Theory Transcendental

(*  Title:      HOL/Transcendental.thy
    Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
    Author:     Lawrence C Paulson
    Author:     Jeremy Avigad
*)

section Power Series, Transcendental Functions etc.

theory Transcendental
imports Series Deriv NthRoot
begin

text A theorem about the factcorial function on the reals.

lemma square_fact_le_2_fact: "fact n * fact n  (fact (2 * n) :: real)"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "(fact (Suc n)) * (fact (Suc n)) = of_nat (Suc n) * of_nat (Suc n) * (fact n * fact n :: real)"
    by (simp add: field_simps)
  also have "  of_nat (Suc n) * of_nat (Suc n) * fact (2 * n)"
    by (rule mult_left_mono [OF Suc]) simp
  also have "  of_nat (Suc (Suc (2 * n))) * of_nat (Suc (2 * n)) * fact (2 * n)"
    by (rule mult_right_mono)+ (auto simp: field_simps)
  also have " = fact (2 * Suc n)" by (simp add: field_simps)
  finally show ?case .
qed

lemma fact_in_Reals: "fact n  "
  by (induction n) auto

lemma of_real_fact [simp]: "of_real (fact n) = fact n"
  by (metis of_nat_fact of_real_of_nat_eq)

lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
  by (simp add: pochhammer_prod)

lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n"
proof -
  have "(fact n :: 'a) = of_real (fact n)"
    by simp
  also have "norm  = fact n"
    by (subst norm_of_real) simp
  finally show ?thesis .
qed

lemma root_test_convergence:
  fixes f :: "nat  'a::banach"
  assumes f: "(λn. root n (norm (f n)))  x" ― ‹could be weakened to lim sup
    and "x < 1"
  shows "summable f"
proof -
  have "0  x"
    by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
  from x < 1 obtain z where z: "x < z" "z < 1"
    by (metis dense)
  from f x < z have "eventually (λn. root n (norm (f n)) < z) sequentially"
    by (rule order_tendstoD)
  then have "eventually (λn. norm (f n)  z^n) sequentially"
    using eventually_ge_at_top
  proof eventually_elim
    fix n
    assume less: "root n (norm (f n)) < z" and n: "1  n"
    from power_strict_mono[OF less, of n] n show "norm (f n)  z ^ n"
      by simp
  qed
  then show "summable f"
    unfolding eventually_sequentially
    using z 0  x by (auto intro!: summable_comparison_test[OF _  summable_geometric])
qed

subsection More facts about binomial coefficients

text 
  These facts could have been proven before, but having real numbers
  makes the proofs a lot easier.


lemma central_binomial_odd:
  "odd n  n choose (Suc (n div 2)) = n choose (n div 2)"
proof -
  assume "odd n"
  hence "Suc (n div 2)  n" by presburger
  hence "n choose (Suc (n div 2)) = n choose (n - Suc (n div 2))"
    by (rule binomial_symmetric)
  also from odd n have "n - Suc (n div 2) = n div 2" by presburger
  finally show ?thesis .
qed

lemma binomial_less_binomial_Suc:
  assumes k: "k < n div 2"
  shows   "n choose k < n choose (Suc k)"
proof -
  from k have k': "k  n" "Suc k  n" by simp_all
  from k' have "real (n choose k) = fact n / (fact k * fact (n - k))"
    by (simp add: binomial_fact)
  also from k' have "n - k = Suc (n - Suc k)" by simp
  also from k' have "fact  = (real n - real k) * fact (n - Suc k)"
    by (subst fact_Suc) (simp_all add: of_nat_diff)
  also from k have "fact k = fact (Suc k) / (real k + 1)" by (simp add: field_simps)
  also have "fact n / (fact (Suc k) / (real k + 1) * ((real n - real k) * fact (n - Suc k))) =
               (n choose (Suc k)) * ((real k + 1) / (real n - real k))"
    using k by (simp add: field_split_simps binomial_fact)
  also from assms have "(real k + 1) / (real n - real k) < 1" by simp
  finally show ?thesis using k by (simp add: mult_less_cancel_left)
qed

lemma binomial_strict_mono:
  assumes "k < k'" "2*k'  n"
  shows   "n choose k < n choose k'"
proof -
  from assms have "k  k' - 1" by simp
  thus ?thesis
  proof (induction rule: inc_induct)
    case base
    with assms binomial_less_binomial_Suc[of "k' - 1" n]
      show ?case by simp
  next
    case (step k)
    from step.prems step.hyps assms have "n choose k < n choose (Suc k)"
      by (intro binomial_less_binomial_Suc) simp_all
    also have " < n choose k'" by (rule step.IH)
    finally show ?case .
  qed
qed

lemma binomial_mono:
  assumes "k  k'" "2*k'  n"
  shows   "n choose k  n choose k'"
  using assms binomial_strict_mono[of k k' n] by (cases "k = k'") simp_all

lemma binomial_strict_antimono:
  assumes "k < k'" "2 * k  n" "k'  n"
  shows   "n choose k > n choose k'"
proof -
  from assms have "n choose (n - k) > n choose (n - k')"
    by (intro binomial_strict_mono) (simp_all add: algebra_simps)
  with assms show ?thesis by (simp add: binomial_symmetric [symmetric])
qed

lemma binomial_antimono:
  assumes "k  k'" "k  n div 2" "k'  n"
  shows   "n choose k  n choose k'"
proof (cases "k = k'")
  case False
  note not_eq = False
  show ?thesis
  proof (cases "k = n div 2  odd n")
    case False
    with assms(2) have "2*k  n" by presburger
    with not_eq assms binomial_strict_antimono[of k k' n]
      show ?thesis by simp
  next
    case True
    have "n choose k'  n choose (Suc (n div 2))"
    proof (cases "k' = Suc (n div 2)")
      case False
      with assms True not_eq have "Suc (n div 2) < k'" by simp
      with assms binomial_strict_antimono[of "Suc (n div 2)" k' n] True
        show ?thesis by auto
    qed simp_all
    also from True have " = n choose k" by (simp add: central_binomial_odd)
    finally show ?thesis .
  qed
qed simp_all

lemma binomial_maximum: "n choose k  n choose (n div 2)"
proof -
  have "k  n div 2  2*k  n" by linarith
  consider "2*k  n" | "2*k  n" "k  n" | "k > n" by linarith
  thus ?thesis
  proof cases
    case 1
    thus ?thesis by (intro binomial_mono) linarith+
  next
    case 2
    thus ?thesis by (intro binomial_antimono) simp_all
  qed (simp_all add: binomial_eq_0)
qed

lemma binomial_maximum': "(2*n) choose k  (2*n) choose n"
  using binomial_maximum[of "2*n"] by simp

lemma central_binomial_lower_bound:
  assumes "n > 0"
  shows   "4^n / (2*real n)  real ((2*n) choose n)"
proof -
  from binomial[of 1 1 "2*n"]
    have "4 ^ n = (k2*n. (2*n) choose k)"
    by (simp add: power_mult power2_eq_square One_nat_def [symmetric] del: One_nat_def)
  also have "{..2*n} = {0<..<2*n}  {0,2*n}" by auto
  also have "(k. (2*n) choose k) =
             (k{0<..<2*n}. (2*n) choose k) + (k{0,2*n}. (2*n) choose k)"
    by (subst sum.union_disjoint) auto
  also have "(k{0,2*n}. (2*n) choose k)  (k1. (n choose k)2)"
    by (cases n) simp_all
  also from assms have "  (kn. (n choose k)2)"
    by (intro sum_mono2) auto
  also have " = (2*n) choose n" by (rule choose_square_sum)
  also have "(k{0<..<2*n}. (2*n) choose k)  (k{0<..<2*n}. (2*n) choose n)"
    by (intro sum_mono binomial_maximum')
  also have " = card {0<..<2*n} * ((2*n) choose n)" by simp
  also have "card {0<..<2*n}  2*n - 1" by (cases n) simp_all
  also have "(2 * n - 1) * (2 * n choose n) + (2 * n choose n) = ((2*n) choose n) * (2*n)"
    using assms by (simp add: algebra_simps)
  finally have "4 ^ n  (2 * n choose n) * (2 * n)" by simp_all
  hence "real (4 ^ n)  real ((2 * n choose n) * (2 * n))"
    by (subst of_nat_le_iff)
  with assms show ?thesis by (simp add: field_simps)
qed


subsection Properties of Power Series

lemma powser_zero [simp]: "(n. f n * 0 ^ n) = f 0"
  for f :: "nat  'a::real_normed_algebra_1"
proof -
  have "(n<1. f n * 0 ^ n) = (n. f n * 0 ^ n)"
    by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
  then show ?thesis by simp
qed

lemma powser_sums_zero: "(λn. a n * 0^n) sums a 0"
  for a :: "nat  'a::real_normed_div_algebra"
  using sums_finite [of "{0}" "λn. a n * 0 ^ n"]
  by simp

lemma powser_sums_zero_iff [simp]: "(λn. a n * 0^n) sums x  a 0 = x"
  for a :: "nat  'a::real_normed_div_algebra"
  using powser_sums_zero sums_unique2 by blast

text 
  Power series has a circle or radius of convergence: if it sums for x›,
  then it sums absolutely for z› with term¦z¦ < ¦x¦.

lemma powser_insidea:
  fixes x z :: "'a::real_normed_div_algebra"
  assumes 1: "summable (λn. f n * x^n)"
    and 2: "norm z < norm x"
  shows "summable (λn. norm (f n * z ^ n))"
proof -
  from 2 have x_neq_0: "x  0" by clarsimp
  from 1 have "(λn. f n * x^n)  0"
    by (rule summable_LIMSEQ_zero)
  then have "convergent (λn. f n * x^n)"
    by (rule convergentI)
  then have "Cauchy (λn. f n * x^n)"
    by (rule convergent_Cauchy)
  then have "Bseq (λn. f n * x^n)"
    by (rule Cauchy_Bseq)
  then obtain K where 3: "0 < K" and 4: "n. norm (f n * x^n)  K"
    by (auto simp: Bseq_def)
  have "N. nN. norm (norm (f n * z ^ n))  K * norm (z ^ n) * inverse (norm (x^n))"
  proof (intro exI allI impI)
    fix n :: nat
    assume "0  n"
    have "norm (norm (f n * z ^ n)) * norm (x^n) =
          norm (f n * x^n) * norm (z ^ n)"
      by (simp add: norm_mult abs_mult)
    also have "  K * norm (z ^ n)"
      by (simp only: mult_right_mono 4 norm_ge_zero)
    also have " = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
      by (simp add: x_neq_0)
    also have " = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
      by (simp only: mult.assoc)
    finally show "norm (norm (f n * z ^ n))  K * norm (z ^ n) * inverse (norm (x^n))"
      by (simp add: mult_le_cancel_right x_neq_0)
  qed
  moreover have "summable (λn. K * norm (z ^ n) * inverse (norm (x^n)))"
  proof -
    from 2 have "norm (norm (z * inverse x)) < 1"
      using x_neq_0
      by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
    then have "summable (λn. norm (z * inverse x) ^ n)"
      by (rule summable_geometric)
    then have "summable (λn. K * norm (z * inverse x) ^ n)"
      by (rule summable_mult)
    then show "summable (λn. K * norm (z ^ n) * inverse (norm (x^n)))"
      using x_neq_0
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
          power_inverse norm_power mult.assoc)
  qed
  ultimately show "summable (λn. norm (f n * z ^ n))"
    by (rule summable_comparison_test)
qed

lemma powser_inside:
  fixes f :: "nat  'a::{real_normed_div_algebra,banach}"
  shows
    "summable (λn. f n * (x^n))  norm z < norm x 
      summable (λn. f n * (z ^ n))"
  by (rule powser_insidea [THEN summable_norm_cancel])

lemma powser_times_n_limit_0:
  fixes x :: "'a::{real_normed_div_algebra,banach}"
  assumes "norm x < 1"
    shows "(λn. of_nat n * x ^ n)  0"
proof -
  have "norm x / (1 - norm x)  0"
    using assms by (auto simp: field_split_simps)
  moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
    using ex_le_of_int by (meson ex_less_of_int)
  ultimately have N0: "N>0"
    by auto
  then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
    using N assms by (auto simp: field_simps)
  have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) 
      real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N  int n" for n :: nat
  proof -
    from that have "real_of_int N * real_of_nat (Suc n)  real_of_nat n * real_of_int (1 + N)"
      by (simp add: algebra_simps)
    then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n)) 
        (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
      using N0 mult_mono by fastforce
    then show ?thesis
      by (simp add: algebra_simps)
  qed
  show ?thesis using *
    by (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
      (simp add: N0 norm_mult field_simps ** del: of_nat_Suc of_int_add)
qed

corollary lim_n_over_pown:
  fixes x :: "'a::{real_normed_field,banach}"
  shows "1 < norm x  ((λn. of_nat n / x^n)  0) sequentially"
  using powser_times_n_limit_0 [of "inverse x"]
  by (simp add: norm_divide field_split_simps)

lemma sum_split_even_odd:
  fixes f :: "nat  real"
  shows "(i<2 * n. if even i then f i else g i) = (i<n. f (2 * i)) + (i<n. g (2 * i + 1))"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "(i<2 * Suc n. if even i then f i else g i) =
    (i<n. f (2 * i)) + (i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
    using Suc.hyps unfolding One_nat_def by auto
  also have " = (i<Suc n. f (2 * i)) + (i<Suc n. g (2 * i + 1))"
    by auto
  finally show ?case .
qed

lemma sums_if':
  fixes g :: "nat  real"
  assumes "g sums x"
  shows "(λ n. if even n then 0 else g ((n - 1) div 2)) sums x"
  unfolding sums_def
proof (rule LIMSEQ_I)
  fix r :: real
  assume "0 < r"
  from g sums x[unfolded sums_def, THEN LIMSEQ_D, OF this]
  obtain no where no_eq: "n. n  no  (norm (sum g {..<n} - x) < r)"
    by blast

  let ?SUM = "λ m. i<m. if even i then 0 else g ((i - 1) div 2)"
  have "(norm (?SUM m - x) < r)" if "m  2 * no" for m
  proof -
    from that have "m div 2  no" by auto
    have sum_eq: "?SUM (2 * (m div 2)) = sum g {..< m div 2}"
      using sum_split_even_odd by auto
    then have "(norm (?SUM (2 * (m div 2)) - x) < r)"
      using no_eq unfolding sum_eq using m div 2  no by auto
    moreover
    have "?SUM (2 * (m div 2)) = ?SUM m"
    proof (cases "even m")
      case True
      then show ?thesis
        by (auto simp: even_two_times_div_two)
    next
      case False
      then have eq: "Suc (2 * (m div 2)) = m" by simp
      then have "even (2 * (m div 2))" using odd m by auto
      have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
      also have " = ?SUM (2 * (m div 2))" using even (2 * (m div 2)) by auto
      finally show ?thesis by auto
    qed
    ultimately show ?thesis by auto
  qed
  then show "no.  m  no. norm (?SUM m - x) < r"
    by blast
qed

lemma sums_if:
  fixes g :: "nat  real"
  assumes "g sums x" and "f sums y"
  shows "(λ n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
proof -
  let ?s = "λ n. if even n then 0 else f ((n - 1) div 2)"
  have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
    for B T E
    by (cases B) auto
  have g_sums: "(λ n. if even n then 0 else g ((n - 1) div 2)) sums x"
    using sums_if'[OF g sums x] .
  have if_eq: "B T E. (if ¬ B then T else E) = (if B then E else T)"
    by auto
  have "?s sums y" using sums_if'[OF f sums y] .
  from this[unfolded sums_def, THEN LIMSEQ_Suc]
  have "(λn. if even n then f (n div 2) else 0) sums y"
    by (simp add: lessThan_Suc_eq_insert_0 sum.atLeast1_atMost_eq image_Suc_lessThan
        if_eq sums_def cong del: if_weak_cong)
  from sums_add[OF g_sums this] show ?thesis
    by (simp only: if_sum)
qed

subsection Alternating series test / Leibniz formula
(* FIXME: generalise these results from the reals via type classes? *)

lemma sums_alternating_upper_lower:
  fixes a :: "nat  real"
  assumes mono: "n. a (Suc n)  a n"
    and a_pos: "n. 0  a n"
    and "a  0"
  shows "l. ((n. (i<2*n. (- 1)^i*a i)  l)  (λ n. i<2*n. (- 1)^i*a i)  l) 
             ((n. l  (i<2*n + 1. (- 1)^i*a i))  (λ n. i<2*n + 1. (- 1)^i*a i)  l)"
  (is "l. ((n. ?f n  l)  _)  ((n. l  ?g n)  _)")
proof (rule nested_sequence_unique)
  have fg_diff: "n. ?f n - ?g n = - a (2 * n)" by auto

  show "n. ?f n  ?f (Suc n)"
  proof
    show "?f n  ?f (Suc n)" for n
      using mono[of "2*n"] by auto
  qed
  show "n. ?g (Suc n)  ?g n"
  proof
    show "?g (Suc n)  ?g n" for n
      using mono[of "Suc (2*n)"] by auto
  qed
  show "n. ?f n  ?g n"
  proof
    show "?f n  ?g n" for n
      using fg_diff a_pos by auto
  qed
  show "(λn. ?f n - ?g n)  0"
    unfolding fg_diff
  proof (rule LIMSEQ_I)
    fix r :: real
    assume "0 < r"
    with a  0[THEN LIMSEQ_D] obtain N where " n. n  N  norm (a n - 0) < r"
      by auto
    then have "n  N. norm (- a (2 * n) - 0) < r"
      by auto
    then show "N. n  N. norm (- a (2 * n) - 0) < r"
      by auto
  qed
qed

lemma summable_Leibniz':
  fixes a :: "nat  real"
  assumes a_zero: "a  0"
    and a_pos: "n. 0  a n"
    and a_monotone: "n. a (Suc n)  a n"
  shows summable: "summable (λ n. (-1)^n * a n)"
    and "n. (i<2*n. (-1)^i*a i)  (i. (-1)^i*a i)"
    and "(λn. i<2*n. (-1)^i*a i)  (i. (-1)^i*a i)"
    and "n. (i. (-1)^i*a i)  (i<2*n+1. (-1)^i*a i)"
    and "(λn. i<2*n+1. (-1)^i*a i)  (i. (-1)^i*a i)"
proof -
  let ?S = "λn. (-1)^n * a n"
  let ?P = "λn. i<n. ?S i"
  let ?f = "λn. ?P (2 * n)"
  let ?g = "λn. ?P (2 * n + 1)"
  obtain l :: real
    where below_l: " n. ?f n  l"
      and "?f  l"
      and above_l: " n. l  ?g n"
      and "?g  l"
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast

  let ?Sa = "λm. n<m. ?S n"
  have "?Sa  l"
  proof (rule LIMSEQ_I)
    fix r :: real
    assume "0 < r"
    with ?f  l[THEN LIMSEQ_D]
    obtain f_no where f: "n. n  f_no  norm (?f n - l) < r"
      by auto
    from 0 < r ?g  l[THEN LIMSEQ_D]
    obtain g_no where g: "n. n  g_no  norm (?g n - l) < r"
      by auto
    have "norm (?Sa n - l) < r" if "n  (max (2 * f_no) (2 * g_no))" for n
    proof -
      from that have "n  2 * f_no" and "n  2 * g_no" by auto
      show ?thesis
      proof (cases "even n")
        case True
        then have n_eq: "2 * (n div 2) = n"
          by (simp add: even_two_times_div_two)
        with n  2 * f_no have "n div 2  f_no"
          by auto
        from f[OF this] show ?thesis
          unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
      next
        case False
        then have "even (n - 1)" by simp
        then have n_eq: "2 * ((n - 1) div 2) = n - 1"
          by (simp add: even_two_times_div_two)
        then have range_eq: "n - 1 + 1 = n"
          using odd_pos[OF False] by auto
        from n_eq n  2 * g_no have "(n - 1) div 2  g_no"
          by auto
        from g[OF this] show ?thesis
          by (simp only: n_eq range_eq)
      qed
    qed
    then show "no. n  no. norm (?Sa n - l) < r" by blast
  qed
  then have sums_l: "(λi. (-1)^i * a i) sums l"
    by (simp only: sums_def)
  then show "summable ?S"
    by (auto simp: summable_def)

  have "l = suminf ?S" by (rule sums_unique[OF sums_l])

  fix n
  show "suminf ?S  ?g n"
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
  show "?f n  suminf ?S"
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
  show "?g  suminf ?S"
    using ?g  l l = suminf ?S by auto
  show "?f  suminf ?S"
    using ?f  l l = suminf ?S by auto
qed

theorem summable_Leibniz:
  fixes a :: "nat  real"
  assumes a_zero: "a  0"
    and "monoseq a"
  shows "summable (λ n. (-1)^n * a n)" (is "?summable")
    and "0 < a 0 
      (n. (i. (- 1)^i*a i)  { i<2*n. (- 1)^i * a i .. i<2*n+1. (- 1)^i * a i})" (is "?pos")
    and "a 0 < 0 
      (n. (i. (- 1)^i*a i)  { i<2*n+1. (- 1)^i * a i .. i<2*n. (- 1)^i * a i})" (is "?neg")
    and "(λn. i<2*n. (- 1)^i*a i)  (i. (- 1)^i*a i)" (is "?f")
    and "(λn. i<2*n+1. (- 1)^i*a i)  (i. (- 1)^i*a i)" (is "?g")
proof -
  have "?summable  ?pos  ?neg  ?f  ?g"
  proof (cases "(n. 0  a n)  (m. nm. a n  a m)")
    case True
    then have ord: "n m. m  n  a n  a m"
      and ge0: "n. 0  a n"
      by auto
    have mono: "a (Suc n)  a n" for n
      using ord[where n="Suc n" and m=n] by auto
    note leibniz = summable_Leibniz'[OF a  0 ge0]
    from leibniz[OF mono]
    show ?thesis using 0  a 0 by auto
  next
    let ?a = "λn. - a n"
    case False
    with monoseq_le[OF monoseq a a  0]
    have "( n. a n  0)  (m. nm. a m  a n)" by auto
    then have ord: "n m. m  n  ?a n  ?a m" and ge0: " n. 0  ?a n"
      by auto
    have monotone: "?a (Suc n)  ?a n" for n
      using ord[where n="Suc n" and m=n] by auto
    note leibniz =
      summable_Leibniz'[OF _ ge0, of "λx. x",
        OF tendsto_minus[OF a  0, unfolded minus_zero] monotone]
    have "summable (λ n. (-1)^n * ?a n)"
      using leibniz(1) by auto
    then obtain l where "(λ n. (-1)^n * ?a n) sums l"
      unfolding summable_def by auto
    from this[THEN sums_minus] have "(λ n. (-1)^n * a n) sums -l"
      by auto
    then have ?summable by (auto simp: summable_def)
    moreover
    have "¦- a - - b¦ = ¦a - b¦" for a b :: real
      unfolding minus_diff_minus by auto

    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
    have move_minus: "(n. - ((- 1) ^ n * a n)) = - (n. (- 1) ^ n * a n)"
      by auto

    have ?pos using 0  ?a 0 by auto
    moreover have ?neg
      using leibniz(2,4)
      unfolding mult_minus_right sum_negf move_minus neg_le_iff_le
      by auto
    moreover have ?f and ?g
      using leibniz(3,5)[unfolded mult_minus_right sum_negf move_minus, THEN tendsto_minus_cancel]
      by auto
    ultimately show ?thesis by auto
  qed
  then show ?summable and ?pos and ?neg and ?f and ?g
    by safe
qed


subsection Term-by-Term Differentiability of Power Series

definition diffs :: "(nat  'a::ring_1)  nat  'a"
  where "diffs c = (λn. of_nat (Suc n) * c (Suc n))"

text Lemma about distributing negation over it.
lemma diffs_minus: "diffs (λn. - c n) = (λn. - diffs c n)"
  by (simp add: diffs_def)

lemma diffs_equiv:
  fixes x :: "'a::{real_normed_vector,ring_1}"
  shows "summable (λn. diffs c n * x^n) 
    (λn. of_nat n * c n * x^(n - Suc 0)) sums (n. diffs c n * x^n)"
  unfolding diffs_def
  by (simp add: summable_sums sums_Suc_imp)

lemma lemma_termdiff1:
  fixes z :: "'a :: {monoid_mult,comm_ring}"
  shows "(p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
    (p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
  by (auto simp: algebra_simps power_add [symmetric])

lemma sumr_diff_mult_const2: "sum f {..<n} - of_nat n * r = (i<n. f i - r)"
  for r :: "'a::ring_1"
  by (simp add: sum_subtractf)

lemma lemma_termdiff2:
  fixes h :: "'a::field"
  assumes h: "h  0"
  shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
         h * (p< n - Suc 0. q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
    (is "?lhs = ?rhs")
proof (cases n)
  case (Suc m)
  have 0: "x k. (n<Suc k. h * (z ^ x * (z ^ (k - n) * (h + z) ^ n))) =
                 (j<Suc k.  h * ((h + z) ^ j * z ^ (x + k - j)))"
    by (auto simp add: power_add [symmetric] mult.commute intro: sum.cong)
  have *: "(i<m. z ^ i * ((z + h) ^ (m - i) - z ^ (m - i))) =
           (i<m. j<m - i. h * ((z + h) ^ j * z ^ (m - Suc j)))"
    by (force simp add: less_iff_Suc_add sum_distrib_left diff_power_eq_sum ac_simps 0
        simp del: sum.lessThan_Suc power_Suc intro: sum.cong)
  have "h * ?lhs = (z + h) ^ n - z ^ n - h * of_nat n * z ^ (n - Suc 0)"
    by (simp add: right_diff_distrib diff_divide_distrib h mult.assoc [symmetric])
  also have "... = h * ((p<Suc m. (z + h) ^ p * z ^ (m - p)) - of_nat (Suc m) * z ^ m)"
    by (simp add: Suc diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc
        del: power_Suc sum.lessThan_Suc of_nat_Suc)
  also have "... = h * ((p<Suc m. (z + h) ^ (m - p) * z ^ p) - of_nat (Suc m) * z ^ m)"
    by (subst sum.nat_diff_reindex[symmetric]) simp
  also have "... = h * (i<Suc m. (z + h) ^ (m - i) * z ^ i - z ^ m)"
    by (simp add: sum_subtractf)
  also have "... = h * ?rhs"
    by (simp add: lemma_termdiff1 sum_distrib_left Suc *)
  finally have "h * ?lhs = h * ?rhs" .
  then show ?thesis
    by (simp add: h)
qed auto


lemma real_sum_nat_ivl_bounded2:
  fixes K :: "'a::linordered_semidom"
  assumes f: "p::nat. p < n  f p  K" and K: "0  K"
  shows "sum f {..<n-k}  of_nat n * K"
proof -
  have "sum f {..<n-k}  (i<n - k. K)"
    by (rule sum_mono [OF f]) auto
  also have "...  of_nat n * K"
    by (auto simp: mult_right_mono K)
  finally show ?thesis .
qed

lemma lemma_termdiff3:
  fixes h z :: "'a::real_normed_field"
  assumes 1: "h  0"
    and 2: "norm z  K"
    and 3: "norm (z + h)  K"
  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) 
    of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
proof -
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
    norm (p<n - Suc 0. q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
    by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
  also have "  of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
  proof (rule mult_right_mono [OF _ norm_ge_zero])
    from norm_ge_zero 2 have K: "0  K"
      by (rule order_trans)
    have le_Kn: "norm ((z + h) ^ i * z ^ j)  K ^ n" if "i + j = n" for i j n
    proof -
      have "norm (z + h) ^ i * norm z ^ j  K ^ i * K ^ j"
        by (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
      also have "... = K^n"
        by (metis power_add that)
      finally show ?thesis
        by (simp add: norm_mult norm_power) 
    qed
    then have "p q.
       p < n; q < n - Suc 0  norm ((z + h) ^ q * z ^ (n - 2 - q))  K ^ (n - 2)"
      by (simp del: subst_all)
    then
    show "norm (p<n - Suc 0. q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) 
        of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
      by (intro order_trans [OF norm_sum]
          real_sum_nat_ivl_bounded2 mult_nonneg_nonneg of_nat_0_le_iff zero_le_power K)
  qed
  also have " = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
    by (simp only: mult.assoc)
  finally show ?thesis .
qed

lemma lemma_termdiff4:
  fixes f :: "'a::real_normed_vector  'b::real_normed_vector"
    and k :: real
  assumes k: "0 < k"
    and le: "h. h  0  norm h < k  norm (f h)  K * norm h"
  shows "f 0 0"
proof (rule tendsto_norm_zero_cancel)
  show "(λh. norm (f h)) 0 0"
  proof (rule real_tendsto_sandwich)
    show "eventually (λh. 0  norm (f h)) (at 0)"
      by simp
    show "eventually (λh. norm (f h)  K * norm h) (at 0)"
      using k by (auto simp: eventually_at dist_norm le)
    show "(λh. 0) (0::'a) (0::real)"
      by (rule tendsto_const)
    have "(λh. K * norm h) (0::'a) K * norm (0::'a)"
      by (intro tendsto_intros)
    then show "(λh. K * norm h) (0::'a) 0"
      by simp
  qed
qed

lemma lemma_termdiff5:
  fixes g :: "'a::real_normed_vector  nat  'b::banach"
    and k :: real
  assumes k: "0 < k"
    and f: "summable f"
    and le: "h n. h  0  norm h < k  norm (g h n)  f n * norm h"
  shows "(λh. suminf (g h)) 0 0"
proof (rule lemma_termdiff4 [OF k])
  fix h :: 'a
  assume "h  0" and "norm h < k"
  then have 1: "n. norm (g h n)  f n * norm h"
    by (simp add: le)
  then have "N. nN. norm (norm (g h n))  f n * norm h"
    by simp
  moreover from f have 2: "summable (λn. f n * norm h)"
    by (rule summable_mult2)
  ultimately have 3: "summable (λn. norm (g h n))"
    by (rule summable_comparison_test)
  then have "norm (suminf (g h))  (n. norm (g h n))"
    by (rule summable_norm)
  also from 1 3 2 have "(n. norm (g h n))  (n. f n * norm h)"
    by (simp add: suminf_le)
  also from f have "(n. f n * norm h) = suminf f * norm h"
    by (rule suminf_mult2 [symmetric])
  finally show "norm (suminf (g h))  suminf f * norm h" .
qed


(* FIXME: Long proofs *)

lemma termdiffs_aux:
  fixes x :: "'a::{real_normed_field,banach}"
  assumes 1: "summable (λn. diffs (diffs c) n * K ^ n)"
    and 2: "norm x < norm K"
  shows "(λh. n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) 0 0"
proof -
  from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K"
    by fast
  from norm_ge_zero r1 have r: "0 < r"
    by (rule order_le_less_trans)
  then have r_neq_0: "r  0" by simp
  show ?thesis
  proof (rule lemma_termdiff5)
    show "0 < r - norm x"
      using r1 by simp
    from r r2 have "norm (of_real r::'a) < norm K"
      by simp
    with 1 have "summable (λn. norm (diffs (diffs c) n * (of_real r ^ n)))"
      by (rule powser_insidea)
    then have "summable (λn. diffs (diffs (λn. norm (c n))) n * r ^ n)"
      using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
    then have "summable (λn. of_nat n * diffs (λn. norm (c n)) n * r ^ (n - Suc 0))"
      by (rule diffs_equiv [THEN sums_summable])
    also have "(λn. of_nat n * diffs (λn. norm (c n)) n * r ^ (n - Suc 0)) =
               (λn. diffs (λm. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
      by (simp add: diffs_def r_neq_0 fun_eq_iff split: nat_diff_split)
    finally have "summable
      (λn. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
      by (rule diffs_equiv [THEN sums_summable])
    also have
      "(λn. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) =
       (λn. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
      by (rule ext) (simp add: r_neq_0 split: nat_diff_split)
    finally show "summable (λn. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
  next
    fix h :: 'a and n
    assume h: "h  0"
    assume "norm h < r - norm x"
    then have "norm x + norm h < r" by simp
    with norm_triangle_ineq 
    have xh: "norm (x + h) < r"
      by (rule order_le_less_trans)
    have "norm (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))
     real n * (real (n - Suc 0) * (r ^ (n - 2) * norm h))"
      by (metis (mono_tags, lifting) h mult.assoc lemma_termdiff3 less_eq_real_def r1 xh)
    then show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) 
      norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
      by (simp only: norm_mult mult.assoc mult_left_mono [OF _ norm_ge_zero])
  qed
qed

lemma termdiffs:
  fixes K x :: "'a::{real_normed_field,banach}"
  assumes 1: "summable (λn. c n * K ^ n)"
    and 2: "summable (λn. (diffs c) n * K ^ n)"
    and 3: "summable (λn. (diffs (diffs c)) n * K ^ n)"
    and 4: "norm x < norm K"
  shows "DERIV (λx. n. c n * x^n) x :> (n. (diffs c) n * x^n)"
  unfolding DERIV_def
proof (rule LIM_zero_cancel)
  show "(λh. (suminf (λn. c n * (x + h) ^ n) - suminf (λn. c n * x^n)) / h
            - suminf (λn. diffs c n * x^n)) 0 0"
  proof (rule LIM_equal2)
    show "0 < norm K - norm x"
      using 4 by (simp add: less_diff_eq)
  next
    fix h :: 'a
    assume "norm (h - 0) < norm K - norm x"
    then have "norm x + norm h < norm K" by simp
    then have 5: "norm (x + h) < norm K"
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
    have "summable (λn. c n * x^n)"
      and "summable (λn. c n * (x + h) ^ n)"
      and "summable (λn. diffs c n * x^n)"
      using 1 2 4 5 by (auto elim: powser_inside)
    then have "((n. c n * (x + h) ^ n) - (n. c n * x^n)) / h - (n. diffs c n * x^n) =
          (n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
      by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
    then show "((n. c n * (x + h) ^ n) - (n. c n * x^n)) / h - (n. diffs c n * x^n) =
          (n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
      by (simp add: algebra_simps)
  next
    show "(λh. n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) 0 0"
      by (rule termdiffs_aux [OF 3 4])
  qed
qed

subsection The Derivative of a Power Series Has the Same Radius of Convergence

lemma termdiff_converges:
  fixes x :: "'a::{real_normed_field,banach}"
  assumes K: "norm x < K"
    and sm: "x. norm x < K  summable(λn. c n * x ^ n)"
  shows "summable (λn. diffs c n * x ^ n)"
proof (cases "x = 0")
  case True
  then show ?thesis
    using powser_sums_zero sums_summable by auto
next
  case False
  then have "K > 0"
    using K less_trans zero_less_norm_iff by blast
  then obtain r :: real where r: "norm x < norm r" "norm r < K" "r > 0"
    using K False
    by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
  have to0: "(λn. of_nat n * (x / of_real r) ^ n)  0"
    using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
  obtain N where N: "n. nN  real_of_nat n * norm x ^ n < r ^ n"
    using r LIMSEQ_D [OF to0, of 1]
    by (auto simp: norm_divide norm_mult norm_power field_simps)
  have "summable (λn. (of_nat n * c n) * x ^ n)"
  proof (rule summable_comparison_test')
    show "summable (λn. norm (c n * of_real r ^ n))"
      apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
      using N r norm_of_real [of "r + K", where 'a = 'a] by auto
    show "n. N  n  norm (of_nat n * c n * x ^ n)  norm (c n * of_real r ^ n)"
      using N r by (fastforce simp add: norm_mult norm_power less_eq_real_def)
  qed
  then have "summable (λn. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
    using summable_iff_shift [of "λn. of_nat n * c n * x ^ n" 1]
    by simp
  then have "summable (λn. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
    using False summable_mult2 [of "λn. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
    by (simp add: mult.assoc) (auto simp: ac_simps)
  then show ?thesis
    by (simp add: diffs_def)
qed

lemma termdiff_converges_all:
  fixes x :: "'a::{real_normed_field,banach}"
  assumes "x. summable (λn. c n * x^n)"
  shows "summable (λn. diffs c n * x^n)"
  by (rule termdiff_converges [where K = "1 + norm x"]) (use assms in auto)

lemma termdiffs_strong:
  fixes K x :: "'a::{real_normed_field,banach}"
  assumes sm: "summable (λn. c n * K ^ n)"
    and K: "norm x < norm K"
  shows "DERIV (λx. n. c n * x^n) x :> (n. diffs c n * x^n)"
proof -
  have "norm K + norm x < norm K + norm K"
    using K by force
  then have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
    by (auto simp: norm_triangle_lt norm_divide field_simps)
  then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
    by simp
  have "summable (λn. c n * (of_real (norm x + norm K) / 2) ^ n)"
    by (metis K2 summable_norm_cancel [OF powser_insidea [OF sm]] add.commute of_real_add)
  moreover have "x. norm x < norm K  summable (λn. diffs c n * x ^ n)"
    by (blast intro: sm termdiff_converges powser_inside)
  moreover have "x. norm x < norm K  summable (λn. diffs(diffs c) n * x ^ n)"
    by (blast intro: sm termdiff_converges powser_inside)
  ultimately show ?thesis
    by (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
       (use K in auto simp: field_simps simp flip: of_real_add)
qed

lemma termdiffs_strong_converges_everywhere:
  fixes K x :: "'a::{real_normed_field,banach}"
  assumes "y. summable (λn. c n * y ^ n)"
  shows "((λx. n. c n * x^n) has_field_derivative (n. diffs c n * x^n)) (at x)"
  using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
  by (force simp del: of_real_add)

lemma termdiffs_strong':
  fixes z :: "'a :: {real_normed_field,banach}"
  assumes "z. norm z < K  summable (λn. c n * z ^ n)"
  assumes "norm z < K"
  shows   "((λz. n. c n * z^n) has_field_derivative (n. diffs c n * z^n)) (at z)"
proof (rule termdiffs_strong)
  define L :: real where "L =  (norm z + K) / 2"
  have "0  norm z" by simp
  also note norm z < K
  finally have K: "K  0" by simp
  from assms K have L: "L  0" "norm z < L" "L < K" by (simp_all add: L_def)
  from L show "norm z < norm (of_real L :: 'a)" by simp
  from L show "summable (λn. c n * of_real L ^ n)" by (intro assms(1)) simp_all
qed

lemma termdiffs_sums_strong:
  fixes z :: "'a :: {banach,real_normed_field}"
  assumes sums: "z. norm z < K  (λn. c n * z ^ n) sums f z"
  assumes deriv: "(f has_field_derivative f') (at z)"
  assumes norm: "norm z < K"
  shows   "(λn. diffs c n * z ^ n) sums f'"
proof -
  have summable: "summable (λn. diffs c n * z^n)"
    by (intro termdiff_converges[OF norm] sums_summable[OF sums])
  from norm have "eventually (λz. z  norm -` {..<K}) (nhds z)"
    by (intro eventually_nhds_in_open open_vimage)
       (simp_all add: continuous_on_norm)
  hence eq: "eventually (λz. (n. c n * z^n) = f z) (nhds z)"
    by eventually_elim (insert sums, simp add: sums_iff)

  have "((λz. n. c n * z^n) has_field_derivative (n. diffs c n * z^n)) (at z)"
    by (intro termdiffs_strong'[OF _ norm] sums_summable[OF sums])
  hence "(f has_field_derivative (n. diffs c n * z^n)) (at z)"
    by (subst (asm) DERIV_cong_ev[OF refl eq refl])
  from this and deriv have "(n. diffs c n * z^n) = f'" by (rule DERIV_unique)
  with summable show ?thesis by (simp add: sums_iff)
qed

lemma isCont_powser:
  fixes K x :: "'a::{real_normed_field,banach}"
  assumes "summable (λn. c n * K ^ n)"
  assumes "norm x < norm K"
  shows "isCont (λx. n. c n * x^n) x"
  using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)

lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]

lemma isCont_powser_converges_everywhere:
  fixes K x :: "'a::{real_normed_field,banach}"
  assumes "y. summable (λn. c n * y ^ n)"
  shows "isCont (λx. n. c n * x^n) x"
  using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
  by (force intro!: DERIV_isCont simp del: of_real_add)

lemma powser_limit_0:
  fixes a :: "nat  'a::{real_normed_field,banach}"
  assumes s: "0 < s"
    and sm: "x. norm x < s  (λn. a n * x ^ n) sums (f x)"
  shows "(f  a 0) (at 0)"
proof -
  have "norm (of_real s / 2 :: 'a) < s"
    using s  by (auto simp: norm_divide)
  then have "summable (λn. a n * (of_real s / 2) ^ n)"
    by (rule sums_summable [OF sm])
  then have "((λx. n. a n * x ^ n) has_field_derivative (n. diffs a n * 0 ^ n)) (at 0)"
    by (rule termdiffs_strong) (use s in auto simp: norm_divide)
  then have "isCont (λx. n. a n * x ^ n) 0"
    by (blast intro: DERIV_continuous)
  then have "((λx. n. a n * x ^ n)  a 0) (at 0)"
    by (simp add: continuous_within)
  moreover have "(λx. f x - (n. a n * x ^ n)) 0 0"
    apply (clarsimp simp: LIM_eq)
    apply (rule_tac x=s in exI)
    using s sm sums_unique by fastforce
  ultimately show ?thesis
    by (rule Lim_transform)
qed

lemma powser_limit_0_strong:
  fixes a :: "nat  'a::{real_normed_field,banach}"
  assumes s: "0 < s"
    and sm: "x. x  0  norm x < s  (λn. a n * x ^ n) sums (f x)"
  shows "(f  a 0) (at 0)"
proof -
  have *: "((λx. if x = 0 then a 0 else f x)  a 0) (at 0)"
    by (rule powser_limit_0 [OF s]) (auto simp: powser_sums_zero sm)
  show ?thesis
    using "*" by (auto cong: Lim_cong_within)
qed


subsection Derivability of power series

lemma DERIV_series':
  fixes f :: "real  nat  real"
  assumes DERIV_f: " n. DERIV (λ x. f x n) x0 :> (f' x0 n)"
    and allf_summable: " x. x  {a <..< b}  summable (f x)"
    and x0_in_I: "x0  {a <..< b}"
    and "summable (f' x0)"
    and "summable L"
    and L_def: "n x y. x  {a <..< b}  y  {a <..< b}  ¦f x n - f y n¦  L n * ¦x - y¦"
  shows "DERIV (λ x. suminf (f x)) x0 :> (suminf (f' x0))"
  unfolding DERIV_def
proof (rule LIM_I)
  fix r :: real
  assume "0 < r" then have "0 < r/3" by auto

  obtain N_L where N_L: " n. N_L  n  ¦  i. L (i + n) ¦ < r/3"
    using suminf_exist_split[OF 0 < r/3 summable L] by auto

  obtain N_f' where N_f': " n. N_f'  n  ¦  i. f' x0 (i + n) ¦ < r/3"
    using suminf_exist_split[OF 0 < r/3 summable (f' x0)] by auto

  let ?N = "Suc (max N_L N_f')"
  have "¦  i. f' x0 (i + ?N) ¦ < r/3" (is "?f'_part < r/3")
    and L_estimate: "¦  i. L (i + ?N) ¦ < r/3"
    using N_L[of "?N"] and N_f' [of "?N"] by auto

  let ?diff = "λi x. (f (x0 + x) i - f x0 i) / x"

  let ?r = "r / (3 * real ?N)"
  from 0 < r have "0 < ?r" by simp

  let ?s = "λn. SOME s. 0 < s  ( x. x  0  ¦ x ¦ < s  ¦ ?diff n x - f' x0 n ¦ < ?r)"
  define S' where "S' = Min (?s ` {..< ?N })"

  have "0 < S'"
    unfolding S'_def
  proof (rule iffD2[OF Min_gr_iff])
    show "x  (?s ` {..< ?N }). 0 < x"
    proof
      fix x
      assume "x  ?s ` {..<?N}"
      then obtain n where "x = ?s n" and "n  {..<?N}"
        using image_iff[THEN iffD1] by blast
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF 0 < ?r, unfolded real_norm_def]
      obtain s where s_bound: "0 < s  (x. x  0  ¦x¦ < s  ¦?diff n x - f' x0 n¦ < ?r)"
        by auto
      have "0 < ?s n"
        by (rule someI2[where a=s]) (auto simp: s_bound simp del: of_nat_Suc)
      then show "0 < x" by (simp only: x = ?s n)
    qed
  qed auto

  define S where "S = min (min (x0 - a) (b - x0)) S'"
  then have "0 < S" and S_a: "S  x0 - a" and S_b: "S  b - x0"
    and "S  S'" using x0_in_I and 0 < S'
    by auto

  have "¦(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)¦ < r"
    if "x  0" and "¦x¦ < S" for x
  proof -
    from that have x_in_I: "x0 + x  {a <..< b}"
      using S_a S_b by auto

    note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
    note div_smbl = summable_divide[OF diff_smbl]
    note all_smbl = summable_diff[OF div_smbl summable (f' x0)]
    note ign = summable_ignore_initial_segment[where k="?N"]
    note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
    note div_shft_smbl = summable_divide[OF diff_shft_smbl]
    note all_shft_smbl = summable_diff[OF div_smbl ign[OF summable (f' x0)]]

    have 1: "¦(¦?diff (n + ?N) x¦)¦  L (n + ?N)" for n
    proof -
      have "¦?diff (n + ?N) x¦  L (n + ?N) * ¦(x0 + x) - x0¦ / ¦x¦"
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
        by (simp only: abs_divide)
      with x  0 show ?thesis by auto
    qed
    note 2 = summable_rabs_comparison_test[OF _ ign[OF summable L]]
    from 1 have "¦  i. ?diff (i + ?N) x ¦  ( i. L (i + ?N))"
      by (metis (lifting) abs_idempotent
          order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF summable L]]])
    then have "¦i. ?diff (i + ?N) x¦  r / 3" (is "?L_part  r/3")
      using L_estimate by auto

    have "¦n<?N. ?diff n x - f' x0 n¦  (n<?N. ¦?diff n x - f' x0 n¦)" ..
    also have " < (n<?N. ?r)"
    proof (rule sum_strict_mono)
      fix n
      assume "n  {..< ?N}"
      have "¦x¦ < S" using ¦x¦ < S .
      also have "S  S'" using S  S' .
      also have "S'  ?s n"
        unfolding S'_def
      proof (rule Min_le_iff[THEN iffD2])
        have "?s n  (?s ` {..<?N})  ?s n  ?s n"
          using n  {..< ?N} by auto
        then show " a  (?s ` {..<?N}). a  ?s n"
          by blast
      qed auto
      finally have "¦x¦ < ?s n" .

      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF 0 < ?r,
          unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
      have "x. x  0  ¦x¦ < ?s n  ¦?diff n x - f' x0 n¦ < ?r" .
      with x  0 and ¦x¦ < ?s n show "¦?diff n x - f' x0 n¦ < ?r"
        by blast
    qed auto
    also have " = of_nat (card {..<?N}) * ?r"
      by (rule sum_constant)
    also have " = real ?N * ?r"
      by simp
    also have " = r/3"
      by (auto simp del: of_nat_Suc)
    finally have "¦n<?N. ?diff n x - f' x0 n ¦ < r / 3" (is "?diff_part < r / 3") .

    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
    have "¦(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)¦ =
        ¦n. ?diff n x - f' x0 n¦"
      unfolding suminf_diff[OF div_smbl summable (f' x0), symmetric]
      using suminf_divide[OF diff_smbl, symmetric] by auto
    also have "  ?diff_part + ¦(n. ?diff (n + ?N) x) - ( n. f' x0 (n + ?N))¦"
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
      unfolding suminf_diff[OF div_shft_smbl ign[OF summable (f' x0)]]
      apply (simp only: add.commute)
      using abs_triangle_ineq by blast
    also have "  ?diff_part + ?L_part + ?f'_part"
      using abs_triangle_ineq4 by auto
    also have " < r /3 + r/3 + r/3"
      using ?diff_part < r/3 ?L_part  r/3 and ?f'_part < r/3
      by (rule add_strict_mono [OF add_less_le_mono])
    finally show ?thesis
      by auto
  qed
  then show "s > 0.  x. x  0  norm (x - 0) < s 
      norm (((n. f (x0 + x) n) - (n. f x0 n)) / x - (n. f' x0 n)) < r"
    using 0 < S by auto
qed

lemma DERIV_power_series':
  fixes f :: "nat  real"
  assumes converges: "x. x  {-R <..< R}  summable (λn. f n * real (Suc n) * x^n)"
    and x0_in_I: "x0  {-R <..< R}"
    and "0 < R"
  shows "DERIV (λx. (n. f n * x^(Suc n))) x0 :> (n. f n * real (Suc n) * x0^n)"
    (is "DERIV (λx. suminf (?f x)) x0 :> suminf (?f' x0)")
proof -
  have for_subinterval: "DERIV (λx. suminf (?f x)) x0 :> suminf (?f' x0)"
    if "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" for R'
  proof -
    from that have "x0  {-R' <..< R'}" and "R'  {-R <..< R}" and "x0  {-R <..< R}"
      by auto
    show ?thesis
    proof (rule DERIV_series')
      show "summable (λ n. ¦f n * real (Suc n) * R'^n¦)"
      proof -
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
          using 0 < R' 0 < R R' < R by (auto simp: field_simps)
        then have in_Rball: "(R' + R) / 2  {-R <..< R}"
          using R' < R by auto
        have "norm R' < norm ((R' + R) / 2)"
          using 0 < R' 0 < R R' < R by (auto simp: field_simps)
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
          by auto
      qed
    next
      fix n x y
      assume "x  {-R' <..< R'}" and "y  {-R' <..< R'}"
      show "¦?f x n - ?f y n¦  ¦f n * real (Suc n) * R'^n¦ * ¦x-y¦"
      proof -
        have "¦f n * x ^ (Suc n) - f n * y ^ (Suc n)¦ =
          (¦f n¦ * ¦<