deriv (fps_radical r (Suc k) a) =
    fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
  using fps_deriv_radical'[of r k a, OF r0 a0]
  by (simp add: fps_of_nat[symmetric])

lemma radical_mult_distrib:
  fixes a :: "'a::field_char_0 fps"
  assumes k: "k > 0"
    and ra0: "r k (a $ 0) ^ k = a $ 0"
    and rb0: "r k (b $ 0) ^ k = b $ 0"
    and a0: "a $ 0 ≠ 0"
    and b0: "b $ 0 ≠ 0"
  shows "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0) ⟷
    fps_radical r k (a * b) = fps_radical r k a * fps_radical r k b"
    (is "?lhs ⟷ ?rhs")
proof
  show ?rhs if r0': ?lhs
  proof -
    from r0' have r0: "(r k ((a * b) $ 0)) ^ k = (a * b) $ 0"
      by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
    show ?thesis
    proof (cases k)
      case 0
      then show ?thesis using r0' by simp
    next
      case (Suc h)
      let ?ra = "fps_radical r (Suc h) a"
      let ?rb = "fps_radical r (Suc h) b"
      have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0"
        using r0' Suc by (simp add: fps_mult_nth)
      have ab0: "(a*b) $ 0 ≠ 0"
        using a0 b0 by (simp add: fps_mult_nth)
      from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded Suc] th0 ab0, symmetric]
        iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded Suc]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded Suc]] Suc r0'
      show ?thesis
        by (auto simp add: power_mult_distrib simp del: power_Suc)
    qed
  qed
  show ?lhs if ?rhs
  proof -
    from that have "(fps_radical r k (a * b)) $ 0 = (fps_radical r k a * fps_radical r k b) $ 0"
      by simp
    then show ?thesis
      using k by (simp add: fps_mult_nth)
  qed
qed

(*
 lemma radical_mult_distrib:
  fixes a:: "'a::field_char_0 fps"
  assumes
  ra0: "r k (a $ 0) ^ k = a $ 0"
  and rb0: "r k (b $ 0) ^ k = b $ 0"
  and r0': "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)"
  and a0: "a$0 ≠ 0"
  and b0: "b$0 ≠ 0"
  shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
 proof-
  from r0' have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0"
  by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
  {assume "k=0" then have ?thesis by simp}
  moreover
  {fix h assume k: "k = Suc h"
  let ?ra = "fps_radical r (Suc h) a"
  let ?rb = "fps_radical r (Suc h) b"
  have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0"
  using r0' k by (simp add: fps_mult_nth)
  have ab0: "(a*b) $ 0 ≠ 0" using a0 b0 by (simp add: fps_mult_nth)
  from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric]
  power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
  have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
 ultimately show ?thesis by (cases k, auto)
 qed
*)

lemma radical_divide:
  fixes a :: "'a::field_char_0 fps"
  assumes kp: "k > 0"
    and ra0: "(r k (a $ 0)) ^ k = a $ 0"
    and rb0: "(r k (b $ 0)) ^ k = b $ 0"
    and a0: "a$0 ≠ 0"
    and b0: "b$0 ≠ 0"
  shows "r k ((a $ 0) / (b$0)) = r k (a$0) / r k (b $ 0) ⟷
    fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b"
  (is "?lhs = ?rhs")
proof
  let ?r = "fps_radical r k"
  from kp obtain h where k: "k = Suc h"
    by (cases k) auto
  have ra0': "r k (a$0) ≠ 0" using a0 ra0 k by auto
  have rb0': "r k (b$0) ≠ 0" using b0 rb0 k by auto

  show ?lhs if ?rhs
  proof -
    from that have "?r (a/b) $ 0 = (?r a / ?r b)$0"
      by simp
    then show ?thesis
      using k a0 b0 rb0' by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse)
  qed
  show ?rhs if ?lhs
  proof -
    from a0 b0 have ab0[simp]: "(a/b)$0 = a$0 / b$0"
      by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def)
    have th0: "r k ((a/b)$0) ^ k = (a/b)$0"
      by (simp add: ‹?lhs› power_divide ra0 rb0)
    from a0 b0 ra0' rb0' kp ‹?lhs›
    have th1: "r k ((a / b) $ 0) = (fps_radical r k a / fps_radical r k b) $ 0"
      by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse)
    from a0 b0 ra0' rb0' kp have ab0': "(a / b) $ 0 ≠ 0"
      by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero)
    note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]]
    note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]]
    from b0 rb0' have th2: "(?r a / ?r b)^k = a/b"
      by (simp add: fps_divide_unit power_mult_distrib fps_inverse_power[symmetric])

    from iffD1[OF radical_unique[where r=r and a="?r a / ?r b" and b="a/b" and k=h], symmetric, unfolded k[symmetric], OF th0 th1 ab0' th2]
    show ?thesis .
  qed
qed

lemma radical_inverse:
  fixes a :: "'a::field_char_0 fps"
  assumes k: "k > 0"
    and ra0: "r k (a $ 0) ^ k = a $ 0"
    and r1: "(r k 1)^k = 1"
    and a0: "a$0 ≠ 0"
  shows "r k (inverse (a $ 0)) = r k 1 / (r k (a $ 0)) ⟷
    fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"
  using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0
  by (simp add: divide_inverse fps_divide_def)


subsection ‹Chain rule›

lemma fps_compose_deriv:
  fixes a :: "'a::idom fps"
  assumes b0: "b$0 = 0"
  shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * fps_deriv b"
proof -
  have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n" for n
  proof -
    have "(fps_deriv (a oo b))$n = sum (λi. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}"
      by (simp add: fps_compose_def field_simps sum_distrib_left del: of_nat_Suc)
    also have "… = sum (λi. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}"
      by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
    also have "… = sum (λi. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}"
      unfolding fps_mult_left_const_nth  by (simp add: field_simps)
    also have "… = sum (λi. of_nat i * a$i * (sum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}"
      unfolding fps_mult_nth ..
    also have "… = sum (λi. of_nat i * a$i * (sum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}"
      by (intro sum.mono_neutral_right) (auto simp add: mult_delta_left not_le)
    also have "… = sum (λi. of_nat (i + 1) * a$(i+1) * (sum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
      unfolding fps_deriv_nth
      by (rule sum.reindex_cong [of Suc]) (simp_all add: mult.assoc)
    finally have th0: "(fps_deriv (a oo b))$n =
      sum (λi. of_nat (i + 1) * a$(i+1) * (sum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" .

    have "(((fps_deriv a) oo b) * (fps_deriv b))$n = sum (λi. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}"
      unfolding fps_mult_nth by (simp add: ac_simps)
    also have "… = sum (λi. sum (λj. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}"
      unfolding fps_deriv_nth fps_compose_nth sum_distrib_left mult.assoc
      by (auto simp: subset_eq b0 startsby_zero_power_prefix sum.mono_neutral_left intro: sum.cong)
    also have "… = sum (λi. of_nat (i + 1) * a$(i+1) * (sum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
      unfolding sum_distrib_left
      by (subst sum.swap) (force intro: sum.cong)
    finally show ?thesis
      unfolding th0 by simp
  qed
  then show ?thesis by (simp add: fps_eq_iff)
qed

lemma fps_poly_sum_fps_X:
  assumes "∀i > n. a$i = 0"
  shows "a = sum (λi. fps_const (a$i) * fps_X^i) {0..n}" (is "a = ?r")
proof -
  have "a$i = ?r$i" for i
    unfolding fps_sum_nth fps_mult_left_const_nth fps_X_power_nth
    by (simp add: mult_delta_right assms)
  then show ?thesis
    unfolding fps_eq_iff by blast
qed


subsection ‹Compositional inverses›

fun compinv :: "'a fps ==> nat ==> 'a::field"
where
  "compinv a 0 = fps_X$0"
| "compinv a (Suc n) =
    (fps_X$ Suc n - sum (λi. (compinv a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"

definition "fps_inv a = Abs_fps (compinv a)"

lemma fps_inv:
  assumes a0: "a$0 = 0"
    and a1: "a$1 ≠ 0"
  shows "fps_inv a oo a = fps_X"
proof -
  let ?i = "fps_inv a oo a"
  have "?i $n = fps_X$n" for n
  proof (induct n rule: nat_less_induct)
    fix n
    assume h: "∀m
    show "?i $ n = fps_X$n"
    proof (cases n)
      case 0
      then show ?thesis using a0
        by (simp add: fps_compose_nth fps_inv_def)
    next
      case (Suc n1)
      have "?i $ n = sum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
        by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
      also have "… = sum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} +
        (fps_X$ Suc n1 - sum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"
        using a0 a1 Suc by (simp add: fps_inv_def)
      also have "… = fps_X$n" using Suc by simp
      finally show ?thesis .
    qed
  qed
  then show ?thesis
    by (simp add: fps_eq_iff)
qed


fun gcompinv :: "'a fps ==> 'a fps ==> nat ==> 'a::field"
where
  "gcompinv b a 0 = b$0"
| "gcompinv b a (Suc n) =
    (b$ Suc n - sum (λi. (gcompinv b a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"

definition "fps_ginv b a = Abs_fps (gcompinv b a)"

lemma fps_ginv:
  assumes a0: "a$0 = 0"
    and a1: "a$1 ≠ 0"
  shows "fps_ginv b a oo a = b"
proof -
  let ?i = "fps_ginv b a oo a"
  have "?i $n = b$n" for n
  proof (induct n rule: nat_less_induct)
    fix n
    assume h: "∀m
    show "?i $ n = b$n"
    proof (cases n)
      case 0
      then show ?thesis using a0
        by (simp add: fps_compose_nth fps_ginv_def)
    next
      case (Suc n1)
      have "?i $ n = sum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1"
        by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
      also have "… = sum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} +
        (b$ Suc n1 - sum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})"
        using a0 a1 Suc by (simp add: fps_ginv_def)
      also have "… = b$n" using Suc by simp
      finally show ?thesis .
    qed
  qed
  then show ?thesis
    by (simp add: fps_eq_iff)
qed

lemma fps_inv_ginv: "fps_inv = fps_ginv fps_X"
proof -
  have "compinv x n = gcompinv fps_X x n" for n and x :: "'a fps"
  proof (induction n rule: nat_less_induct)
    case (1 n)
    then show ?case
      by (cases n) auto
  qed
  then show ?thesis
    by (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def)
qed

lemma fps_compose_1[simp]: "1 oo a = 1"
  by (simp add: fps_eq_iff fps_compose_nth mult_delta_left)

lemma fps_compose_0[simp]: "0 oo a = 0"
  by (simp add: fps_eq_iff fps_compose_nth)

lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a $ 0)"
  by (simp add: fps_eq_iff fps_compose_nth power_0_left sum.neutral)

lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
  by (simp add: fps_eq_iff fps_compose_nth field_simps sum.distrib)

lemma fps_compose_sum_distrib: "(sum f S) oo a = sum (λi. f i oo a) S"
proof (cases "finite S")
  case True
  show ?thesis
  proof (rule finite_induct[OF True])
    show "sum f {} oo a = (∑i∈{}. f i oo a)"
      by simp
  next
    fix x F
    assume fF: "finite F"
      and xF: "x ∉ F"
      and h: "sum f F oo a = sum (λi. f i oo a) F"
    show "sum f (insert x F) oo a = sum (λi. f i oo a) (insert x F)"
      using fF xF h by (simp add: fps_compose_add_distrib)
  qed
next
  case False
  then show ?thesis by simp
qed

lemma convolution_eq:
  "sum (λi. a (i :: nat) * b (n - i)) {0 .. n} =
    sum (λ(i,j). a i * b j) {(i,j). i ≤ n ∧ j ≤ n ∧ i + j = n}"
  by (rule sum.reindex_bij_witness[where i=fst and j="λi. (i, n - i)"]) auto

lemma product_composition_lemma:
  assumes c0: "c$0 = (0::'a::idom)"
    and d0: "d$0 = 0"
  shows "((a oo c) * (b oo d))$n =
    sum (λ(k,m). a$k * b$m * (c^k * d^m) $ n) {(k,m). k + m ≤ n}"  (is "?l = ?r")
proof -
  let ?S = "{(k::nat, m::nat). k + m ≤ n}"
  have s: "?S ⊆ {0..n} × {0..n}" by (simp add: subset_eq)
  have f: "finite {(k::nat, m::nat). k + m ≤ n}"
    by (auto intro: finite_subset[OF s])
  have "?r = (∑(k, m) ∈ {(k, m). k + m ≤ n}. ∑j = 0..n. a $ k * b $ m * (c ^ k $ j * d ^ m $ (n - j)))"
    by (simp add: fps_mult_nth sum_distrib_left)
  also have "… = (∑i = 0..n. ∑(k,m)∈{(k,m). k+m ≤ n}. a $ k * c ^ k $ i * b $ m * d ^ m $ (n-i))"
    unfolding sum.swap [where A = "{0..n}"] by (auto simp add: field_simps intro: sum.cong)
  also have "... = (∑i = 0..n.
                    ∑q = 0..i. ∑j = 0..n - i. a $ q * c ^ q $ i * (b $ j * d ^ j $ (n - i)))"
    apply (rule sum.cong [OF refl])
    apply (simp add: sum.cartesian_product mult.assoc)
    apply (rule sum.mono_neutral_right[OF f], force)
    by clarsimp (meson c0 d0 leI startsby_zero_power_prefix)
  also have "… = ?l"
    by (simp add: fps_mult_nth fps_compose_nth sum_product)
  finally show ?thesis by simp
qed

lemma sum_pair_less_iff:
  "sum (λ((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m ≤ n} =
    sum (λs. sum (λi. a i * b (s - i) * c s) {0..s}) {0..n}"
  (is "?l = ?r")
proof -
  have th0: "{(k, m). k + m ≤ n} = (∪s∈{0..n}. ∪i∈{0..s}. {(i, s - i)})"
    by auto
  show "?l = ?r"
    unfolding th0
    by (simp add: sum.UNION_disjoint eq_diff_iff disjoint_iff)
qed

lemma fps_compose_mult_distrib_lemma:
  assumes c0: "c$0 = (0::'a::idom)"
  shows "((a oo c) * (b oo c))$n = sum (λs. sum (λi. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}"
  unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
  unfolding sum_pair_less_iff[where a = "λk. a$k" and b="λm. b$m" and c="λs. (c ^ s)$n" and n = n] ..

lemma fps_compose_mult_distrib:
  assumes c0: "c $ 0 = (0::'a::idom)"
  shows "(a * b) oo c = (a oo c) * (b oo c)"
proof (clarsimp simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0])
  show "(a * b oo c) $ n = (∑s = 0..n. ∑i = 0..s. a $ i * b $ (s - i) * c ^ s $ n)" for n
    by (simp add: fps_compose_nth fps_mult_nth sum_distrib_right)
qed


lemma fps_compose_prod_distrib:
  assumes c0: "c$0 = (0::'a::idom)"
  shows "prod a S oo c = prod (λk. a k oo c) S"
proof (induct S rule: infinite_finite_induct)
next
  case (insert)
  then show ?case   
    by (simp add: fps_compose_mult_distrib[OF c0])
qed auto

lemma fps_compose_divide:
  assumes [simp]: "g dvd f" "h $ 0 = 0"
  shows   "fps_compose f h = fps_compose (f / g :: 'a :: field fps) h * fps_compose g h"
proof -
  have "f = (f / g) * g" by simp
  also have "fps_compose … h = fps_compose (f / g) h * fps_compose g h"
    by (subst fps_compose_mult_distrib) simp_all
  finally show ?thesis .
qed

lemma fps_compose_divide_distrib:
  assumes "g dvd f" "h $ 0 = 0" "fps_compose g h ≠ 0"
  shows   "fps_compose (f / g :: 'a :: field fps) h = fps_compose f h / fps_compose g h"
  using fps_compose_divide[OF assms(1,2)] assms(3) by simp

lemma fps_compose_power:
  assumes c0: "c$0 = (0::'a::idom)"
  shows "(a oo c)^n = a^n oo c"
proof (cases n)
  case 0
  then show ?thesis by simp
next
  case (Suc m)
  have "(∏n = 0..m. a) oo c = (∏n = 0..m. a oo c)"
    using c0 fps_compose_prod_distrib by blast
  moreover have th0: "a^n = prod (λk. a) {0..m}" "(a oo c) ^ n = prod (λk. a oo c) {0..m}"
    by (simp_all add: prod_constant Suc)
  ultimately show ?thesis
    by presburger
qed

lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
  by (simp add: fps_eq_iff fps_compose_nth field_simps sum_negf[symmetric])
    
lemma fps_compose_sub_distrib: "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
  using fps_compose_add_distrib [of a "- b" c] by (simp add: fps_compose_uminus)

lemma fps_X_fps_compose: "fps_X oo a = Abs_fps (λn. if n = 0 then (0::'a::comm_ring_1) else a$n)"
  by (simp add: fps_eq_iff fps_compose_nth mult_delta_left)

lemma fps_compose_eq_0_iff:
  fixes F G :: "'a :: idom fps"
  assumes "fps_nth G 0 = 0"
  shows "fps_compose F G = 0 ⟷ F = 0 ∨ (G = 0 ∧ fps_nth F 0 = 0)"
proof safe
  assume *: "fps_compose F G = 0" "F ≠ 0"
  have "fps_nth (fps_compose F G) 0 = fps_nth F 0"
    by simp
  also have "fps_compose F G = 0"
    by (simp add: *)
  finally show "fps_nth F 0 = 0"
    by simp
  show "G = 0"
  proof (rule ccontr)
    assume "G ≠ 0"
    hence "subdegree G > 0" using assms
      using subdegree_eq_0_iff by blast
    define N where "N = subdegree F * subdegree G"
    have "fps_nth (fps_compose F G) N = (∑i = 0..N. fps_nth F i * fps_nth (G ^ i) N)"
      unfolding fps_compose_def by (simp add: N_def)
    also have "… = (∑i∈{subdegree F}. fps_nth F i * fps_nth (G ^ i) N)"
    proof (intro sum.mono_neutral_right ballI)
      fix i assume i: "i ∈ {0..N} - {subdegree F}"
      show "fps_nth F i * fps_nth (G ^ i) N = 0"
      proof (cases i "subdegree F" rule: linorder_cases)
        assume "i > subdegree F"
        hence "fps_nth (G ^ i) N = 0"
          using i ‹subdegree G > 0› by (intro fps_pow_nth_below_subdegree) (auto simp: N_def)
        thus ?thesis by simp
      qed (use i in ‹auto simp: N_def›)
    qed (use ‹subdegree G > 0› in ‹auto simp: N_def›)
    also have "… = fps_nth F (subdegree F) * fps_nth (G ^ subdegree F) N"
      by simp
    also have "… ≠ 0"
      using ‹G ≠ 0› ‹F ≠ 0› by (auto simp: N_def)
    finally show False using * by auto
  qed
qed auto

lemma subdegree_fps_compose [simp]:
  fixes F G :: "'a :: idom fps"
  assumes [simp]: "fps_nth G 0 = 0"
  shows "subdegree (fps_compose F G) = subdegree F * subdegree G"
proof (cases "G = 0"; cases "F = 0")
  assume [simp]: "G ≠ 0" "F ≠ 0"
  define m where "m = subdegree F"
  define F' where "F' = fps_shift m F"
  have F_eq: "F = F' * fps_X ^ m"
    unfolding F'_def by (simp add: fps_shift_times_fps_X_power m_def)
  have [simp]: "F' ≠ 0"
    using ‹F ≠ 0› unfolding F_eq by auto
  have "subdegree (fps_compose F G) = subdegree (fps_compose F' G) + m * subdegree G"
    by (simp add: F_eq fps_compose_mult_distrib fps_compose_eq_0_iff flip: fps_compose_power)
  also have "subdegree (fps_compose F' G) = 0"
    by (intro subdegree_eq_0) (auto simp: F'_def m_def)
  finally show ?thesis by (simp add: m_def)
qed auto

lemma fps_inverse_compose:
  assumes b0: "(b$0 :: 'a::field) = 0"
    and a0: "a$0 ≠ 0"
  shows "inverse a oo b = inverse (a oo b)"
proof -
  let ?ia = "inverse a"
  let ?ab = "a oo b"
  let ?iab = "inverse ?ab"

  from a0 have ia0: "?ia $ 0 ≠ 0" by simp
  from a0 have ab0: "?ab $ 0 ≠ 0" by (simp add: fps_compose_def)
  have "(?ia oo b) * (a oo b) = 1"
    unfolding fps_compose_mult_distrib[OF b0, symmetric]
    unfolding inverse_mult_eq_1[OF a0]
    fps_compose_1 ..

  then have "(?ia oo b) * (a oo b) * ?iab = 1 * ?iab" by simp
  then have "(?ia oo b) * (?iab * (a oo b)) = ?iab" by simp
  then show ?thesis unfolding inverse_mult_eq_1[OF ab0] by simp
qed

lemma fps_divide_compose:
  assumes c0: "(c$0 :: 'a::field) = 0"
    and b0: "b$0 ≠ 0"
  shows "(a/b) oo c = (a oo c) / (b oo c)"
    using b0 c0 by (simp add: fps_divide_unit fps_inverse_compose fps_compose_mult_distrib)

lemma gp:
  assumes a0: "a$0 = (0::'a::field)"
  shows "(Abs_fps (λn. 1)) oo a = 1/(1 - a)"
    (is "?one oo a = _")
proof -
  have o0: "?one $ 0 ≠ 0" by simp
  have th0: "(1 - fps_X) $ 0 ≠ (0::'a)" by simp
  from fps_inverse_gp[where ?'a = 'a]
  have "inverse ?one = 1 - fps_X" by (simp add: fps_eq_iff)
  then have "inverse (inverse ?one) = inverse (1 - fps_X)" by simp
  then have th: "?one = 1/(1 - fps_X)" unfolding fps_inverse_idempotent[OF o0]
    by (simp add: fps_divide_def)
  show ?thesis
    unfolding th
    unfolding fps_divide_compose[OF a0 th0]
    fps_compose_1 fps_compose_sub_distrib fps_X_fps_compose_startby0[OF a0] ..
qed

lemma fps_compose_radical:
  assumes b0: "b$0 = (0::'a::field_char_0)"
    and ra0: "r (Suc k) (a$0) ^ Suc k = a$0"
    and a0: "a$0 ≠ 0"
  shows "fps_radical r (Suc k) a oo b = fps_radical r (Suc k) (a oo b)"
proof -
  let ?r = "fps_radical r (Suc k)"
  let ?ab = "a oo b"
  have ab0: "?ab $ 0 = a$0"
    by (simp add: fps_compose_def)
  from ab0 a0 ra0 have rab0: "?ab $ 0 ≠ 0" "r (Suc k) (?ab $ 0) ^ Suc k = ?ab $ 0"
    by simp_all
  have th00: "r (Suc k) ((a oo b) $ 0) = (fps_radical r (Suc k) a oo b) $ 0"
    by (simp add: ab0 fps_compose_def)
  have th0: "(?r a oo b) ^ (Suc k) = a oo b"
    unfolding fps_compose_power[OF b0]
    unfolding iffD1[OF power_radical[of a r k], OF a0 ra0]  ..
  from iffD1[OF radical_unique[where r=r and k=k and b= ?ab and a = "?r a oo b", OF rab0(2) th00 rab0(1)], OF th0]
  show ?thesis  .
qed

lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b"
  by (simp add: fps_eq_iff fps_compose_nth sum_distrib_left mult.assoc)

lemma fps_const_mult_apply_right:
  "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
  by (simp add: fps_const_mult_apply_left mult.commute)

lemma fps_compose_assoc:
  assumes c0: "c$0 = (0::'a::idom)"
    and b0: "b$0 = 0"
  shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
proof -
  have "?l$n = ?r$n" for n
  proof -
    have "?l$n = (sum (λi. (fps_const (a$i) * b^i) oo c) {0..n})$n"
      by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left
        sum_distrib_left mult.assoc fps_sum_nth)
    also have "… = ((sum (λi. fps_const (a$i) * b^i) {0..n}) oo c)$n"
      by (simp add: fps_compose_sum_distrib)
    also have "... = (∑i = 0..n. ∑j = 0..n. a $ j * (b ^ j $ i * c ^ i $ n))"
      by (simp add: fps_compose_nth fps_sum_nth sum_distrib_right mult.assoc)
    also have "... = (∑i = 0..n. ∑j = 0..i. a $ j * (b ^ j $ i * c ^ i $ n))"
      by (intro sum.cong [OF refl] sum.mono_neutral_right; simp add: b0 startsby_zero_power_prefix)
    also have "… = ?r$n"
      by (simp add: fps_compose_nth  sum_distrib_right mult.assoc)
    finally show ?thesis .
  qed
  then show ?thesis
    by (simp add: fps_eq_iff)
qed


lemma fps_X_power_compose:
  assumes a0: "a$0=0"
  shows "fps_X^k oo a = (a::'a::idom fps)^k"
  (is "?l = ?r")
proof (cases k)
  case 0
  then show ?thesis by simp
next
  case (Suc h)
  have "?l $ n = ?r $n" for n
  proof -
    consider "k > n" | "k ≤ n" by arith
    then show ?thesis
    proof cases
      case 1
      then show ?thesis
        using a0 startsby_zero_power_prefix[OF a0] Suc
        by (simp add: fps_compose_nth del: power_Suc)
    next
      case 2
      then show ?thesis
        by (simp add: fps_compose_nth mult_delta_left)
    qed
  qed
  then show ?thesis
    unfolding fps_eq_iff by blast
qed

lemma fps_inv_right:
  assumes a0: "a$0 = 0"
    and a1: "a$1 ≠ 0"
  shows "a oo fps_inv a = fps_X"
proof -
  let ?ia = "fps_inv a"
  let ?iaa = "a oo fps_inv a"
  have th0: "?ia $ 0 = 0"
    by (simp add: fps_inv_def)
  have th1: "?iaa $ 0 = 0"
    using a0 a1 by (simp add: fps_inv_def fps_compose_nth)
  have th2: "fps_X$0 = 0"
    by simp
  from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo fps_X"
    by simp
  then have "(a oo fps_inv a) oo a = fps_X oo a"
    by (simp add: fps_compose_assoc[OF a0 th0] fps_X_fps_compose_startby0[OF a0])
  with fps_compose_inj_right[OF a0 a1] show ?thesis
    by simp
qed

lemma fps_inv_deriv:
  assumes a0: "a$0 = (0::'a::field)"
    and a1: "a$1 ≠ 0"
  shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
proof -
  let ?ia = "fps_inv a"
  let ?d = "fps_deriv a oo ?ia"
  let ?dia = "fps_deriv ?ia"
  have ia0: "?ia$0 = 0"
    by (simp add: fps_inv_def)
  have th0: "?d$0 ≠ 0"
    using a1 by (simp add: fps_compose_nth)
  from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
    by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
  then have "inverse ?d * ?d * ?dia = inverse ?d * 1"
    by simp
  with inverse_mult_eq_1 [OF th0] show "?dia = inverse ?d"
    by simp
qed

lemma fps_inv_idempotent:
  assumes a0: "a$0 = 0"
    and a1: "a$1 ≠ 0"
  shows "fps_inv (fps_inv a) = a"
proof -
  let ?r = "fps_inv"
  have ra0: "?r a $ 0 = 0"
    by (simp add: fps_inv_def)
  from a1 have ra1: "?r a $ 1 ≠ 0"
    by (simp add: fps_inv_def field_simps)
  have fps_X0: "fps_X$0 = 0"
    by simp
  from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = fps_X" .
  then have "?r (?r a) oo ?r a oo a = fps_X oo a"
    by simp
  then have "?r (?r a) oo (?r a oo a) = a"
    unfolding fps_X_fps_compose_startby0[OF a0]
    unfolding fps_compose_assoc[OF a0 ra0, symmetric] .
  then show ?thesis
    unfolding fps_inv[OF a0 a1] by simp
qed

lemma fps_ginv_ginv:
  assumes a0: "a$0 = 0"
    and a1: "a$1 ≠ 0"
    and c0: "c$0 = 0"
    and  c1: "c$1 ≠ 0"
  shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c"
proof -
  let ?r = "fps_ginv"
  from c0 have rca0: "?r c a $0 = 0"
    by (simp add: fps_ginv_def)
  from a1 c1 have rca1: "?r c a $ 1 ≠ 0"
    by (simp add: fps_ginv_def field_simps)
  from fps_ginv[OF rca0 rca1]
  have "?r b (?r c a) oo ?r c a = b" .
  then have "?r b (?r c a) oo ?r c a oo a = b oo a"
    by simp
  then have "?r b (?r c a) oo (?r c a oo a) = b oo a"
    by (simp add: a0 fps_compose_assoc rca0)
  then have "?r b (?r c a) oo c = b oo a"
    unfolding fps_ginv[OF a0 a1] .
  then have "?r b (?r c a) oo c oo fps_inv c= b oo a oo fps_inv c"
    by simp
  then have "?r b (?r c a) oo (c oo fps_inv c) = b oo a oo fps_inv c"
    by (metis c0 c1 fps_compose_assoc fps_compose_nth_0 fps_inv fps_inv_right)
  then show ?thesis
    unfolding fps_inv_right[OF c0 c1] by simp
qed

lemma fps_ginv_deriv:
  assumes a0:"a$0 = (0::'a::field)"
    and a1: "a$1 ≠ 0"
  shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv fps_X a"
proof -
  let ?ia = "fps_ginv b a"
  let ?ifps_Xa = "fps_ginv fps_X a"
  let ?d = "fps_deriv"
  let ?dia = "?d ?ia"
  have ifps_Xa0: "?ifps_Xa $ 0 = 0"
    by (simp add: fps_ginv_def)
  have da0: "?d a $ 0 ≠ 0"
    using a1 by simp
  from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b"
    by simp
  then have "(?d ?ia oo a) * ?d a = ?d b"
    unfolding fps_compose_deriv[OF a0] .
  then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)"
    by simp
  with a1 have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a"
    by (simp add: fps_divide_unit)
  then have "(?d ?ia oo a) oo ?ifps_Xa = (?d b / ?d a) oo ?ifps_Xa"
    unfolding inverse_mult_eq_1[OF da0] by simp
  then have "?d ?ia oo (a oo ?ifps_Xa) = (?d b / ?d a) oo ?ifps_Xa"
    unfolding fps_compose_assoc[OF ifps_Xa0 a0] .
  then show ?thesis unfolding fps_inv_ginv[symmetric]
    unfolding fps_inv_right[OF a0 a1] by simp
qed

lemma fps_compose_linear:
  "fps_compose (f :: 'a :: comm_ring_1 fps) (fps_const c * fps_X) = Abs_fps (λn. c^n * f $ n)"
  by (simp add: fps_eq_iff fps_compose_def power_mult_distrib
                if_distrib cong: if_cong)
              
lemma fps_compose_uminus': 
  "fps_compose f (-fps_X :: 'a :: comm_ring_1 fps) = Abs_fps (λn. (-1)^n * f $ n)"
  using fps_compose_linear[of f "-1"] 
  by (simp only: fps_const_neg [symmetric] fps_const_1_eq_1) simp
lemma fps_nth_compose_linear [simp]:
  fixes f :: "'a :: comm_ring_1 fps"
  shows "fps_nth (fps_compose f (fps_const c * fps_X)) n = c ^ n * fps_nth f n"
proof -
  have "fps_nth (fps_compose f (fps_const c * fps_X)) n =
        (∑i∈{n}. fps_nth f i * fps_nth ((fps_const c * fps_X) ^ i) n)"
    unfolding fps_compose_nth
    by (intro sum.mono_neutral_cong_right) (auto simp: power_mult_distrib)
  also have "… = c ^ n * fps_nth f n"
    by (simp add: power_mult_distrib)
  finally show ?thesis .
qed

subsection ‹Elementary series›

subsubsection ‹Exponential series›

definition "fps_exp x = Abs_fps (λn. x^n / of_nat (fact n))"

lemma fps_exp_deriv[simp]: "fps_deriv (fps_exp a) = fps_const (a::'a::field_char_0) * fps_exp a" 
  (is "?l = ?r")
proof -
  have "?l$n = ?r $ n" for n
    using of_nat_neq_0 by (auto simp add: fps_exp_def divide_simps)
  then show ?thesis
    by (simp add: fps_eq_iff)
qed

lemma fps_exp_unique_ODE:
  "fps_deriv a = fps_const c * a ⟷ a = fps_const (a$0) * fps_exp (c::'a::field_char_0)"
  (is "?lhs ⟷ ?rhs")
proof
  show ?rhs if ?lhs
  proof -
    from that have th: "∧n. a $ Suc n = c * a$n / of_nat (Suc n)"
      by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
    have th': "a$n = a$0 * c ^ n/ (fact n)" for n
    proof (induct n)
      case 0
      then show ?case by simp
    next
      case Suc
      then show ?case
        by (simp add: th divide_simps)
    qed
    show ?thesis
      by (auto simp add: fps_eq_iff fps_const_mult_left fps_exp_def intro: th')
  qed
  show ?lhs if ?rhs
    using that by (metis fps_exp_deriv fps_deriv_mult_const_left mult.left_commute)
qed

lemma fps_exp_add_mult: "fps_exp (a + b) = fps_exp (a::'a::field_char_0) * fps_exp b" (is "?l = ?r")
proof -
  have "fps_deriv ?r = fps_const (a + b) * ?r"
    by (simp add: fps_const_add[symmetric] field_simps del: fps_const_add)
  then have "?r = ?l"
    by (simp only: fps_exp_unique_ODE) (simp add: fps_mult_nth fps_exp_def)
  then show ?thesis ..
qed

lemma fps_exp_nth[simp]: "fps_exp a $ n = a^n / of_nat (fact n)"
  by (simp add: fps_exp_def)

lemma fps_exp_0[simp]: "fps_exp (0::'a::field) = 1"
  by (simp add: fps_eq_iff power_0_left)

lemma fps_exp_neg: "fps_exp (- a) = inverse (fps_exp (a::'a::field_char_0))"
proof -
  from fps_exp_add_mult[of a "- a"] have th0: "fps_exp a * fps_exp (- a) = 1" by simp
  from fps_inverse_unique[OF th0] show ?thesis by simp
qed

lemma fps_exp_nth_deriv[simp]: 
  "fps_nth_deriv n (fps_exp (a::'a::field_char_0)) = (fps_const a)^n * (fps_exp a)"
  by (induct n) auto

lemma fps_X_compose_fps_exp[simp]: "fps_X oo fps_exp (a::'a::field) = fps_exp a - 1"
  by (simp add: fps_eq_iff fps_X_fps_compose)

lemma fps_inv_fps_exp_compose:
  assumes a: "a ≠ 0"
  shows "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X"
    and "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X"
proof -
  let ?b = "fps_exp a - 1"
  have b0: "?b $ 0 = 0"
    by simp
  have b1: "?b $ 1 ≠ 0"
    by (simp add: a)
  from fps_inv[OF b0 b1] show "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X" .
  from fps_inv_right[OF b0 b1] show "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X" .
qed

lemma fps_exp_power_mult: "(fps_exp (c::'a::field_char_0))^n = fps_exp (of_nat n * c)"
  by (induct n) (simp_all add: field_simps fps_exp_add_mult)

lemma radical_fps_exp:
  assumes r: "r (Suc k) 1 = 1"
  shows "fps_radical r (Suc k) (fps_exp (c::'a::field_char_0)) = fps_exp (c / of_nat (Suc k))"
proof -
  let ?ck = "(c / of_nat (Suc k))"
  let ?r = "fps_radical r (Suc k)"
  have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c"
    by (simp_all del: of_nat_Suc)
  have th0: "fps_exp ?ck ^ (Suc k) = fps_exp c" unfolding fps_exp_power_mult eq0 ..
  have th: "r (Suc k) (fps_exp c $0) ^ Suc k = fps_exp c $ 0"
    "r (Suc k) (fps_exp c $ 0) = fps_exp ?ck $ 0" "fps_exp c $ 0 ≠ 0" using r by simp_all
  from th0 radical_unique[where r=r and k=k, OF th] show ?thesis
    by auto
qed

lemma fps_exp_compose_linear [simp]: 
  "fps_exp (d::'a::field_char_0) oo (fps_const c * fps_X) = fps_exp (c * d)"
  by (simp add: fps_compose_linear fps_exp_def fps_eq_iff power_mult_distrib)
  
lemma fps_fps_exp_compose_minus [simp]: 
  "fps_compose (fps_exp c) (-fps_X) = fps_exp (-c :: 'a :: field_char_0)"
  using fps_exp_compose_linear[of c "-1 :: 'a"] 
  unfolding fps_const_neg [symmetric] fps_const_1_eq_1 by simp

lemma fps_exp_eq_iff [simp]: "fps_exp c = fps_exp d ⟷ c = (d :: 'a :: field_char_0)"
proof
  assume "fps_exp c = fps_exp d"
  from arg_cong[of _ _ "λF. F $ 1", OF this] show "c = d" by simp
qed simp_all

lemma fps_exp_eq_fps_const_iff [simp]: 
  "fps_exp (c :: 'a :: field_char_0) = fps_const c' ⟷ c = 0 ∧ c' = 1"
proof
  assume "c = 0 ∧ c' = 1"
  thus "fps_exp c = fps_const c'" by (simp add: fps_eq_iff)
next
  assume "fps_exp c = fps_const c'"
  from arg_cong[of _ _ "λF. F $ 1", OF this] arg_cong[of _ _ "λF. F $ 0", OF this] 
    show "c = 0 ∧ c' = 1" by simp_all
qed

lemma fps_exp_neq_0 [simp]: "¬fps_exp (c :: 'a :: field_char_0) = 0"
  unfolding fps_const_0_eq_0 [symmetric] fps_exp_eq_fps_const_iff by simp  

lemma fps_exp_eq_1_iff [simp]: "fps_exp (c :: 'a :: field_char_0) = 1 ⟷ c = 0"
  unfolding fps_const_1_eq_1 [symmetric] fps_exp_eq_fps_const_iff by simp
    
lemma fps_exp_neq_numeral_iff [simp]: 
  "fps_exp (c :: 'a :: field_char_0) = numeral n ⟷ c = 0 ∧ n = Num.One"
  unfolding numeral_fps_const fps_exp_eq_fps_const_iff by simp


subsubsection ‹Logarithmic series›

lemma Abs_fps_if_0:
  "Abs_fps (λn. if n = 0 then (v::'a::ring_1) else f n) =
    fps_const v + fps_X * Abs_fps (λn. f (Suc n))"
  by (simp add: fps_eq_iff)

definition fps_ln :: "'a::field_char_0 ==> 'a fps"
  where "fps_ln c = fps_const (1/c) * Abs_fps (λn. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)"

lemma fps_ln_deriv: "fps_deriv (fps_ln c) = fps_const (1/c) * inverse (1 + fps_X)"
  unfolding fps_inverse_fps_X_plus1
  by (simp add: fps_ln_def fps_eq_iff del: of_nat_Suc)

lemma fps_ln_nth: "fps_ln c $ n = (if n = 0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))"
  by (simp add: fps_ln_def field_simps)

lemma fps_ln_0 [simp]: "fps_ln c $ 0 = 0" by (simp add: fps_ln_def)

lemma fps_ln_fps_exp_inv:
  fixes a :: "'a::field_char_0"
  assumes a: "a ≠ 0"
  shows "fps_ln a = fps_inv (fps_exp a - 1)"  (is "?l = ?r")
proof -
  let ?b = "fps_exp a - 1"
  have b0: "?b $ 0 = 0" by simp
  have b1: "?b $ 1 ≠ 0" by (simp add: a)
  have "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) =
    (fps_const a * (fps_exp a - 1) + fps_const a) oo fps_inv (fps_exp a - 1)"
    by (simp add: field_simps)
  also have "… = fps_const a * (fps_X + 1)"
    by (simp add: fps_compose_add_distrib fps_inv_right[OF b0 b1] distrib_left flip: fps_const_mult_apply_left)
  finally have eq: "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_const a * (fps_X + 1)" .
  from fps_inv_deriv[OF b0 b1, unfolded eq]
  have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (fps_X + 1)"
    using a by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
  then have "fps_deriv ?l = fps_deriv ?r"
    by (simp add: fps_ln_deriv add.commute fps_divide_def divide_inverse)
  then show ?thesis unfolding fps_deriv_eq_iff
    by (simp add: fps_ln_nth fps_inv_def)
qed

lemma fps_ln_mult_add:
  assumes c0: "c≠0"
    and d0: "d≠0"
  shows "fps_ln c + fps_ln d = fps_const (c+d) * fps_ln (c*d)"
  (is "?r = ?l")
proof-
  from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps)
  have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + fps_X)"
    by (simp add: fps_ln_deriv fps_const_add[symmetric] algebra_simps del: fps_const_add)
  also have "… = fps_deriv ?l"
    by (simp add: eq fps_ln_deriv)
  finally show ?thesis
    unfolding fps_deriv_eq_iff by simp
qed

lemma fps_X_dvd_fps_ln [simp]: "fps_X dvd fps_ln c"
proof -
  have "fps_ln c = fps_X * Abs_fps (λn. (-1) ^ n / (of_nat (Suc n) * c))"
    by (intro fps_ext) (simp add: fps_ln_def of_nat_diff)
  thus ?thesis by simp
qed


subsubsection ‹Binomial series›

definition "fps_binomial a = Abs_fps (λn. a gchoose n)"

lemma fps_binomial_nth[simp]: "fps_binomial a $ n = a gchoose n"
  by (simp add: fps_binomial_def)

lemma fps_binomial_ODE_unique:
  fixes c :: "'a::field_char_0"
  shows "fps_deriv a = (fps_const c * a) / (1 + fps_X) ⟷ a = fps_const (a$0) * fps_binomial c"
  (is "?lhs ⟷ ?rhs")
proof
  let ?da = "fps_deriv a"
  let ?x1 = "(1 + fps_X):: 'a fps"
  let ?l = "?x1 * ?da"
  let ?r = "fps_const c * a"

  have eq: "?l = ?r ⟷ ?lhs"
  proof -
    have x10: "?x1 $ 0 ≠ 0" by simp
    have "?l = ?r ⟷ inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
    also have "… ⟷ ?da = (fps_const c * a) / ?x1"
      unfolding fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10]
      by (simp add: field_simps)
    finally show ?thesis .
  qed

  show ?rhs if ?lhs
  proof -
    from eq that have h: "?l = ?r" ..
    have th0: "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n" for n
    proof -
      from h have "?l $ n = ?r $ n" by simp
      then show ?thesis
        by (simp add: field_simps del: of_nat_Suc split: if_split_asm)
    qed
    have th1: "a $ n = (c gchoose n) * a $ 0" for n
    proof (induct n)
      case 0
      then show ?case by simp
    next
      case (Suc m)
      have "(c - of_nat m) * (c gchoose m) = (c gchoose Suc m) * of_nat (Suc m)"
        by (metis gbinomial_absorb_comp gbinomial_absorption mult.commute)
      with Suc show ?case
        unfolding th0
        by (simp add: divide_simps del: of_nat_Suc)
    qed
    show ?thesis
      by (metis expand_fps_eq fps_binomial_nth fps_mult_right_const_nth mult.commute th1)
  qed

  show ?lhs if ?rhs
  proof -
    have th00: "x * (a $ 0 * y) = a $ 0 * (x * y)" for x y
      by (simp add: mult.commute)
    have "?l = (1 + fps_X) * fps_deriv (fps_const (a $ 0) * fps_binomial c)"
      using that by auto
    also have "... = fps_const c * (fps_const (a $ 0) * fps_binomial c)"
    proof (clarsimp simp add: fps_eq_iff algebra_simps)
      show "a $ 0 * (c gchoose Suc n) + (of_nat n * ((c gchoose n) * a $ 0) + of_nat n * (a $ 0 * (c gchoose Suc n)))
         = c * ((c gchoose n) * a $ 0)" for n
      unfolding mult.assoc[symmetric]  
      by (simp add: field_simps gbinomial_mult_1)
  qed
    also have "... = ?r"
      using that by auto
    finally have "?l = ?r" .
    with eq show ?thesis ..
  qed
qed

lemma fps_binomial_ODE_unique':
  "(fps_deriv a = fps_const c * a / (1 + fps_X) ∧ a $ 0 = 1) ⟷ (a = fps_binomial c)"
  by (subst fps_binomial_ODE_unique) auto

lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + fps_X)"
proof -
  let ?a = "fps_binomial c"
  have th0: "?a = fps_const (?a$0) * ?a" by (simp)
  from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
qed

lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
proof -
  let ?P = "?r - ?l"
  let ?b = "fps_binomial"
  let ?db = "λx. fps_deriv (?b x)"
  have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)"  by simp
  also have "… = inverse (1 + fps_X) *
      (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
    unfolding fps_binomial_deriv
    by (simp add: fps_divide_def field_simps)
  also have "… = (fps_const (c + d)/ (1 + fps_X)) * ?P"
    by (simp add: field_simps fps_divide_unit fps_const_add[symmetric] del: fps_const_add)
  finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + fps_X)"
    by (simp add: fps_divide_def)
  have "?P = fps_const (?P$0) * ?b (c + d)"
    unfolding fps_binomial_ODE_unique[symmetric]
    using th0 by simp
  then have "?P = 0" by (simp add: fps_mult_nth)
  then show ?thesis by simp
qed

lemma fps_binomial_minus_one: "fps_binomial (- 1) = inverse (1 + fps_X)"
  (is "?l = inverse ?r")
proof-
  have th: "?r$0 ≠ 0" by simp
  have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + fps_X)"
    by (simp add: fps_inverse_deriv[OF th] fps_divide_def
      power2_eq_square mult.commute fps_const_neg[symmetric] del: fps_const_neg)
  have eq: "inverse ?r $ 0 = 1"
    by (simp add: fps_inverse_def)
  from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + fps_X)" "- 1"] th'] eq
  show ?thesis by (simp add: fps_inverse_def)
qed

lemma fps_binomial_of_nat: "fps_binomial (of_nat n) = (1 + fps_X :: 'a :: field_char_0 fps) ^ n"
proof (cases "n = 0")
  case [simp]: True
  have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = 0" by simp
  also have "… = fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)" by (simp add: fps_binomial_def)
  finally show ?thesis by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) simp_all
next
  case False
  have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = fps_const (of_nat n) * (1 + fps_X) ^ (n - 1)"
    by (simp add: fps_deriv_power)
  also have "(1 + fps_X :: 'a fps) $ 0 ≠ 0" by simp
  hence "(1 + fps_X :: 'a fps) ≠ 0" by (intro notI) (simp only: , simp)
  with False have "(1 + fps_X :: 'a fps) ^ (n - 1) = (1 + fps_X) ^ n / (1 + fps_X)"
    by (cases n) (simp_all )
  also have "fps_const (of_nat n :: 'a) * ((1 + fps_X) ^ n / (1 + fps_X)) =
               fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)"
    by (simp add: unit_div_mult_swap)
  finally show ?thesis
    by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) (simp_all add: fps_power_nth)
qed

lemma fps_binomial_0 [simp]: "fps_binomial 0 = 1"
  using fps_binomial_of_nat[of 0] by simp
  
lemma fps_binomial_power: "fps_binomial a ^ n = fps_binomial (of_nat n * a)"
  by (induction n) (simp_all add: fps_binomial_add_mult ring_distribs)

lemma fps_binomial_1: "fps_binomial 1 = 1 + fps_X"
  using fps_binomial_of_nat[of 1] by simp

lemma fps_binomial_minus_of_nat:
  "fps_binomial (- of_nat n) = inverse ((1 + fps_X :: 'a :: field_char_0 fps) ^ n)"
  by (rule sym, rule fps_inverse_unique)
     (simp add: fps_binomial_of_nat [symmetric] fps_binomial_add_mult [symmetric])

lemma one_minus_const_fps_X_power:
  "c ≠ 0 ==> (1 - fps_const c * fps_X) ^ n =
     fps_compose (fps_binomial (of_nat n)) (-fps_const c * fps_X)"
  by (subst fps_binomial_of_nat)
     (simp add: fps_compose_power [symmetric] fps_compose_add_distrib fps_const_neg [symmetric] 
           del: fps_const_neg)

lemma one_minus_fps_X_const_neg_power:
  "inverse ((1 - fps_const c * fps_X) ^ n) =
       fps_compose (fps_binomial (-of_nat n)) (-fps_const c * fps_X)"
proof (cases "c = 0")
  case False
  thus ?thesis
  by (subst fps_binomial_minus_of_nat)
     (simp add: fps_compose_power [symmetric] fps_inverse_compose fps_compose_add_distrib
                fps_const_neg [symmetric] del: fps_const_neg)
qed simp

lemma fps_X_plus_const_power:
  "c ≠ 0 ==> (fps_X + fps_const c) ^ n =
     fps_const (c^n) * fps_compose (fps_binomial (of_nat n)) (fps_const (inverse c) * fps_X)"
  by (subst fps_binomial_of_nat)
     (simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib
                fps_const_power [symmetric] power_mult_distrib [symmetric] 
                algebra_simps inverse_mult_eq_1' del: fps_const_power)

lemma fps_X_plus_const_neg_power:
  "c ≠ 0 ==> inverse ((fps_X + fps_const c) ^ n) =
     fps_const (inverse c^n) * fps_compose (fps_binomial (-of_nat n)) (fps_const (inverse c) * fps_X)"
  by (subst fps_binomial_minus_of_nat)
     (simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib
                fps_const_power [symmetric] power_mult_distrib [symmetric] fps_inverse_compose 
                algebra_simps fps_const_inverse [symmetric] fps_inverse_mult [symmetric]
                fps_inverse_power [symmetric] inverse_mult_eq_1'
           del: fps_const_power)


lemma one_minus_const_fps_X_neg_power':
  fixes c :: "'a :: field_char_0"
  assumes "n > 0"
  shows "inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (λk. of_nat ((n + k - 1) choose k) * c^k)"
proof -
  have 🍋: "∧j. Abs_fps (λna. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ j =
          Abs_fps (λk. of_nat (n + k - 1 choose k) * c ^ k) $ j"
    using assms
    by (simp add: gbinomial_minus binomial_gbinomial of_nat_diff flip: power_mult_distrib mult.assoc)
  show ?thesis
    apply (rule fps_ext)
    using 🍋 
    by (metis (no_types, lifting) one_minus_fps_X_const_neg_power fps_const_neg fps_compose_linear fps_nth_Abs_fps)
qed

text ‹Vandermonde's Identity as a consequence.›
lemma gbinomial_Vandermonde:
  "sum (λk. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
proof -
  let ?ba = "fps_binomial a"
  let ?bb = "fps_binomial b"
  let ?bab = "fps_binomial (a + b)"
  from fps_binomial_add_mult[of a b] have "?bab $ n = (?ba * ?bb)$n" by simp
  then show ?thesis by (simp add: fps_mult_nth)
qed

lemma binomial_Vandermonde:
  "sum (λk. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
  using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
  by (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric]
                 of_nat_sum[symmetric] of_nat_add[symmetric] of_nat_eq_iff)

lemma binomial_Vandermonde_same: "sum (λk. (n choose k)🪙2) {0..n} = (2 * n) choose n"
  using binomial_Vandermonde[of n n n, symmetric]
  unfolding mult_2
  by (metis atMost_atLeast0 choose_square_sum mult_2)

lemma Vandermonde_pochhammer_lemma:
  fixes a :: "'a::field_char_0"
  assumes b: "∧j. j==> b ≠ of_nat j"
  shows "sum (λk. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) /
      (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} =
    pochhammer (- (a + b)) n / pochhammer (- b) n"
  (is "?l = ?r")
proof -
  let ?m1 = "λm. (- 1 :: 'a) ^ m"
  let ?f = "λm. of_nat (fact m)"
  let ?p = "λ(x::'a). pochhammer (- x)"
  from b have bn0: "?p b n ≠ 0"
    unfolding pochhammer_eq_0_iff by simp
  have th00:
    "b gchoose (n - k) =
        (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
      (is ?gchoose)
    "pochhammer (1 + b - of_nat n) k ≠ 0"
      (is ?pochhammer)
    if kn: "k ∈ {0..n}" for k
  proof -
    from kn have "k ≤ n" by simp
    have nz: "pochhammer (1 + b - of_nat n) n ≠ 0"
    proof
      assume "pochhammer (1 + b - of_nat n) n = 0"
      then have c: "pochhammer (b - of_nat n + 1) n = 0"
        by (simp add: algebra_simps)
      then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j"
        unfolding pochhammer_eq_0_iff by blast
      from j have "b = of_nat n - of_nat j - of_nat 1"
        by (simp add: algebra_simps)
      then show False 
        using ‹j 🚫› j b
        by (metis bn0 c mult_cancel_right2 pochhammer_minus)
    qed

    from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k ≠ 0"
      by (rule pochhammer_neq_0_mono)

    consider "k = 0 ∨ n = 0" | "k ≠ 0" "n ≠ 0"
      by blast
    then have "b gchoose (n - k) =
      (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
    proof cases
      case 1
      then show ?thesis
        using kn by (cases "k = 0") (simp_all add: gbinomial_pochhammer)
    next
      case neq: 2
      then obtain m where m: "n = Suc m"
        by (cases n) auto
      from neq(1) obtain h where h: "k = Suc h"
        by (cases k) auto
      show ?thesis
      proof (cases "k = n")
        case True
        with pochhammer_minus'[where k=k and b=b] bn0 show ?thesis       
          by (simp add: pochhammer_same)
      next
        case False
        with kn have kn': "k < n"
          by simp
        have "h ≤ m"
          using ‹k ≤ n› h m by blast
        have m1nk: "?m1 n = prod (λi. - 1) {..m}" "?m1 k = prod (λi. - 1) {0..h}"
          by (simp_all add: m h)
        have bnz0: "pochhammer (b - of_nat n + 1) k ≠ 0"
          using bn0 kn
          unfolding pochhammer_eq_0_iff
          by (metis add.commute add_diff_eq nz' pochhammer_eq_0_iff)
        have eq1: "prod (λk. (1::'a) + of_nat m - of_nat k) {..h} =
          prod of_nat {Suc (m - h) .. Suc m}"
          using kn' h m
          by (intro prod.reindex_bij_witness[where i="λk. Suc m - k" and j="λk. Suc m - k"])
             (auto simp: of_nat_diff)
        have "(∏i = 0..∏x = n - k..
          using ‹k ≤ n›
          using prod.atLeastLessThan_shift_bounds [where ?'a = 'a, of "λi. 1 + of_nat i" 0 "n - k" k]
          by (auto simp add: of_nat_diff field_simps)
        then have "fact (n - k) * pochhammer ((1::'a) + of_nat n - of_nat k) k = fact n"
          using ‹k ≤ n›
          by (auto simp add: fact_split [of k n] pochhammer_prod field_simps)
        then have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
          by (simp add: pochhammer_minus field_simps)
        have "?m1 n * ?p b n = pochhammer (b - of_nat m) (Suc m)"
          by (simp add: pochhammer_minus field_simps m)
        also have "... = (∏i = 0..m. b - of_nat i)"
          by (auto simp add: pochhammer_prod_rev of_nat_diff prod.atLeast_Suc_atMost_Suc_shift simp del: prod.cl_ivl_Suc)
        finally have th20: "?m1 n * ?p b n = prod (λi. b - of_nat i) {0..m}" .
        have "(∏x = 0..h. b - of_nat m + of_nat (h - x)) = (∏i = m - h..m. b - of_nat i)"
          using ‹h ≤ m› prod.atLeastAtMost_shift_0 [of "m - h" m, where ?'a = 'a]
          by (auto simp add: of_nat_diff field_simps)
        then have th21:"pochhammer (b - of_nat n + 1) k = prod (λi. b - of_nat i) {n - k .. n - 1}"
          using kn by (simp add: pochhammer_prod_rev m h prod.atLeast_Suc_atMost_Suc_shift del: prod.op_ivl_Suc del: prod.cl_ivl_Suc)
        have "?m1 n * ?p b n =
          prod (λi. b - of_nat i) {0.. n - k - 1} * pochhammer (b - of_nat n + 1) k"
          using kn' m h unfolding th20 th21 
          by (auto simp flip: prod.union_disjoint intro: prod.cong)
        then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k =
          prod (λi. b - of_nat i) {0.. n - k - 1}"
          using nz' by (simp add: field_simps)
        have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) =
          ((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"
          using bnz0
          by (simp add: field_simps)
        also have "… = b gchoose (n - k)"
          unfolding th1 th2
          using kn' m h
          by (auto simp: field_simps gbinomial_mult_fact intro: prod.cong)
        finally show ?thesis by simp
      qed
    qed
    then show ?gchoose and ?pochhammer
      using nz' by force+
  qed
  have "?r = ((a + b) gchoose n) * (of_nat (fact n) / (?m1 n * pochhammer (- b) n))"
    unfolding gbinomial_pochhammer
    using bn0 by (auto simp add: field_simps)
  also have "… = ?l"
    using bn0
    unfolding gbinomial_Vandermonde[symmetric]
    apply (simp add: th00)
    by (simp add: gbinomial_pochhammer sum_distrib_right sum_distrib_left field_simps)
  finally show ?thesis by simp
qed

lemma Vandermonde_pochhammer:
  fixes a :: "'a::field_char_0"
  assumes c: "∀i ∈ {0..< n}. c ≠ - of_nat i"
  shows "sum (λk. (pochhammer a k * pochhammer (- (of_nat n)) k) /
    (of_nat (fact k) * pochhammer c k)) {0..n} = pochhammer (c - a) n / pochhammer c n"
proof -
  let ?a = "- a"
  let ?b = "c + of_nat n - 1"
  have h: "?b ≠ of_nat j" if "j < n" for j
  proof -
    have "c ≠ - of_nat (n - j - 1)"
      using c that by (auto simp: dest!: bspec [where x = "n-j-1"])
    with that show ?thesis
      by (auto simp add: algebra_simps of_nat_diff)
  qed
  have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n"
    unfolding pochhammer_minus
    by (simp add: algebra_simps)
  have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n"
    unfolding pochhammer_minus
    by simp
  have nz: "pochhammer c n ≠ 0" using c
    by (simp add: pochhammer_eq_0_iff)
  from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1]
  show ?thesis
    using nz by (simp add: field_simps sum_distrib_left)
qed


subsubsection ‹Trigonometric functions›

definition "fps_sin (c::'a::field_char_0) =
  Abs_fps (λn. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"

definition "fps_cos (c::'a::field_char_0) =
  Abs_fps (λn. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"

lemma fps_sin_0 [simp]: "fps_sin 0 = 0"
  by (intro fps_ext) (auto simp: fps_sin_def elim!: oddE)

lemma fps_cos_0 [simp]: "fps_cos 0 = 1"
  by (intro fps_ext) (simp add: fps_cos_def)

lemma fps_sin_deriv:
  "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
  (is "?lhs = ?rhs")
proof (rule fps_ext)
  fix n :: nat
  show "?lhs $ n = ?rhs $ n"
  proof (cases "even n")
    case True
    have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp
    also have "… = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
      using True by (simp add: fps_sin_def)
    also have "… = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
      unfolding fact_Suc of_nat_mult
      by (simp add: field_simps del: of_nat_add of_nat_Suc)
    also have "… = (- 1)^(n div 2) * c^Suc n / of_nat (fact n)"
      by (simp add: field_simps del: of_nat_add of_nat_Suc)
    finally show ?thesis
      using True by (simp add: fps_cos_def field_simps)
  next
    case False
    then show ?thesis
      by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
  qed
qed

lemma fps_cos_deriv: "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
  (is "?lhs = ?rhs")
proof (rule fps_ext)
  have th0: "- ((- 1::'a) ^ n) = (- 1)^Suc n" for n
    by simp
  show "?lhs $ n = ?rhs $ n" for n
  proof (cases "even n")
    case False
    then have n0: "n ≠ 0" by presburger
    from False have th1: "Suc ((n - 1) div 2) = Suc n div 2"
      by (cases n) simp_all
    have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp
    also have "… = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))"
      using False by (simp add: fps_cos_def)
    also have "… = (- 1)^((n + 1) div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
      unfolding fact_Suc of_nat_mult
      by (simp add: field_simps del: of_nat_add of_nat_Suc)
    also have "… = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
      by (simp add: field_simps del: of_nat_add of_nat_Suc)
    also have "… = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
      unfolding th0 unfolding th1 by simp
    finally show ?thesis
      using False by (simp add: fps_sin_def field_simps)
  next
    case True
    then show ?thesis
      by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
  qed
qed

lemma fps_sin_cos_sum_of_squares: "(fps_cos c)🪙2 + (fps_sin c)🪙2 = 1"
  (is "?lhs = _")
proof -
  have "fps_deriv ?lhs = 0"
    by (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv field_simps flip: fps_const_neg)
  then have "?lhs = fps_const (?lhs $ 0)"
    unfolding fps_deriv_eq_0_iff .
  also have "… = 1"
    by (simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
  finally show ?thesis .
qed

lemma fps_sin_nth_0 [simp]: "fps_sin c $ 0 = 0"
  unfolding fps_sin_def by simp

lemma fps_sin_nth_1 [simp]: "fps_sin c $ Suc 0 = c"
  unfolding fps_sin_def by simp

lemma fps_sin_nth_add_2:
    "fps_sin c $ (n + 2) = - (c * c * fps_sin c $ n / (of_nat (n + 1) * of_nat (n + 2)))"
proof (cases n)
  case (Suc n')
  then show ?thesis
    unfolding fps_sin_def by (simp add: field_simps)
qed (auto simp: fps_sin_def)


lemma fps_cos_nth_0 [simp]: "fps_cos c $ 0 = 1"
  unfolding fps_cos_def by simp

lemma fps_cos_nth_1 [simp]: "fps_cos c $ Suc 0 = 0"
  unfolding fps_cos_def by simp

lemma fps_cos_nth_add_2:
  "fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat (n + 1) * of_nat (n + 2)))"
proof (cases n)
  case (Suc n')
  then show ?thesis
    unfolding fps_cos_def by (simp add: field_simps)
qed (auto simp: fps_cos_def)

lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
  by simp

lemma eq_fps_sin:
  assumes a0: "a $ 0 = 0"
    and a1: "a $ 1 = c"
    and a2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
  shows "fps_sin c = a"
proof (rule fps_ext)
  fix n
  show "fps_sin c $ n = a $ n"
  proof (induction n rule: nat_induct2)
    case (step n)
    then have "of_nat (n + 1) * (of_nat (n + 2) * a $ (n + 2)) =
     - (c * c * fps_sin c $ n)"
      using a2
      by (metis fps_const_mult fps_deriv_nth fps_mult_left_const_nth fps_neg_nth nat_add_1_add_1)
    with step show ?case
      by (metis (no_types, lifting) a0 add.commute add.inverse_inverse fps_sin_nth_0 fps_sin_nth_add_2 mult_divide_mult_cancel_left_if mult_minus_right nonzero_mult_div_cancel_left not_less_zero of_nat_eq_0_iff plus_1_eq_Suc zero_less_Suc)
  qed (use assms in auto)
qed

lemma eq_fps_cos:
  assumes a0: "a $ 0 = 1"
    and a1: "a $ 1 = 0"
    and a2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
  shows "fps_cos c = a"
proof (rule fps_ext)
  fix n
  show "fps_cos c $ n = a $ n"
  proof (induction n rule: nat_induct2)
    case (step n)
    then have "of_nat (n + 1) * (of_nat (n + 2) * a $ (n + 2)) =
     - (c * c * fps_cos c $ n)"
      using a2
      by (metis fps_const_mult fps_deriv_nth fps_mult_left_const_nth fps_neg_nth nat_add_1_add_1)
    with step show ?case
      by (metis (no_types, lifting) a0 add.commute add.inverse_inverse fps_cos_nth_0 fps_cos_nth_add_2 mult_divide_mult_cancel_left_if mult_minus_right nonzero_mult_div_cancel_left not_less_zero of_nat_eq_0_iff plus_1_eq_Suc zero_less_Suc)
  qed (use assms in auto)
qed

lemma fps_sin_add: "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b"
proof -
  have "fps_deriv (fps_deriv (fps_sin a * fps_cos b + fps_cos a * fps_sin b)) =
         - (fps_const (a + b) * fps_const (a + b) * (fps_sin a * fps_cos b + fps_cos a * fps_sin b))"
    by (simp flip: fps_const_neg fps_const_add fps_const_mult
        add: fps_sin_deriv fps_cos_deriv algebra_simps)
  then show ?thesis
    by (auto intro: eq_fps_sin)
qed


lemma fps_cos_add: "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b"
proof -
  have "fps_deriv
     (fps_deriv (fps_cos a * fps_cos b - fps_sin a * fps_sin b)) =
    - (fps_const (a + b) * fps_const (a + b) *
       (fps_cos a * fps_cos b - fps_sin a * fps_sin b))"
    by (simp flip: fps_const_neg fps_const_add fps_const_mult
        add: fps_sin_deriv fps_cos_deriv algebra_simps)
  then show ?thesis
    by (auto intro: eq_fps_cos)
qed

lemma fps_sin_even: "fps_sin (- c) = - fps_sin c"
  by (simp add: fps_eq_iff fps_sin_def)

lemma fps_cos_odd: "fps_cos (- c) = fps_cos c"
  by (simp add: fps_eq_iff fps_cos_def)

definition "fps_tan c = fps_sin c / fps_cos c"

lemma fps_tan_0 [simp]: "fps_tan 0 = 0"
  by (simp add: fps_tan_def)

lemma fps_tan_deriv: "fps_deriv (fps_tan c) = fps_const c / (fps_cos c)🪙2"
proof -
  have th0: "fps_cos c $ 0 ≠ 0" by (simp add: fps_cos_def)
  from this have "fps_cos c ≠ 0" by (intro notI) simp
  hence "fps_deriv (fps_tan c) =
           fps_const c * (fps_cos c^2 + fps_sin c^2) / (fps_cos c^2)"
    by (simp add: fps_tan_def fps_divide_deriv power2_eq_square algebra_simps
                  fps_sin_deriv fps_cos_deriv fps_const_neg[symmetric] div_mult_swap
             del: fps_const_neg)
  also note fps_sin_cos_sum_of_squares
  finally show ?thesis by simp
qed

text ‹Connection to @{const "fps_exp"} over the complex numbers --- Euler and de Moivre.›

lemma fps_exp_ii_sin_cos: "fps_exp (i * c) = fps_cos c + fps_const i * fps_sin c"
  (is "?l = ?r")
proof -
  have "?l $ n = ?r $ n" for n
  proof (cases "even n")
    case True
    then obtain m where m: "n = 2 * m" ..
    show ?thesis
      by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus [of "c ^ 2"])
  next
    case False
    then obtain m where m: "n = 2 * m + 1" ..
    show ?thesis
      by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
        power_mult power_minus [of "c ^ 2"])
  qed
  then show ?thesis
    by (simp add: fps_eq_iff)
qed

lemma fps_exp_minus_ii_sin_cos: "fps_exp (- (i * c)) = fps_cos c - fps_const i * fps_sin c"
  unfolding minus_mult_right fps_exp_ii_sin_cos by (simp add: fps_sin_even fps_cos_odd)

lemma fps_cos_fps_exp_ii: "fps_cos c = (fps_exp (i * c) + fps_exp (- i * c)) / fps_const 2"
proof -
  have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2"
    by (simp add: numeral_fps_const)
  show ?thesis
    unfolding fps_exp_ii_sin_cos minus_mult_commute
    by (simp add: fps_sin_even fps_cos_odd numeral_fps_const fps_divide_unit fps_const_inverse th)
qed

lemma fps_sin_fps_exp_ii: "fps_sin c = (fps_exp (i * c) - fps_exp (- i * c)) / fps_const (2*i)"
proof -
  have th: "fps_const i * fps_sin c + fps_const i * fps_sin c = fps_sin c * fps_const (2 * i)"
    by (simp add: fps_eq_iff numeral_fps_const)
  show ?thesis
    unfolding fps_exp_ii_sin_cos minus_mult_commute
    by (simp add: fps_sin_even fps_cos_odd fps_divide_unit fps_const_inverse th)
qed

lemma fps_tan_fps_exp_ii:
  "fps_tan c = (fps_exp (i * c) - fps_exp (- i * c)) /
      (fps_const i * (fps_exp (i * c) + fps_exp (- i * c)))"
  unfolding fps_tan_def fps_sin_fps_exp_ii fps_cos_fps_exp_ii
  by (simp add: fps_divide_unit fps_inverse_mult fps_const_inverse)

lemma fps_demoivre:
  "(fps_cos a + fps_const i * fps_sin a)^n =
    fps_cos (of_nat n * a) + fps_const i * fps_sin (of_nat n * a)"
  unfolding fps_exp_ii_sin_cos[symmetric] fps_exp_power_mult
  by (simp add: ac_simps)


subsection ‹Hypergeometric series›

definition "fps_hypergeo as bs (c::'a::field_char_0) =
  Abs_fps (λn. (foldl (λr a. r* pochhammer a n) 1 as * c^n) /
    (foldl (λr b. r * pochhammer b n) 1 bs * of_nat (fact n)))"

lemma fps_hypergeo_nth[simp]: "fps_hypergeo as bs c $ n =
  (foldl (λr a. r* pochhammer a n) 1 as * c^n) /
    (foldl (λr b. r * pochhammer b n) 1 bs * of_nat (fact n))"
  by (simp add: fps_hypergeo_def)

lemma foldl_mult_start:
  fixes v :: "'a::comm_ring_1"
  shows "foldl (λr x. r * f x) v as * x = foldl (λr x. r * f x) (v * x) as "
  by (induct as arbitrary: x v) (auto simp add: algebra_simps)

lemma foldr_mult_foldl:
  fixes v :: "'a::comm_ring_1"
  shows "foldr (λx r. r * f x) as v = foldl (λr x. r * f x) v as"
  by (induct as arbitrary: v) (simp_all add: foldl_mult_start)

lemma fps_hypergeo_nth_alt:
  "fps_hypergeo as bs c $ n = foldr (λa r. r * pochhammer a n) as (c ^ n) /
    foldr (λb r. r * pochhammer b n) bs (of_nat (fact n))"
  by (simp add: foldl_mult_start foldr_mult_foldl)

lemma fps_hypergeo_fps_exp[simp]: "fps_hypergeo [] [] c = fps_exp c"
  by (simp add: fps_eq_iff)

lemma fps_hypergeo_1_0[simp]: "fps_hypergeo [1] [] c = 1/(1 - fps_const c * fps_X)"
proof -
  let ?a = "(Abs_fps (λn. 1)) oo (fps_const c * fps_X)"
  have th0: "(fps_const c * fps_X) $ 0 = 0" by simp
  show ?thesis unfolding gp[OF th0, symmetric]
    by (simp add: fps_eq_iff pochhammer_fact[symmetric]
      fps_compose_nth power_mult_distrib if_distrib cong del: if_weak_cong)
qed

lemma fps_hypergeo_B[simp]: "fps_hypergeo [-a] [] (- 1) = fps_binomial a"
  by (simp add: fps_eq_iff gbinomial_pochhammer algebra_simps)

lemma fps_hypergeo_0[simp]: "fps_hypergeo as bs c $ 0 = 1"
proof -
  have "foldl (λ(r::'a) (a::'a). r) 1 as = 1" for as
    by (induction as) auto
  then show ?thesis
    by auto
qed

lemma foldl_prod_prod:
  "foldl (λ(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (λr x. r * g x) w as =
    foldl (λr x. r * f x * g x) (v * w) as"
  by (induct as arbitrary: v w) (simp_all add: algebra_simps)


lemma fps_hypergeo_rec:
  "fps_hypergeo as bs c $ Suc n = ((foldl (λr a. r* (a + of_nat n)) c as) /
    (foldl (λr b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * fps_hypergeo as bs c $ n"
  apply (simp add: foldl_mult_start del: of_nat_Suc of_nat_add fact_Suc)
  unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc
  by (simp add: algebra_simps)

lemma fps_XD_nth[simp]: "fps_XD a $ n = of_nat n * a$n"
  by (simp add: fps_XD_def)

lemma fps_XD_0th[simp]: "fps_XD a $ 0 = 0"
  by simp
lemma fps_XD_Suc[simp]:" fps_XD a $ Suc n = of_nat (Suc n) * a $ Suc n"
  by simp

definition "fps_XDp c a = fps_XD a + fps_const c * a"

lemma fps_XDp_nth[simp]: "fps_XDp c a $ n = (c + of_nat n) * a$n"
  by (simp add: fps_XDp_def algebra_simps)

lemma fps_XDp_commute: "fps_XDp b ∘ fps_XDp (c::'a::comm_ring_1) = fps_XDp c ∘ fps_XDp b"
  by (simp add: fps_XDp_def fun_eq_iff fps_eq_iff algebra_simps)

lemma fps_XDp0 [simp]: "fps_XDp 0 = fps_XD"
  by (simp add: fun_eq_iff fps_eq_iff)

lemma fps_XDp_fps_integral [simp]:
  fixes  a :: "'a::{division_ring,ring_char_0} fps"
  shows  "fps_XDp 0 (fps_integral a c) = fps_X * a"
  using  fps_deriv_fps_integral[of a c]
  by     (simp add: fps_XD_def)

lemma fps_hypergeo_minus_nat:
  "fps_hypergeo [- of_nat n] [- of_nat (n + m)] (c::'a::field_char_0) $ k =
    (if k ≤ n then
      pochhammer (- of_nat n) k * c ^ k / (pochhammer (- of_nat (n + m)) k * of_nat (fact k))
     else 0)"
  "fps_hypergeo [- of_nat m] [- of_nat (m + n)] (c::'a::field_char_0) $ k =
    (if k ≤ m then
      pochhammer (- of_nat m) k * c ^ k / (pochhammer (- of_nat (m + n)) k * of_nat (fact k))
     else 0)"
  by (simp_all add: pochhammer_eq_0_iff)

lemma pochhammer_rec_if: "pochhammer a n = (if n = 0 then 1 else a * pochhammer (a + 1) (n - 1))"
  by (cases n) (simp_all add: pochhammer_rec)

lemma fps_XDp_foldr_nth [simp]: "foldr (λc r. fps_XDp c ∘ r) cs (λc. fps_XDp c a) c0 $ n =
    foldr (λc r. (c + of_nat n) * r) cs (c0 + of_nat n) * a$n"
  by (induct cs arbitrary: c0) (simp_all add: algebra_simps)

lemma genric_fps_XDp_foldr_nth:
  assumes f: "∀n c a. f c a $ n = (of_nat n + k c) * a$n"
  shows "foldr (λc r. f c ∘ r) cs (λc. g c a) c0 $ n =
    foldr (λc r. (k c + of_nat n) * r) cs (g c0 a $ n)"
  by (induct cs arbitrary: c0) (simp_all add: algebra_simps f)

lemma dist_less_imp_nth_equal:
  assumes "dist f g < inverse (2 ^ i)"
    and"j ≤ i"
  shows "f $ j = g $ j"
proof (rule ccontr)
  assume "f $ j ≠ g $ j"
  hence "f ≠ g" by auto
  with assms have "i < subdegree (f - g)"
    by (simp add: if_split_asm dist_fps_def)
  also have "… ≤ j"
    using ‹f $ j ≠ g $ j› by (intro subdegree_leI) simp_all
  finally show False using ‹j ≤ i› by simp
qed

lemma nth_equal_imp_dist_less:
  assumes "∧j. j ≤ i ==> f $ j = g $ j"
  shows "dist f g < inverse (2 ^ i)"
proof (cases "f = g")
  case True
  then show ?thesis by simp
next
  case False
  with assms have "dist f g = inverse (2 ^ subdegree (f - g))"
    by (simp add: if_split_asm dist_fps_def)
  moreover
  from assms and False have "i < subdegree (f - g)"
    by (intro subdegree_greaterI) simp_all
  ultimately show ?thesis by simp
qed

lemma dist_less_eq_nth_equal: "dist f g < inverse (2 ^ i) ⟷ (∀j ≤ i. f $ j = g $ j)"
  using dist_less_imp_nth_equal nth_equal_imp_dist_less by blast

instance fps :: (comm_ring_1) complete_space
proof
  fix fps_X :: "nat ==> 'a fps"
  assume "Cauchy fps_X"
  obtain M where M: "∀i. ∀m ≥ M i. ∀j ≤ i. fps_X (M i) $ j = fps_X m $ j"
  proof -
    have "∃M. ∀m ≥ M. ∀j≤i. fps_X M $ j = fps_X m $ j" for i
    proof -
      have "0 < inverse ((2::real)^i)" by simp
      from metric_CauchyD[OF ‹Cauchy fps_X› this] dist_less_imp_nth_equal
      show ?thesis by blast
    qed
    then show ?thesis using that by metis
  qed

  show "convergent fps_X"
  proof (rule convergentI)
    show "fps_X <---- Abs_fps (λi. fps_X (M i) $ i)"
      unfolding tendsto_iff
    proof safe
      fix e::real assume e: "0 < e"
      have "(λn. inverse (2 ^ n) :: real) <---- 0" by (rule LIMSEQ_inverse_realpow_zero) simp_all
      from this and e have "eventually (λi. inverse (2 ^ i) < e) sequentially"
        by (rule order_tendstoD)
      then obtain i where "inverse (2 ^ i) < e"
        by (auto simp: eventually_sequentially)
      have "eventually (λx. M i ≤ x) sequentially"
        by (auto simp: eventually_sequentially)
      then show "eventually (λx. dist (fps_X x) (Abs_fps (λi. fps_X (M i) $ i)) < e) sequentially"
      proof eventually_elim
        fix x
        assume x: "M i ≤ x"
        have "fps_X (M i) $ j = fps_X (M j) $ j" if "j ≤ i" for j
          using M that by (metis nat_le_linear)
        with x have "dist (fps_X x) (Abs_fps (λj. fps_X (M j) $ j)) < inverse (2 ^ i)"
          using M by (force simp: dist_less_eq_nth_equal)
        also note ‹inverse (2 ^ i) 🚫›
        finally show "dist (fps_X x) (Abs_fps (λj. fps_X (M j) $ j)) < e" .
      qed
    qed
  qed
qed

(* TODO: Figure out better notation for this thing *)
bundle fps_syntax
begin
notation fps_nth (infixl ‹$› 75)
end

unbundle no fps_syntax

end