src/HOL/Transcendental.thy

add lemma

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

*)

header{*Power Series, Transcendental Functions etc.*}

theory Transcendental
imports Fact Series Deriv NthRoot
begin

subsection {* Properties of Power Series *}

lemma lemma_realpow_diff:
  fixes y :: "'a::monoid_mult"
  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
proof -
  assume "p \<le> n"
  hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
  thus ?thesis by (simp add: power_commutes)
qed

lemma lemma_realpow_diff_sumr:
  fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
      y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
         del: setsum_op_ivl_Suc)

lemma lemma_realpow_diff_sumr2:
  fixes y :: "'a::{comm_ring,monoid_mult}" shows
     "x ^ (Suc n) - y ^ (Suc n) =
      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
apply (induct n, simp)
apply (simp del: setsum_op_ivl_Suc)
apply (subst setsum_op_ivl_Suc)
apply (subst lemma_realpow_diff_sumr)
apply (simp add: distrib_left del: setsum_op_ivl_Suc)
apply (subst mult_left_commute [of "x - y"])
apply (erule subst)
apply (simp add: algebra_simps)
done

lemma lemma_realpow_rev_sumr:
     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
apply (rule inj_onI, simp)
apply auto
apply (rule_tac x="n - x" in image_eqI, simp, simp)
done

text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}

lemma powser_insidea:
  fixes x z :: "'a::real_normed_field"
  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
  assumes 2: "norm z < norm x"
  shows "summable (\<lambda>n. norm (f n * z ^ n))"
proof -
  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
    by (rule summable_LIMSEQ_zero)
  hence "convergent (\<lambda>n. f n * x ^ n)"
    by (rule convergentI)
  hence "Cauchy (\<lambda>n. f n * x ^ n)"
    by (rule convergent_Cauchy)
  hence "Bseq (\<lambda>n. f n * x ^ n)"
    by (rule Cauchy_Bseq)
  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
    by (simp add: Bseq_def, safe)
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
                   K * norm (z ^ n) * inverse (norm (x ^ n))"
  proof (intro exI allI impI)
    fix n::nat assume "0 \<le> 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 "\<dots> \<le> K * norm (z ^ n)"
      by (simp only: mult_right_mono 4 norm_ge_zero)
    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
      by (simp add: x_neq_0)
    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
      by (simp only: mult_assoc)
    finally show "norm (norm (f n * z ^ n)) \<le>
                  K * norm (z ^ n) * inverse (norm (x ^ n))"
      by (simp add: mult_le_cancel_right x_neq_0)
  qed
  moreover have "summable (\<lambda>n. 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: nonzero_norm_divide divide_inverse [symmetric])
    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
      by (rule summable_geometric)
    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
      by (rule summable_mult)
    thus "summable (\<lambda>n. 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 (\<lambda>n. norm (f n * z ^ n))"
    by (rule summable_comparison_test)
qed

lemma powser_inside:
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
     "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
      ==> summable (%n. f(n) * (z ^ n))"
by (rule powser_insidea [THEN summable_norm_cancel])

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

lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
  shows "(\<lambda> 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: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast

  let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
  { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
    have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
      using sum_split_even_odd by auto
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
    moreover
    have "?SUM (2 * (m div 2)) = ?SUM m"
    proof (cases "even m")
      case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
    next
      case False hence "even (Suc m)" by auto
      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
      have eq: "Suc (2 * (m div 2)) = m" by auto
      hence "even (2 * (m div 2))" using `odd m` by auto
      have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
      also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
      finally show ?thesis by auto
    qed
    ultimately have "(norm (?SUM m - x) < r)" by auto
  }
  thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
qed

lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
  shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
proof -
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
  { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
      by (cases B) auto } note if_sum = this
  have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
  {
    have "?s 0 = 0" by auto
    have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
    have if_eq: "\<And>B T E. (if \<not> 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 "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
                even_Suc Suc_m1 if_eq .
  } from sums_add[OF g_sums this]
  show ?thesis unfolding if_sum .
qed

subsection {* Alternating series test / Leibniz formula *}

lemma sums_alternating_upper_lower:
  fixes a :: "nat \<Rightarrow> real"
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
  shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
             ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
  (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
proof -
  have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto

  have "\<forall> n. ?f n \<le> ?f (Suc n)"
  proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
  moreover
  have "\<forall> n. ?g (Suc n) \<le> ?g n"
  proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
    unfolding One_nat_def by auto qed
  moreover
  have "\<forall> n. ?f n \<le> ?g n"
  proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
    unfolding One_nat_def by auto qed
  moreover
  have "(\<lambda> n. ?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 "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
    hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
    thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
  qed
  ultimately
  show ?thesis by (rule nested_sequence_unique)
qed

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

  let ?Sa = "\<lambda> m. \<Sum> n = 0..<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: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto

    from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
    obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto

    { fix n :: nat
      assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
      have "norm (?Sa n - l) < r"
      proof (cases "even n")
        case True from even_nat_div_two_times_two[OF this]
        have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
        from f[OF this]
        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
      next
        case False hence "even (n - 1)" by simp
        from even_nat_div_two_times_two[OF this]
        have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
        hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto

        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
        from g[OF this]
        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
      qed
    }
    thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
  qed
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
  thus "summable ?S" using summable_def by auto

  have "l = suminf ?S" using sums_unique[OF sums_l] .

  { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
  { fix n show "?f n \<le> 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 \<Rightarrow> real"
  assumes a_zero: "a ----> 0" and "monoseq a"
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
  and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
  and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
  and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
  and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
proof -
  have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
  proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
    case True
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
    { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
    note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
    from leibniz[OF mono]
    show ?thesis using `0 \<le> a 0` by auto
  next
    let ?a = "\<lambda> n. - a n"
    case False
    with monoseq_le[OF `monoseq a` `a ----> 0`]
    have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
    { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
    note monotone = this
    note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
    have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
    from this[THEN sums_minus]
    have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
    hence ?summable unfolding summable_def by auto
    moreover
    have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto

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

    have ?pos using `0 \<le> ?a 0` by auto
    moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
    moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] by auto
    ultimately show ?thesis by auto
  qed
  from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
       this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
  show ?summable and ?pos and ?neg and ?f and ?g .
qed

subsection {* Term-by-Term Differentiability of Power Series *}

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

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

lemma sums_Suc_imp:
  assumes f: "f 0 = 0"
  shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
unfolding sums_def
apply (rule LIMSEQ_imp_Suc)
apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
apply (simp only: setsum_shift_bounds_Suc_ivl)
done

lemma diffs_equiv:
  fixes x :: "'a::{real_normed_vector, ring_1}"
  shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
unfolding diffs_def
apply (drule summable_sums)
apply (rule sums_Suc_imp, simp_all)
done

lemma lemma_termdiff1:
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
by(auto simp add: algebra_simps power_add [symmetric])

lemma sumr_diff_mult_const2:
  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
by (simp add: setsum_subtractf)

lemma lemma_termdiff2:
  fixes h :: "'a :: {field}"
  assumes h: "h \<noteq> 0" shows
  "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
   h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
        (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
apply (simp add: right_diff_distrib diff_divide_distrib h)
apply (simp add: mult_assoc [symmetric])
apply (cases "n", simp)
apply (simp add: lemma_realpow_diff_sumr2 h
                 right_diff_distrib [symmetric] mult_assoc
            del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
apply (subst lemma_realpow_rev_sumr)
apply (subst sumr_diff_mult_const2)
apply simp
apply (simp only: lemma_termdiff1 setsum_right_distrib)
apply (rule setsum_cong [OF refl])
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
apply (clarify)
apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
            del: setsum_op_ivl_Suc power_Suc)
apply (subst mult_assoc [symmetric], subst power_add [symmetric])
apply (simp add: mult_ac)
done

lemma real_setsum_nat_ivl_bounded2:
  fixes K :: "'a::linordered_semidom"
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
  assumes K: "0 \<le> K"
  shows "setsum f {0..<n-k} \<le> of_nat n * K"
apply (rule order_trans [OF setsum_mono])
apply (rule f, simp)
apply (simp add: mult_right_mono K)
done

lemma lemma_termdiff3:
  fixes h z :: "'a::{real_normed_field}"
  assumes 1: "h \<noteq> 0"
  assumes 2: "norm z \<le> K"
  assumes 3: "norm (z + h) \<le> K"
  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
          \<le> 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 (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
    apply (subst lemma_termdiff2 [OF 1])
    apply (subst norm_mult)
    apply (rule mult_commute)
    done
  also have "\<dots> \<le> 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 \<le> K" by (rule order_trans)
    have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
      apply (erule subst)
      apply (simp only: norm_mult norm_power power_add)
      apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
      done
    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
              (z + h) ^ q * z ^ (n - 2 - q))
          \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
      apply (intro
         order_trans [OF norm_setsum]
         real_setsum_nat_ivl_bounded2
         mult_nonneg_nonneg
         of_nat_0_le_iff
         zero_le_power K)
      apply (rule le_Kn, simp)
      done
  qed
  also have "\<dots> = 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_field} \<Rightarrow>
              'b::real_normed_vector"
  assumes k: "0 < (k::real)"
  assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
  shows "f -- 0 --> 0"
unfolding LIM_eq diff_0_right
proof (safe)
  let ?h = "of_real (k / 2)::'a"
  have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
  hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
  hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
  hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)

  fix r::real assume r: "0 < r"
  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
  proof (cases)
    assume "K = 0"
    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
      by simp
    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
  next
    assume K_neq_zero: "K \<noteq> 0"
    with zero_le_K have K: "0 < K" by simp
    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
    proof (rule exI, safe)
      from k r K show "0 < min k (r * inverse K / 2)"
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
    next
      fix x::'a
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
        by simp_all
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
        by (rule mult_strict_left_mono)
      also have "\<dots> = r / 2"
        using K_neq_zero by simp
      also have "r / 2 < r"
        using r by simp
      finally show "norm (f x) < r" .
    qed
  qed
qed

lemma lemma_termdiff5:
  fixes g :: "'a::{real_normed_field} \<Rightarrow>
              nat \<Rightarrow> 'b::banach"
  assumes k: "0 < (k::real)"
  assumes f: "summable f"
  assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
proof (rule lemma_termdiff4 [OF k])
  fix h::'a assume "h \<noteq> 0" and "norm h < k"
  hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
    by (simp add: le)
  hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
    by simp
  moreover from f have B: "summable (\<lambda>n. f n * norm h)"
    by (rule summable_mult2)
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
    by (rule summable_comparison_test)
  hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
    by (rule summable_norm)
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
    by (rule summable_le)
  also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
    by (rule suminf_mult2 [symmetric])
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
qed


text{* FIXME: Long proofs*}

lemma termdiffs_aux:
  fixes x :: "'a::{real_normed_field,banach}"
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
  assumes 2: "norm x < norm K"
  shows "(\<lambda>h. \<Sum>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)
  hence r_neq_0: "r \<noteq> 0" by simp
  show ?thesis
  proof (rule lemma_termdiff5)
    show "0 < r - norm x" using r1 by simp
  next
    from r r2 have "norm (of_real r::'a) < norm K"
      by simp
    with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
      by (rule powser_insidea)
    hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
      using r
      by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
    hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
      by (rule diffs_equiv [THEN sums_summable])
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
      = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
      apply (rule ext)
      apply (simp add: diffs_def)
      apply (case_tac n, simp_all add: r_neq_0)
      done
    finally have "summable
      (\<lambda>n. 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
      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
           r ^ (n - Suc 0)) =
       (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
      apply (rule ext)
      apply (case_tac "n", simp)
      apply (case_tac "nat", simp)
      apply (simp add: r_neq_0)
      done
    finally show
      "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
  next
    fix h::'a and n::nat
    assume h: "h \<noteq> 0"
    assume "norm h < r - norm x"
    hence "norm x + norm h < r" by simp
    with norm_triangle_ineq have xh: "norm (x + h) < r"
      by (rule order_le_less_trans)
    show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
      apply (simp only: norm_mult mult_assoc)
      apply (rule mult_left_mono [OF _ norm_ge_zero])
      apply (simp (no_asm) add: mult_assoc [symmetric])
      apply (rule lemma_termdiff3)
      apply (rule h)
      apply (rule r1 [THEN order_less_imp_le])
      apply (rule xh [THEN order_less_imp_le])
      done
  qed
qed

lemma termdiffs:
  fixes K x :: "'a::{real_normed_field,banach}"
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
  assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
  assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
  assumes 4: "norm x < norm K"
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
unfolding deriv_def
proof (rule LIM_zero_cancel)
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
            - suminf (\<lambda>n. 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 "h \<noteq> 0"
    assume "norm (h - 0) < norm K - norm x"
    hence "norm x + norm h < norm K" by simp
    hence 5: "norm (x + h) < norm K"
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
    have A: "summable (\<lambda>n. c n * x ^ n)"
      by (rule powser_inside [OF 1 4])
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
      by (rule powser_inside [OF 1 5])
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
      by (rule powser_inside [OF 2 4])
    show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
             - (\<Sum>n. diffs c n * x ^ n) =
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
      apply (subst sums_unique [OF diffs_equiv [OF C]])
      apply (subst suminf_diff [OF B A])
      apply (subst suminf_divide [symmetric])
      apply (rule summable_diff [OF B A])
      apply (subst suminf_diff)
      apply (rule summable_divide)
      apply (rule summable_diff [OF B A])
      apply (rule sums_summable [OF diffs_equiv [OF C]])
      apply (rule arg_cong [where f="suminf"], rule ext)
      apply (simp add: algebra_simps)
      done
  next
    show "(\<lambda>h. \<Sum>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 {* Derivability of power series *}

lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
  and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
  and "summable (f' x0)"
  and "summable L" and L_def: "\<And> n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar> f x n - f y n \<bar> \<le> L n * \<bar> x - y \<bar>"
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
  unfolding deriv_def
proof (rule LIM_I)
  fix r :: real assume "0 < r" hence "0 < r/3" by auto

  obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
    using suminf_exist_split[OF `0 < r/3` `summable L`] by auto

  obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
    using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto

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

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

  let ?r = "r / (3 * real ?N)"
  have "0 < 3 * real ?N" by auto
  from divide_pos_pos[OF `0 < r` this]
  have "0 < ?r" .

  let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
  def S' \<equiv> "Min (?s ` { 0 ..< ?N })"

  have "0 < S'" unfolding S'_def
  proof (rule iffD2[OF Min_gr_iff])
    show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
    proof (rule ballI)
      fix x assume "x \<in> ?s ` {0..<?N}"
      then obtain n where "x = ?s n" and "n \<in> {0..<?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 \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)" by auto
      have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
      thus "0 < x" unfolding `x = ?s n` .
    qed
  qed auto

  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
  hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`
    by auto

  { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
    hence x_in_I: "x0 + x \<in> { 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)`]]

    { fix n
      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
      hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
    have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
    hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto

    have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
    also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
    proof (rule setsum_strict_mono)
      fix n assume "n \<in> { 0 ..< ?N}"
      have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
      also have "S \<le> S'" using `S \<le> S'` .
      also have "S' \<le> ?s n" unfolding S'_def
      proof (rule Min_le_iff[THEN iffD2])
        have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
        thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
      qed auto
      finally have "\<bar> x \<bar> < ?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 "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
      with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
      show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
    qed auto
    also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
    also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
    also have "\<dots> = r/3" by auto
    finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .

    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
    have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
                    \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
    also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
    also have "\<dots> < r /3 + r/3 + r/3"
      using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
      by (rule add_strict_mono [OF add_less_le_mono])
    finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
      by auto
  } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
    unfolding real_norm_def diff_0_right by blast
qed

lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
  and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
  shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
  (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
proof -
  { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
    have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
    proof (rule DERIV_series')
      show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
      proof -
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
        have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
      qed
      { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
        show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
        proof -
          have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
          also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
          proof (rule mult_left_mono)
            have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
            also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
            proof (rule setsum_mono)
              fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
              { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
                hence "\<bar>x\<bar> \<le> R'"  by auto
                hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
              } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
            qed
            also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
            finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
          qed
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra
          finally show ?thesis .
        qed }
      { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
          by (auto intro!: DERIV_intros simp del: power_Suc) }
      { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
        have "summable (\<lambda> n. f n * x^n)"
        proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
          fix n
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
          show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
            by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
        qed
        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
        show "summable (?f x)" by auto }
      show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
      show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
    qed
  } note for_subinterval = this
  let ?R = "(R + \<bar>x0\<bar>) / 2"
  have "\<bar>x0\<bar> < ?R" using assms by auto
  hence "- ?R < x0"
  proof (cases "x0 < 0")
    case True
    hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
    thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
  next
    case False
    have "- ?R < 0" using assms by auto
    also have "\<dots> \<le> x0" using False by auto
    finally show ?thesis .
  qed
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
  from for_subinterval[OF this]
  show ?thesis .
qed

subsection {* Exponential Function *}

definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
  "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"

lemma summable_exp_generic:
  fixes x :: "'a::{real_normed_algebra_1,banach}"
  defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
  shows "summable S"
proof -
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
    unfolding S_def by (simp del: mult_Suc)
  obtain r :: real where r0: "0 < r" and r1: "r < 1"
    using dense [OF zero_less_one] by fast
  obtain N :: nat where N: "norm x < real N * r"
    using reals_Archimedean3 [OF r0] by fast
  from r1 show ?thesis
  proof (rule ratio_test [rule_format])
    fix n :: nat
    assume n: "N \<le> n"
    have "norm x \<le> real N * r"
      using N by (rule order_less_imp_le)
    also have "real N * r \<le> real (Suc n) * r"
      using r0 n by (simp add: mult_right_mono)
    finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
      using norm_ge_zero by (rule mult_right_mono)
    hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
      by (rule order_trans [OF norm_mult_ineq])
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
      by (simp add: pos_divide_le_eq mult_ac)
    thus "norm (S (Suc n)) \<le> r * norm (S n)"
      by (simp add: S_Suc inverse_eq_divide)
  qed
qed

lemma summable_norm_exp:
  fixes x :: "'a::{real_normed_algebra_1,banach}"
  shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
proof (rule summable_norm_comparison_test [OF exI, rule_format])
  show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
    by (rule summable_exp_generic)
next
  fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
    by (simp add: norm_power_ineq)
qed

lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
by (insert summable_exp_generic [where x=x], simp)

lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])


lemma exp_fdiffs:
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
         del: mult_Suc of_nat_Suc)

lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
by (simp add: diffs_def)

lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
unfolding exp_def scaleR_conv_of_real
apply (rule DERIV_cong)
apply (rule termdiffs [where K="of_real (1 + norm x)"])
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
apply (simp del: of_real_add)
done

declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]

lemma isCont_exp: "isCont exp x"
  by (rule DERIV_exp [THEN DERIV_isCont])

lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
  by (rule isCont_o2 [OF _ isCont_exp])

lemma tendsto_exp [tendsto_intros]:
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
  by (rule isCont_tendsto_compose [OF isCont_exp])

lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
  unfolding continuous_def by (rule tendsto_exp)

lemma continuous_on_exp [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
  unfolding continuous_on_def by (auto intro: tendsto_exp)

subsubsection {* Properties of the Exponential Function *}

lemma powser_zero:
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
proof -
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
  thus ?thesis unfolding One_nat_def by simp
qed

lemma exp_zero [simp]: "exp 0 = 1"
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)

lemma setsum_cl_ivl_Suc2:
  "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
         del: setsum_cl_ivl_Suc)

lemma exp_series_add:
  fixes x y :: "'a::{real_field}"
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
proof (induct n)
  case 0
  show ?case
    unfolding S_def by simp
next
  case (Suc n)
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
    unfolding S_def by (simp del: mult_Suc)
  hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
    by simp

  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
    by (simp only: times_S)
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
    by (simp only: Suc)
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
    by (rule distrib_right)
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
    by (simp only: setsum_right_distrib mult_ac)
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
                + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
    by (simp add: times_S Suc_diff_le)
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
             (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
    by (subst setsum_cl_ivl_Suc2, simp)
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
    by (subst setsum_cl_ivl_Suc, simp)
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
             (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
              real_of_nat_add [symmetric], simp)
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
    by (simp only: scaleR_right.setsum)
  finally show
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
    by (simp del: setsum_cl_ivl_Suc)
qed

lemma exp_add: "exp (x + y) = exp x * exp y"
unfolding exp_def
by (simp only: Cauchy_product summable_norm_exp exp_series_add)

lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
by (rule exp_add [symmetric])

lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
unfolding exp_def
apply (subst suminf_of_real)
apply (rule summable_exp_generic)
apply (simp add: scaleR_conv_of_real)
done

lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
proof
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
  also assume "exp x = 0"
  finally show "False" by simp
qed

lemma exp_minus: "exp (- x) = inverse (exp x)"
by (rule inverse_unique [symmetric], simp add: mult_exp_exp)

lemma exp_diff: "exp (x - y) = exp x / exp y"
  unfolding diff_minus divide_inverse
  by (simp add: exp_add exp_minus)


subsubsection {* Properties of the Exponential Function on Reals *}

text {* Comparisons of @{term "exp x"} with zero. *}

text{*Proof: because every exponential can be seen as a square.*}
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
proof -
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
  thus ?thesis by (simp add: exp_add [symmetric])
qed

lemma exp_gt_zero [simp]: "0 < exp (x::real)"
by (simp add: order_less_le)

lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
by (simp add: not_less)

lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
by (simp add: not_le)

lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
by simp

lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
apply (induct "n")
apply (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
done

text {* Strict monotonicity of exponential. *}

lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
apply (drule order_le_imp_less_or_eq, auto)
apply (simp add: exp_def)
apply (rule order_trans)
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
done

lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
proof -
  assume x: "0 < x"
  hence "1 < 1 + x" by simp
  also from x have "1 + x \<le> exp x"
    by (simp add: exp_ge_add_one_self_aux)
  finally show ?thesis .
qed

lemma exp_less_mono:
  fixes x y :: real
  assumes "x < y" shows "exp x < exp y"
proof -
  from `x < y` have "0 < y - x" by simp
  hence "1 < exp (y - x)" by (rule exp_gt_one)
  hence "1 < exp y / exp x" by (simp only: exp_diff)
  thus "exp x < exp y" by simp
qed

lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
apply (simp add: linorder_not_le [symmetric])
apply (auto simp add: order_le_less exp_less_mono)
done

lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
by (auto intro: exp_less_mono exp_less_cancel)

lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
by (auto simp add: linorder_not_less [symmetric])

lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
by (simp add: order_eq_iff)

text {* Comparisons of @{term "exp x"} with one. *}

lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
  using exp_less_cancel_iff [where x=0 and y=x] by simp

lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
  using exp_less_cancel_iff [where x=x and y=0] by simp

lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
  using exp_le_cancel_iff [where x=0 and y=x] by simp

lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
  using exp_le_cancel_iff [where x=x and y=0] by simp

lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
  using exp_inj_iff [where x=x and y=0] by simp

lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
proof (rule IVT)
  assume "1 \<le> y"
  hence "0 \<le> y - 1" by simp
  hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
  thus "y \<le> exp (y - 1)" by simp
qed (simp_all add: le_diff_eq)

lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
proof (rule linorder_le_cases [of 1 y])
  assume "1 \<le> y" thus "\<exists>x. exp x = y"
    by (fast dest: lemma_exp_total)
next
  assume "0 < y" and "y \<le> 1"
  hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
  then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
  hence "exp (- x) = y" by (simp add: exp_minus)
  thus "\<exists>x. exp x = y" ..
qed


subsection {* Natural Logarithm *}

definition ln :: "real \<Rightarrow> real" where
  "ln x = (THE u. exp u = x)"

lemma ln_exp [simp]: "ln (exp x) = x"
  by (simp add: ln_def)

lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
  by (auto dest: exp_total)

lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
  by (metis exp_gt_zero exp_ln)

lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
  by (erule subst, rule ln_exp)

lemma ln_one [simp]: "ln 1 = 0"
  by (rule ln_unique, simp)

lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
  by (rule ln_unique, simp add: exp_add)

lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
  by (rule ln_unique, simp add: exp_minus)

lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
  by (rule ln_unique, simp add: exp_diff)

lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
  by (rule ln_unique, simp add: exp_real_of_nat_mult)

lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
  by (subst exp_less_cancel_iff [symmetric], simp)

lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
  by (simp add: linorder_not_less [symmetric])

lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
  by (simp add: order_eq_iff)

lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
  apply (rule exp_le_cancel_iff [THEN iffD1])
  apply (simp add: exp_ge_add_one_self_aux)
  done

lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
  by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all

lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
  using ln_le_cancel_iff [of 1 x] by simp

lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
  using ln_le_cancel_iff [of 1 x] by simp

lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
  using ln_le_cancel_iff [of 1 x] by simp

lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
  using ln_less_cancel_iff [of x 1] by simp

lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
  using ln_less_cancel_iff [of 1 x] by simp

lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
  using ln_less_cancel_iff [of 1 x] by simp

lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
  using ln_less_cancel_iff [of 1 x] by simp

lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
  using ln_inj_iff [of x 1] by simp

lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
  by simp

lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
  apply (subgoal_tac "isCont ln (exp (ln x))", simp)
  apply (rule isCont_inverse_function [where f=exp], simp_all)
  done

lemma tendsto_ln [tendsto_intros]:
  "\<lbrakk>(f ---> a) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
  by (rule isCont_tendsto_compose [OF isCont_ln])

lemma continuous_ln:
  "continuous F f \<Longrightarrow> 0 < f (Lim F (\<lambda>x. x)) \<Longrightarrow> continuous F (\<lambda>x. ln (f x))"
  unfolding continuous_def by (rule tendsto_ln)

lemma isCont_ln' [continuous_intros]:
  "continuous (at x) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))"
  unfolding continuous_at by (rule tendsto_ln)

lemma continuous_within_ln [continuous_intros]:
  "continuous (at x within s) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))"
  unfolding continuous_within by (rule tendsto_ln)

lemma continuous_on_ln [continuous_on_intros]:
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. 0 < f x) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"