(*  Title       : Transcendental.thy

    Author      : Jacques D. Fleuriot

    Copyright   : 1998,1999 University of Cambridge

                  1999,2001 University of Edinburgh

    Conversion to Isar and new proofs by Lawrence C Paulson, 2004

*)

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 cong: strong_setsum_cong)

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: right_distrib del: setsum_op_ivl_Suc)

apply (subst mult_left_commute [where a="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,banach}"

  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 (simp add: Cauchy_convergent_iff)

  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

    { fix B T E have "(if \<not> B then T else E) = (if B then E else T)" by auto } note if_eq = this

    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 lemma_nest_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)" using even_num_iff odd_pos by auto 

	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 LIMSEQ_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 LIMSEQ_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:

     "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] cong: strong_setsum_cong)

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::ordered_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

         zero_le_imp_of_nat

         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{* Some properties of factorials *}

lemma real_of_nat_fact_not_zero [simp]: "real (fact n) \<noteq> 0"

by auto

lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact n)"

by auto

lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact n)"

by simp

lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact n))"

by (auto simp add: positive_imp_inverse_positive)

lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact n))"

by (auto intro: order_less_imp_le)

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 auto

    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 real_mult_assoc[symmetric] by algebra

	  finally show ?thesis .

	qed }

      { fix n

	from DERIV_pow[of "Suc n" x0, THEN DERIV_cmult[where c="f n"]]

	show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)" unfolding real_mult_assoc by auto }

      { 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 real_mult_assoc, unfolded real_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 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 norm_scaleR 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_scaleR 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 lemma_exp_ext: "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"

by (auto intro!: ext simp add: exp_def)

lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"

apply (simp add: exp_def)

apply (subst lemma_exp_ext)

apply (subgoal_tac "DERIV (\<lambda>u. \<Sum>n. of_real (inverse (real (fact n))) * u ^ n) x :> (\<Sum>n. diffs (\<lambda>n. of_real (inverse (real (fact n)))) n * x ^ n)")

apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs)

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

lemma isCont_exp [simp]: "isCont exp x"

by (rule DERIV_exp [THEN DERIV_isCont])

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 left_distrib)

  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 add: scaleR_cancel_left 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 of_real.suminf)

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)