src/HOL/Transcendental.thy
author immler
Fri, 10 Mar 2017 23:16:40 +0100
changeset 65204 d23eded35a33
parent 65109 a79c1080f1e9
child 65552 f533820e7248
permissions -rw-r--r--
modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
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(*  Title:      HOL/Transcendental.thy
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    Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
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    Author:     Lawrence C Paulson
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    Author:     Jeremy Avigad
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*)
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section \<open>Power Series, Transcendental Functions etc.\<close>
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theory Transcendental
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imports Binomial Series Deriv NthRoot
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begin
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text \<open>A fact theorem on reals.\<close>
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lemma square_fact_le_2_fact: "fact n * fact n \<le> (fact (2 * n) :: real)"
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proof (induct n)
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  case 0
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  then show ?case by simp
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next
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  case (Suc n)
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  have "(fact (Suc n)) * (fact (Suc n)) = of_nat (Suc n) * of_nat (Suc n) * (fact n * fact n :: real)"
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    by (simp add: field_simps)
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  also have "\<dots> \<le> of_nat (Suc n) * of_nat (Suc n) * fact (2 * n)"
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    by (rule mult_left_mono [OF Suc]) simp
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  also have "\<dots> \<le> of_nat (Suc (Suc (2 * n))) * of_nat (Suc (2 * n)) * fact (2 * n)"
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    by (rule mult_right_mono)+ (auto simp: field_simps)
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  also have "\<dots> = fact (2 * Suc n)" by (simp add: field_simps)
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  finally show ?case .
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qed
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lemma fact_in_Reals: "fact n \<in> \<real>"
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  by (induction n) auto
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lemma of_real_fact [simp]: "of_real (fact n) = fact n"
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  by (metis of_nat_fact of_real_of_nat_eq)
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ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
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ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
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lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
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  by (simp add: pochhammer_prod)
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lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n"
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proof -
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  have "(fact n :: 'a) = of_real (fact n)"
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    by simp
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  also have "norm \<dots> = fact n"
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    by (subst norm_of_real) simp
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  finally show ?thesis .
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qed
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lemma root_test_convergence:
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  fixes f :: "nat \<Rightarrow> 'a::banach"
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  assumes f: "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> x" \<comment> "could be weakened to lim sup"
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    and "x < 1"
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  shows "summable f"
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proof -
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  have "0 \<le> x"
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    by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
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  from \<open>x < 1\<close> obtain z where z: "x < z" "z < 1"
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    by (metis dense)
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  from f \<open>x < z\<close> have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
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    by (rule order_tendstoD)
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  then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
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    using eventually_ge_at_top
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  proof eventually_elim
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    fix n
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    assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
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    from power_strict_mono[OF less, of n] n show "norm (f n) \<le> z ^ n"
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      by simp
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  qed
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  then show "summable f"
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    unfolding eventually_sequentially
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    using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _  summable_geometric])
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qed
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subsection \<open>More facts about binomial coefficients\<close>
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text \<open>
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  These facts could have been proven before, but having real numbers 
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  makes the proofs a lot easier.
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\<close>
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lemma central_binomial_odd:
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  "odd n \<Longrightarrow> n choose (Suc (n div 2)) = n choose (n div 2)"
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proof -
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  assume "odd n"
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  hence "Suc (n div 2) \<le> n" by presburger
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  hence "n choose (Suc (n div 2)) = n choose (n - Suc (n div 2))"
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    by (rule binomial_symmetric)
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  also from \<open>odd n\<close> have "n - Suc (n div 2) = n div 2" by presburger
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  finally show ?thesis .
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qed
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lemma binomial_less_binomial_Suc:
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  assumes k: "k < n div 2"
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  shows   "n choose k < n choose (Suc k)"
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    95
proof -
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    96
  from k have k': "k \<le> n" "Suc k \<le> n" by simp_all
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  from k' have "real (n choose k) = fact n / (fact k * fact (n - k))"
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    by (simp add: binomial_fact)
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  also from k' have "n - k = Suc (n - Suc k)" by simp
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  also from k' have "fact \<dots> = (real n - real k) * fact (n - Suc k)"
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    by (subst fact_Suc) (simp_all add: of_nat_diff)
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  also from k have "fact k = fact (Suc k) / (real k + 1)" by (simp add: field_simps)
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   103
  also have "fact n / (fact (Suc k) / (real k + 1) * ((real n - real k) * fact (n - Suc k))) =
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               (n choose (Suc k)) * ((real k + 1) / (real n - real k))"
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   105
    using k by (simp add: divide_simps binomial_fact)
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   106
  also from assms have "(real k + 1) / (real n - real k) < 1" by simp
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  finally show ?thesis using k by (simp add: mult_less_cancel_left)
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   108
qed
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   109
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lemma binomial_strict_mono:
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  assumes "k < k'" "2*k' \<le> n"
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   112
  shows   "n choose k < n choose k'"
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   113
proof -
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   114
  from assms have "k \<le> k' - 1" by simp
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   115
  thus ?thesis
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   116
  proof (induction rule: inc_induct)
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   117
    case base
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    with assms binomial_less_binomial_Suc[of "k' - 1" n] 
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   119
      show ?case by simp
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  next
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    case (step k)
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   122
    from step.prems step.hyps assms have "n choose k < n choose (Suc k)" 
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   123
      by (intro binomial_less_binomial_Suc) simp_all
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   124
    also have "\<dots> < n choose k'" by (rule step.IH)
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   125
    finally show ?case .
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   126
  qed
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   127
qed
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   128
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lemma binomial_mono:
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  assumes "k \<le> k'" "2*k' \<le> n"
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   131
  shows   "n choose k \<le> n choose k'"
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   132
  using assms binomial_strict_mono[of k k' n] by (cases "k = k'") simp_all
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   133
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lemma binomial_strict_antimono:
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  assumes "k < k'" "2 * k \<ge> n" "k' \<le> n"
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   136
  shows   "n choose k > n choose k'"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   137
proof -
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   138
  from assms have "n choose (n - k) > n choose (n - k')"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   139
    by (intro binomial_strict_mono) (simp_all add: algebra_simps)
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   140
  with assms show ?thesis by (simp add: binomial_symmetric [symmetric])
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   141
qed
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   142
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   143
lemma binomial_antimono:
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   144
  assumes "k \<le> k'" "k \<ge> n div 2" "k' \<le> n"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   145
  shows   "n choose k \<ge> n choose k'"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   146
proof (cases "k = k'")
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   147
  case False
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   148
  note not_eq = False
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   149
  show ?thesis
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   150
  proof (cases "k = n div 2 \<and> odd n")
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   151
    case False
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   152
    with assms(2) have "2*k \<ge> n" by presburger
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   153
    with not_eq assms binomial_strict_antimono[of k k' n] 
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   154
      show ?thesis by simp
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   155
  next
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   156
    case True
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   157
    have "n choose k' \<le> n choose (Suc (n div 2))"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   158
    proof (cases "k' = Suc (n div 2)") 
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   159
      case False
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   160
      with assms True not_eq have "Suc (n div 2) < k'" by simp
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   161
      with assms binomial_strict_antimono[of "Suc (n div 2)" k' n] True
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   162
        show ?thesis by auto
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   163
    qed simp_all
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   164
    also from True have "\<dots> = n choose k" by (simp add: central_binomial_odd)
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   165
    finally show ?thesis .
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   166
  qed
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   167
qed simp_all
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   168
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   169
lemma binomial_maximum: "n choose k \<le> n choose (n div 2)"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   170
proof -
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   171
  have "k \<le> n div 2 \<longleftrightarrow> 2*k \<le> n" by linarith
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   172
  consider "2*k \<le> n" | "2*k \<ge> n" "k \<le> n" | "k > n" by linarith
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   173
  thus ?thesis
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   174
  proof cases
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   175
    case 1
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   176
    thus ?thesis by (intro binomial_mono) linarith+
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   177
  next
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   178
    case 2
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   179
    thus ?thesis by (intro binomial_antimono) simp_all
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   180
  qed (simp_all add: binomial_eq_0)
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   181
qed
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   182
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   183
lemma binomial_maximum': "(2*n) choose k \<le> (2*n) choose n"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   184
  using binomial_maximum[of "2*n"] by simp
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   185
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   186
lemma central_binomial_lower_bound:
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   187
  assumes "n > 0"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   188
  shows   "4^n / (2*real n) \<le> real ((2*n) choose n)"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   189
proof -
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   190
  from binomial[of 1 1 "2*n"]
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   191
    have "4 ^ n = (\<Sum>k=0..2*n. (2*n) choose k)"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   192
    by (simp add: power_mult power2_eq_square One_nat_def [symmetric] del: One_nat_def)
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   193
  also have "{0..2*n} = {0<..<2*n} \<union> {0,2*n}" by auto
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   194
  also have "(\<Sum>k\<in>\<dots>. (2*n) choose k) = 
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   195
               (\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) + (\<Sum>k\<in>{0,2*n}. (2*n) choose k)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   196
    by (subst sum.union_disjoint) auto
63766
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   197
  also have "(\<Sum>k\<in>{0,2*n}. (2*n) choose k) \<le> (\<Sum>k\<le>1. (n choose k)\<^sup>2)" 
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   198
    by (cases n) simp_all
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   199
  also from assms have "\<dots> \<le> (\<Sum>k\<le>n. (n choose k)\<^sup>2)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   200
    by (intro sum_mono3) auto
63766
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   201
  also have "\<dots> = (2*n) choose n" by (rule choose_square_sum)
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   202
  also have "(\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) \<le> (\<Sum>k\<in>{0<..<2*n}. (2*n) choose n)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   203
    by (intro sum_mono binomial_maximum')
63766
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   204
  also have "\<dots> = card {0<..<2*n} * ((2*n) choose n)" by simp
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   205
  also have "card {0<..<2*n} \<le> 2*n - 1" by (cases n) simp_all
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   206
  also have "(2 * n - 1) * (2 * n choose n) + (2 * n choose n) = ((2*n) choose n) * (2*n)"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   207
    using assms by (simp add: algebra_simps)
63834
6a757f36997e tuned proofs;
wenzelm
parents: 63766
diff changeset
   208
  finally have "4 ^ n \<le> (2 * n choose n) * (2 * n)" by simp_all
63766
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   209
  hence "real (4 ^ n) \<le> real ((2 * n choose n) * (2 * n))"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   210
    by (subst of_nat_le_iff)
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   211
  with assms show ?thesis by (simp add: field_simps)
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   212
qed
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   213
63467
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   214
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   215
subsection \<open>Properties of Power Series\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   216
63467
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   217
lemma powser_zero [simp]: "(\<Sum>n. f n * 0 ^ n) = f 0"
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   218
  for f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   219
proof -
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   220
  have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   221
    by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   222
  then show ?thesis by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   223
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   224
63467
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   225
lemma powser_sums_zero: "(\<lambda>n. a n * 0^n) sums a 0"
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   226
  for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   227
  using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   228
  by simp
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   229
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   230
lemma powser_sums_zero_iff [simp]: "(\<lambda>n. a n * 0^n) sums x \<longleftrightarrow> a 0 = x"
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   231
  for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   232
  using powser_sums_zero sums_unique2 by blast
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   233
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   234
text \<open>
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   235
  Power series has a circle or radius of convergence: if it sums for \<open>x\<close>,
f3781c5fb03f misc tuning and modernization;
wenzelm
parents: 63417
diff changeset
   236
  then it sums absolutely for \<open>z\<close> with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   237
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   238
lemma powser_insidea:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   239
  fixes x z :: "'a::real_normed_div_algebra"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   240
  assumes 1: "summable (\<lambda>n. f n * x^n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   241
    and 2: "norm z < norm x"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   242
  shows "summable (\<lambda>n. norm (f n * z ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   243
proof -
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   244
  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   245
  from 1 have "(\<lambda>n. f n * x^n) \<longlonglongrightarrow> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   246
    by (rule summable_LIMSEQ_zero)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   247
  then have "convergent (\<lambda>n. f n * x^n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   248
    by (rule convergentI)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   249
  then have "Cauchy (\<lambda>n. f n * x^n)"
44726
8478eab380e9 generalize some lemmas
huffman
parents: 44725
diff changeset
   250
    by (rule convergent_Cauchy)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   251
  then have "Bseq (\<lambda>n. f n * x^n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   252
    by (rule Cauchy_Bseq)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   253
  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   254
    by (auto simp add: Bseq_def)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   255
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   256
  proof (intro exI allI impI)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   257
    fix n :: nat
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   258
    assume "0 \<le> n"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   259
    have "norm (norm (f n * z ^ n)) * norm (x^n) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   260
          norm (f n * x^n) * norm (z ^ n)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   261
      by (simp add: norm_mult abs_mult)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   262
    also have "\<dots> \<le> K * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   263
      by (simp only: mult_right_mono 4 norm_ge_zero)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   264
    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   265
      by (simp add: x_neq_0)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   266
    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   267
      by (simp only: mult.assoc)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   268
    finally show "norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   269
      by (simp add: mult_le_cancel_right x_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   270
  qed
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   271
  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   272
  proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   273
    from 2 have "norm (norm (z * inverse x)) < 1"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   274
      using x_neq_0
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   275
      by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   276
    then have "summable (\<lambda>n. norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   277
      by (rule summable_geometric)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   278
    then have "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   279
      by (rule summable_mult)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   280
    then show "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   281
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   282
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   283
          power_inverse norm_power mult.assoc)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   284
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   285
  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   286
    by (rule summable_comparison_test)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   287
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   288
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   289
lemma powser_inside:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   290
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   291
  shows
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   292
    "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   293
      summable (\<lambda>n. f n * (z ^ n))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   294
  by (rule powser_insidea [THEN summable_norm_cancel])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   295
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   296
lemma powser_times_n_limit_0:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   297
  fixes x :: "'a::{real_normed_div_algebra,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   298
  assumes "norm x < 1"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   299
    shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   300
proof -
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   301
  have "norm x / (1 - norm x) \<ge> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   302
    using assms by (auto simp: divide_simps)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   303
  moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   304
    using ex_le_of_int by (meson ex_less_of_int)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
   305
  ultimately have N0: "N>0"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   306
    by auto
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
   307
  then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   308
    using N assms by (auto simp: field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   309
  have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) \<le>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   310
      real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N \<le> int n" for n :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   311
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   312
    from that have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   313
      by (simp add: algebra_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   314
    then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n)) \<le>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   315
        (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   316
      using N0 mult_mono by fastforce
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   317
    then show ?thesis
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   318
      by (simp add: algebra_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   319
  qed
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   320
  show ?thesis using *
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   321
    by (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   322
      (simp add: N0 norm_mult field_simps ** del: of_nat_Suc of_int_add)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   323
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   324
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   325
corollary lim_n_over_pown:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   326
  fixes x :: "'a::{real_normed_field,banach}"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   327
  shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   328
  using powser_times_n_limit_0 [of "inverse x"]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   329
  by (simp add: norm_divide divide_simps)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   330
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   331
lemma sum_split_even_odd:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   332
  fixes f :: "nat \<Rightarrow> real"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   333
  shows "(\<Sum>i<2 * n. if even i then f i else g i) = (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   334
proof (induct n)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   335
  case 0
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   336
  then show ?case by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   337
next
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   338
  case (Suc n)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   339
  have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   340
    (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   341
    using Suc.hyps unfolding One_nat_def by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   342
  also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   343
    by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   344
  finally show ?case .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   345
qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   346
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   347
lemma sums_if':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   348
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   349
  assumes "g sums x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   350
  shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   351
  unfolding sums_def
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   352
proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   353
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   354
  assume "0 < r"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   355
  from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this]
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   356
  obtain no where no_eq: "\<And>n. n \<ge> no \<Longrightarrow> (norm (sum g {..<n} - x) < r)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   357
    by blast
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   358
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   359
  let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   360
  have "(norm (?SUM m - x) < r)" if "m \<ge> 2 * no" for m
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   361
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   362
    from that have "m div 2 \<ge> no" by auto
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   363
    have sum_eq: "?SUM (2 * (m div 2)) = sum g {..< m div 2}"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   364
      using sum_split_even_odd by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   365
    then have "(norm (?SUM (2 * (m div 2)) - x) < r)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   366
      using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   367
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   368
    have "?SUM (2 * (m div 2)) = ?SUM m"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   369
    proof (cases "even m")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   370
      case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   371
      then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   372
        by (auto simp add: even_two_times_div_two)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   373
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   374
      case False
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58740
diff changeset
   375
      then have eq: "Suc (2 * (m div 2)) = m" by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   376
      then have "even (2 * (m div 2))" using \<open>odd m\<close> by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   377
      have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   378
      also have "\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   379
      finally show ?thesis by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   380
    qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   381
    ultimately show ?thesis by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   382
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   383
  then show "\<exists>no. \<forall> m \<ge> no. norm (?SUM m - x) < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   384
    by blast
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   385
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   386
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   387
lemma sums_if:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   388
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   389
  assumes "g sums x" and "f sums y"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   390
  shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   391
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   392
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   393
  have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   394
    for B T E
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   395
    by (cases B) auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   396
  have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   397
    using sums_if'[OF \<open>g sums x\<close>] .
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   398
  have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   399
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   400
  have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] .
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   401
  from this[unfolded sums_def, THEN LIMSEQ_Suc]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   402
  have "(\<lambda>n. if even n then f (n div 2) else 0) sums y"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   403
    by (simp add: lessThan_Suc_eq_insert_0 sum_atLeast1_atMost_eq image_Suc_lessThan
63566
e5abbdee461a more accurate cong del;
wenzelm
parents: 63558
diff changeset
   404
        if_eq sums_def cong del: if_weak_cong)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   405
  from sums_add[OF g_sums this] show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   406
    by (simp only: if_sum)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   407
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   408
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   409
subsection \<open>Alternating series test / Leibniz formula\<close>
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   410
(* FIXME: generalise these results from the reals via type classes? *)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   411
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   412
lemma sums_alternating_upper_lower:
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   413
  fixes a :: "nat \<Rightarrow> real"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   414
  assumes mono: "\<And>n. a (Suc n) \<le> a n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   415
    and a_pos: "\<And>n. 0 \<le> a n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   416
    and "a \<longlonglongrightarrow> 0"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   417
  shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> l) \<and>
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   418
             ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   419
  (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   420
proof (rule nested_sequence_unique)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   421
  have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   422
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   423
  show "\<forall>n. ?f n \<le> ?f (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   424
  proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   425
    show "?f n \<le> ?f (Suc n)" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   426
      using mono[of "2*n"] by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   427
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   428
  show "\<forall>n. ?g (Suc n) \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   429
  proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   430
    show "?g (Suc n) \<le> ?g n" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   431
      using mono[of "Suc (2*n)"] by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   432
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   433
  show "\<forall>n. ?f n \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   434
  proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   435
    show "?f n \<le> ?g n" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   436
      using fg_diff a_pos by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   437
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   438
  show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   439
    unfolding fg_diff
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   440
  proof (rule LIMSEQ_I)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   441
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   442
    assume "0 < r"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   443
    with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   444
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   445
    then have "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   446
      by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   447
    then show "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   448
      by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   449
  qed
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   450
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   451
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   452
lemma summable_Leibniz':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   453
  fixes a :: "nat \<Rightarrow> real"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   454
  assumes a_zero: "a \<longlonglongrightarrow> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   455
    and a_pos: "\<And>n. 0 \<le> a n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   456
    and a_monotone: "\<And>n. a (Suc n) \<le> a n"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   457
  shows summable: "summable (\<lambda> n. (-1)^n * a n)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   458
    and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   459
    and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   460
    and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   461
    and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   462
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   463
  let ?S = "\<lambda>n. (-1)^n * a n"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   464
  let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   465
  let ?f = "\<lambda>n. ?P (2 * n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   466
  let ?g = "\<lambda>n. ?P (2 * n + 1)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   467
  obtain l :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   468
    where below_l: "\<forall> n. ?f n \<le> l"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   469
      and "?f \<longlonglongrightarrow> l"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   470
      and above_l: "\<forall> n. l \<le> ?g n"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   471
      and "?g \<longlonglongrightarrow> l"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   472
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   473
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   474
  let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   475
  have "?Sa \<longlonglongrightarrow> l"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   476
  proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   477
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   478
    assume "0 < r"
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   479
    with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   480
    obtain f_no where f: "\<And>n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   481
      by auto
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   482
    from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   483
    obtain g_no where g: "\<And>n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   484
      by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   485
    have "norm (?Sa n - l) < r" if "n \<ge> (max (2 * f_no) (2 * g_no))" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   486
    proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   487
      from that have "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   488
      show ?thesis
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   489
      proof (cases "even n")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   490
        case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   491
        then have n_eq: "2 * (n div 2) = n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   492
          by (simp add: even_two_times_div_two)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   493
        with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   494
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   495
        from f[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   496
          unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   497
      next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   498
        case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   499
        then have "even (n - 1)" by simp
58710
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58709
diff changeset
   500
        then have n_eq: "2 * ((n - 1) div 2) = n - 1"
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58709
diff changeset
   501
          by (simp add: even_two_times_div_two)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   502
        then have range_eq: "n - 1 + 1 = n"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   503
          using odd_pos[OF False] by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   504
        from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   505
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   506
        from g[OF this] show ?thesis
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   507
          by (simp only: n_eq range_eq)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   508
      qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   509
    qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   510
    then show "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   511
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   512
  then have sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   513
    by (simp only: sums_def)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   514
  then show "summable ?S"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   515
    by (auto simp: summable_def)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   516
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   517
  have "l = suminf ?S" by (rule sums_unique[OF sums_l])
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   518
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   519
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   520
  show "suminf ?S \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   521
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   522
  show "?f n \<le> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   523
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   524
  show "?g \<longlonglongrightarrow> suminf ?S"
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   525
    using \<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   526
  show "?f \<longlonglongrightarrow> suminf ?S"
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   527
    using \<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   528
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   529
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   530
theorem summable_Leibniz:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   531
  fixes a :: "nat \<Rightarrow> real"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   532
  assumes a_zero: "a \<longlonglongrightarrow> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   533
    and "monoseq a"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   534
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   535
    and "0 < a 0 \<longrightarrow>
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   536
      (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   537
    and "a 0 < 0 \<longrightarrow>
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   538
      (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg")
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   539
    and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?f")
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   540
    and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?g")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   541
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   542
  have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   543
  proof (cases "(\<forall>n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   544
    case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   545
    then have ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   546
      and ge0: "\<And>n. 0 \<le> a n"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   547
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   548
    have mono: "a (Suc n) \<le> a n" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   549
      using ord[where n="Suc n" and m=n] by auto
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   550
    note leibniz = summable_Leibniz'[OF \<open>a \<longlonglongrightarrow> 0\<close> ge0]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   551
    from leibniz[OF mono]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   552
    show ?thesis using \<open>0 \<le> a 0\<close> by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   553
  next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   554
    let ?a = "\<lambda>n. - a n"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   555
    case False
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   556
    with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   557
    have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   558
    then have ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   559
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   560
    have monotone: "?a (Suc n) \<le> ?a n" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   561
      using ord[where n="Suc n" and m=n] by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   562
    note leibniz =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   563
      summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   564
        OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   565
    have "summable (\<lambda> n. (-1)^n * ?a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   566
      using leibniz(1) by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   567
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   568
      unfolding summable_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   569
    from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   570
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   571
    then have ?summable by (auto simp: summable_def)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   572
    moreover
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   573
    have "\<bar>- a - - b\<bar> = \<bar>a - b\<bar>" for a b :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   574
      unfolding minus_diff_minus by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   575
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   576
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   577
    have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   578
      by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   579
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   580
    have ?pos using \<open>0 \<le> ?a 0\<close> by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   581
    moreover have ?neg
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   582
      using leibniz(2,4)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   583
      unfolding mult_minus_right sum_negf move_minus neg_le_iff_le
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   584
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   585
    moreover have ?f and ?g
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   586
      using leibniz(3,5)[unfolded mult_minus_right sum_negf move_minus, THEN tendsto_minus_cancel]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   587
      by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   588
    ultimately show ?thesis by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   589
  qed
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   590
  then show ?summable and ?pos and ?neg and ?f and ?g
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   591
    by safe
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   592
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   593
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   594
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   595
subsection \<open>Term-by-Term Differentiability of Power Series\<close>
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   596
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   597
definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   598
  where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   599
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   600
text \<open>Lemma about distributing negation over it.\<close>
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   601
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   602
  by (simp add: diffs_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   603
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   604
lemma diffs_equiv:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   605
  fixes x :: "'a::{real_normed_vector,ring_1}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   606
  shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   607
    (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   608
  unfolding diffs_def
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   609
  by (simp add: summable_sums sums_Suc_imp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   610
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   611
lemma lemma_termdiff1:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   612
  fixes z :: "'a :: {monoid_mult,comm_ring}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   613
  shows "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   614
    (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   615
  by (auto simp add: algebra_simps power_add [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   616
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   617
lemma sumr_diff_mult_const2: "sum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   618
  for r :: "'a::ring_1"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   619
  by (simp add: sum_subtractf)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   620
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
   621
lemma lemma_realpow_rev_sumr:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   622
  "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) = (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   623
  by (subst nat_diff_sum_reindex[symmetric]) simp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
   624
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   625
lemma lemma_termdiff2:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   626
  fixes h :: "'a::field"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   627
  assumes h: "h \<noteq> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   628
  shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   629
    h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   630
    (is "?lhs = ?rhs")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   631
  apply (subgoal_tac "h * ?lhs = h * ?rhs")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   632
   apply (simp add: h)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   633
  apply (simp add: right_diff_distrib diff_divide_distrib h)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   634
  apply (simp add: mult.assoc [symmetric])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   635
  apply (cases n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   636
  apply simp
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   637
  apply (simp add: diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   638
      del: power_Suc sum_lessThan_Suc of_nat_Suc)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   639
  apply (subst lemma_realpow_rev_sumr)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   640
  apply (subst sumr_diff_mult_const2)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   641
  apply simp
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   642
  apply (simp only: lemma_termdiff1 sum_distrib_left)
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   643
  apply (rule sum.cong [OF refl])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   644
  apply (simp add: less_iff_Suc_add)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   645
  apply clarify
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   646
  apply (simp add: sum_distrib_left diff_power_eq_sum ac_simps
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   647
      del: sum_lessThan_Suc power_Suc)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   648
  apply (subst mult.assoc [symmetric], subst power_add [symmetric])
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   649
  apply (simp add: ac_simps)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   650
  done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   651
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   652
lemma real_sum_nat_ivl_bounded2:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   653
  fixes K :: "'a::linordered_semidom"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   654
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   655
    and K: "0 \<le> K"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   656
  shows "sum f {..<n-k} \<le> of_nat n * K"
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   657
  apply (rule order_trans [OF sum_mono])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   658
   apply (rule f)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   659
   apply simp
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   660
  apply (simp add: mult_right_mono K)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   661
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   662
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   663
lemma lemma_termdiff3:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   664
  fixes h z :: "'a::real_normed_field"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   665
  assumes 1: "h \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   666
    and 2: "norm z \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   667
    and 3: "norm (z + h) \<le> K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   668
  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \<le>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   669
    of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   670
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   671
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   672
    norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   673
    by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   674
  also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   675
  proof (rule mult_right_mono [OF _ norm_ge_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   676
    from norm_ge_zero 2 have K: "0 \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   677
      by (rule order_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   678
    have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   679
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   680
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   681
      apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   682
      done
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   683
    show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) \<le>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   684
        of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   685
      apply (intro
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   686
          order_trans [OF norm_sum]
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   687
          real_sum_nat_ivl_bounded2
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   688
          mult_nonneg_nonneg
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   689
          of_nat_0_le_iff
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   690
          zero_le_power K)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   691
      apply (rule le_Kn)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   692
      apply simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   693
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   694
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   695
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   696
    by (simp only: mult.assoc)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   697
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   698
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   699
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   700
lemma lemma_termdiff4:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   701
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   702
    and k :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   703
  assumes k: "0 < k"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   704
    and le: "\<And>h. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (f h) \<le> K * norm h"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   705
  shows "f \<midarrow>0\<rightarrow> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   706
proof (rule tendsto_norm_zero_cancel)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   707
  show "(\<lambda>h. norm (f h)) \<midarrow>0\<rightarrow> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   708
  proof (rule real_tendsto_sandwich)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   709
    show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   710
      by simp
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   711
    show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   712
      using k by (auto simp add: eventually_at dist_norm le)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   713
    show "(\<lambda>h. 0) \<midarrow>(0::'a)\<rightarrow> (0::real)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   714
      by (rule tendsto_const)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   715
    have "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> K * norm (0::'a)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   716
      by (intro tendsto_intros)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   717
    then show "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   718
      by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   719
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   720
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   721
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   722
lemma lemma_termdiff5:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   723
  fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   724
    and k :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   725
  assumes k: "0 < k"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   726
    and f: "summable f"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   727
    and le: "\<And>h n. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (g h n) \<le> f n * norm h"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   728
  shows "(\<lambda>h. suminf (g h)) \<midarrow>0\<rightarrow> 0"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   729
proof (rule lemma_termdiff4 [OF k])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   730
  fix h :: 'a
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   731
  assume "h \<noteq> 0" and "norm h < k"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   732
  then have 1: "\<forall>n. norm (g h n) \<le> f n * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   733
    by (simp add: le)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   734
  then have "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   735
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   736
  moreover from f have 2: "summable (\<lambda>n. f n * norm h)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   737
    by (rule summable_mult2)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   738
  ultimately have 3: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   739
    by (rule summable_comparison_test)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   740
  then have "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   741
    by (rule summable_norm)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   742
  also from 1 3 2 have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
   743
    by (rule suminf_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   744
  also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   745
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   746
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   747
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   748
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   749
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   750
(* FIXME: Long proofs *)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   751
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   752
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   753
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   754
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   755
    and 2: "norm x < norm K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   756
  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   757
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   758
  from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   759
    by fast
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   760
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   761
    by (rule order_le_less_trans)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   762
  then have r_neq_0: "r \<noteq> 0" by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   763
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   764
  proof (rule lemma_termdiff5)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   765
    show "0 < r - norm x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   766
      using r1 by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   767
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   768
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   769
    with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   770
      by (rule powser_insidea)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   771
    then have "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   772
      using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   773
    then have "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   774
      by (rule diffs_equiv [THEN sums_summable])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   775
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   776
      (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   777
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   778
      apply (simp add: diffs_def)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   779
      apply (case_tac n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   780
       apply (simp_all add: r_neq_0)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   781
      done
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   782
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   783
      (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   784
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   785
    also have
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   786
      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   787
       (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   788
      apply (rule ext)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   789
      apply (case_tac n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   790
       apply simp
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 54576
diff changeset
   791
      apply (rename_tac nat)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   792
      apply (case_tac nat)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   793
       apply simp
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   794
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   795
      done
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   796
    finally show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   797
  next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   798
    fix h :: 'a
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   799
    fix n :: nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   800
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   801
    assume "norm h < r - norm x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   802
    then have "norm x + norm h < r" by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   803
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   804
      by (rule order_le_less_trans)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   805
    show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<le>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   806
      norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   807
      apply (simp only: norm_mult mult.assoc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   808
      apply (rule mult_left_mono [OF _ norm_ge_zero])
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   809
      apply (simp add: mult.assoc [symmetric])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   810
      apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   811
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   812
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   813
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   814
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   815
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   816
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   817
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   818
    and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   819
    and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   820
    and 4: "norm x < norm K"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   821
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   822
  unfolding DERIV_def
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   823
proof (rule LIM_zero_cancel)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   824
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   825
            - suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   826
  proof (rule LIM_equal2)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   827
    show "0 < norm K - norm x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   828
      using 4 by (simp add: less_diff_eq)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   829
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   830
    fix h :: 'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   831
    assume "norm (h - 0) < norm K - norm x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   832
    then have "norm x + norm h < norm K" by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   833
    then have 5: "norm (x + h) < norm K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   834
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   835
    have "summable (\<lambda>n. c n * x^n)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   836
      and "summable (\<lambda>n. c n * (x + h) ^ n)"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   837
      and "summable (\<lambda>n. diffs c n * x^n)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   838
      using 1 2 4 5 by (auto elim: powser_inside)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   839
    then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   840
          (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   841
      by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   842
    then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   843
          (\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   844
      by (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   845
  next
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   846
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   847
      by (rule termdiffs_aux [OF 3 4])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   848
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   849
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   850
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   851
subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close>
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   852
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   853
lemma termdiff_converges:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   854
  fixes x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   855
  assumes K: "norm x < K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   856
    and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   857
  shows "summable (\<lambda>n. diffs c n * x ^ n)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   858
proof (cases "x = 0")
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   859
  case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   860
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   861
    using powser_sums_zero sums_summable by auto
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   862
next
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   863
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   864
  then have "K > 0"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   865
    using K less_trans zero_less_norm_iff by blast
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   866
  then obtain r :: real where r: "norm x < norm r" "norm r < K" "r > 0"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   867
    using K False
61738
c4f6031f1310 New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents: 61694
diff changeset
   868
    by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
   869
  have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> 0"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   870
    using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   871
  then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   872
    using r unfolding LIMSEQ_iff
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   873
    apply (drule_tac x=1 in spec)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   874
    apply (auto simp: norm_divide norm_mult norm_power field_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   875
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   876
  have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   877
    apply (rule summable_comparison_test' [of "\<lambda>n. norm(c n * (of_real r) ^ n)" N])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   878
     apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   879
    using N r norm_of_real [of "r + K", where 'a = 'a]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   880
      apply (auto simp add: norm_divide norm_mult norm_power field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   881
    apply (fastforce simp: less_eq_real_def)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   882
    done
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   883
  then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   884
    using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1]
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   885
    by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   886
  then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   887
    using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60762
diff changeset
   888
    by (simp add: mult.assoc) (auto simp: ac_simps)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
   889
  then show ?thesis
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   890
    by (simp add: diffs_def)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   891
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   892
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   893
lemma termdiff_converges_all:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   894
  fixes x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   895
  assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   896
  shows "summable (\<lambda>n. diffs c n * x^n)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   897
  apply (rule termdiff_converges [where K = "1 + norm x"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   898
  using assms
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   899
   apply auto
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   900
  done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   901
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   902
lemma termdiffs_strong:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   903
  fixes K x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   904
  assumes sm: "summable (\<lambda>n. c n * K ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   905
    and K: "norm x < norm K"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   906
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   907
proof -
60762
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
   908
  have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   909
    using K
61738
c4f6031f1310 New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents: 61694
diff changeset
   910
    apply (auto simp: norm_divide field_simps)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   911
    apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   912
     apply (auto simp: mult_2_right norm_triangle_mono)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   913
    done
60762
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
   914
  then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
   915
    by simp
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   916
  have "summable (\<lambda>n. c n * (of_real (norm x + norm K) / 2) ^ n)"
60762
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
   917
    by (metis K2 summable_norm_cancel [OF powser_insidea [OF sm]] add.commute of_real_add)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   918
  moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs c n * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   919
    by (blast intro: sm termdiff_converges powser_inside)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   920
  moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs(diffs c) n * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   921
    by (blast intro: sm termdiff_converges powser_inside)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   922
  ultimately show ?thesis
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   923
    apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   924
      apply (auto simp: field_simps)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   925
    using K
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   926
    apply (simp_all add: of_real_add [symmetric] del: of_real_add)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   927
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   928
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   929
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   930
lemma termdiffs_strong_converges_everywhere:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   931
  fixes K x :: "'a::{real_normed_field,banach}"
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   932
  assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   933
  shows "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)"
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   934
  using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   935
  by (force simp del: of_real_add)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
   936
63721
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   937
lemma termdiffs_strong':
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   938
  fixes z :: "'a :: {real_normed_field,banach}"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   939
  assumes "\<And>z. norm z < K \<Longrightarrow> summable (\<lambda>n. c n * z ^ n)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   940
  assumes "norm z < K"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   941
  shows   "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   942
proof (rule termdiffs_strong)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   943
  define L :: real where "L =  (norm z + K) / 2"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   944
  have "0 \<le> norm z" by simp
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   945
  also note \<open>norm z < K\<close>
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   946
  finally have K: "K \<ge> 0" by simp
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   947
  from assms K have L: "L \<ge> 0" "norm z < L" "L < K" by (simp_all add: L_def)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   948
  from L show "norm z < norm (of_real L :: 'a)" by simp
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   949
  from L show "summable (\<lambda>n. c n * of_real L ^ n)" by (intro assms(1)) simp_all
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   950
qed
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   951
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   952
lemma termdiffs_sums_strong:
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   953
  fixes z :: "'a :: {banach,real_normed_field}"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   954
  assumes sums: "\<And>z. norm z < K \<Longrightarrow> (\<lambda>n. c n * z ^ n) sums f z"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   955
  assumes deriv: "(f has_field_derivative f') (at z)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   956
  assumes norm: "norm z < K"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   957
  shows   "(\<lambda>n. diffs c n * z ^ n) sums f'"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   958
proof -
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   959
  have summable: "summable (\<lambda>n. diffs c n * z^n)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   960
    by (intro termdiff_converges[OF norm] sums_summable[OF sums])
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   961
  from norm have "eventually (\<lambda>z. z \<in> norm -` {..<K}) (nhds z)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   962
    by (intro eventually_nhds_in_open open_vimage) 
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   963
       (simp_all add: continuous_on_norm continuous_on_id)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   964
  hence eq: "eventually (\<lambda>z. (\<Sum>n. c n * z^n) = f z) (nhds z)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   965
    by eventually_elim (insert sums, simp add: sums_iff)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   966
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   967
  have "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   968
    by (intro termdiffs_strong'[OF _ norm] sums_summable[OF sums])
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   969
  hence "(f has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   970
    by (subst (asm) DERIV_cong_ev[OF refl eq refl])
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   971
  from this and deriv have "(\<Sum>n. diffs c n * z^n) = f'" by (rule DERIV_unique)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   972
  with summable show ?thesis by (simp add: sums_iff)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   973
qed
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   974
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   975
lemma isCont_powser:
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   976
  fixes K x :: "'a::{real_normed_field,banach}"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   977
  assumes "summable (\<lambda>n. c n * K ^ n)"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   978
  assumes "norm x < norm K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   979
  shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   980
  using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
   981
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   982
lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   983
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   984
lemma isCont_powser_converges_everywhere:
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   985
  fixes K x :: "'a::{real_normed_field,banach}"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   986
  assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   987
  shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   988
  using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   989
  by (force intro!: DERIV_isCont simp del: of_real_add)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   990
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
   991
lemma powser_limit_0:
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   992
  fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   993
  assumes s: "0 < s"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   994
    and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   995
  shows "(f \<longlongrightarrow> a 0) (at 0)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   996
proof -
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   997
  have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   998
    apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   999
    using s
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1000
    apply (auto simp: norm_divide)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1001
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1002
  then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1003
    apply (rule termdiffs_strong)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1004
    using s
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1005
    apply (auto simp: norm_divide)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1006
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1007
  then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1008
    by (blast intro: DERIV_continuous)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1009
  then have "((\<lambda>x. \<Sum>n. a n * x ^ n) \<longlongrightarrow> a 0) (at 0)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1010
    by (simp add: continuous_within)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
  1011
  then show ?thesis
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1012
    apply (rule Lim_transform)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1013
    apply (auto simp add: LIM_eq)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1014
    apply (rule_tac x="s" in exI)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
  1015
    using s
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1016
    apply (auto simp: sm [THEN sums_unique])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1017
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1018
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1019
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
  1020
lemma powser_limit_0_strong:
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1021
  fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1022
  assumes s: "0 < s"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1023
    and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1024
  shows "(f \<longlongrightarrow> a 0) (at 0)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1025
proof -
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1026
  have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1027
    apply (rule powser_limit_0 [OF s])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1028
    apply (case_tac "x = 0")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1029
     apply (auto simp add: powser_sums_zero sm)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1030
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1031
  show ?thesis
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1032
    apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1033
     apply (simp_all add: *)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1034
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1035
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1036
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1037
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1038
subsection \<open>Derivability of power series\<close>
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1039
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1040
lemma DERIV_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1041
  fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1042
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1043
    and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1044
    and x0_in_I: "x0 \<in> {a <..< b}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1045
    and "summable (f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1046
    and "summable L"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1047
    and L_def: "\<And>n x y. x \<in> {a <..< b} \<Longrightarrow> y \<in> {a <..< b} \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1048
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  1049
  unfolding DERIV_def
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1050
proof (rule LIM_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1051
  fix r :: real
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1052
  assume "0 < r" then have "0 < r/3" by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1053
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1054
  obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1055
    using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1056
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1057
  obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1058
    using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1059
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1060
  let ?N = "Suc (max N_L N_f')"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1061
  have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1062
    and L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1063
    using N_L[of "?N"] and N_f' [of "?N"] by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1064
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1065
  let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1066
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1067
  let ?r = "r / (3 * real ?N)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1068
  from \<open>0 < r\<close> have "0 < ?r" by simp
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1069
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1070
  let ?s = "\<lambda>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)"
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62949
diff changeset
  1071
  define S' where "S' = Min (?s ` {..< ?N })"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1072
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1073
  have "0 < S'"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1074
    unfolding S'_def
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1075
  proof (rule iffD2[OF Min_gr_iff])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1076
    show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1077
    proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1078
      fix x
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1079
      assume "x \<in> ?s ` {..<?N}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1080
      then obtain n where "x = ?s n" and "n \<in> {..<?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1081
        using image_iff[THEN iffD1] by blast
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1082
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1083
      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)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1084
        by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1085
      have "0 < ?s n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1086
        by (rule someI2[where a=s]) (auto simp add: s_bound simp del: of_nat_Suc)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1087
      then show "0 < x" by (simp only: \<open>x = ?s n\<close>)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1088
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1089
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1090
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62949
diff changeset
  1091
  define S where "S = min (min (x0 - a) (b - x0)) S'"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1092
  then have "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1093
    and "S \<le> S'" using x0_in_I and \<open>0 < S'\<close>
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1094
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1095
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1096
  have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1097
    if "x \<noteq> 0" and "\<bar>x\<bar> < S" for x
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1098
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1099
    from that have x_in_I: "x0 + x \<in> {a <..< b}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1100
      using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1101
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1102
    note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1103
    note div_smbl = summable_divide[OF diff_smbl]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1104
    note all_smbl = summable_diff[OF div_smbl \<open>summable (f' x0)\<close>]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1105
    note ign = summable_ignore_initial_segment[where k="?N"]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1106
    note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1107
    note div_shft_smbl = summable_divide[OF diff_shft_smbl]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1108
    note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1109
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1110
    have 1: "\<bar>(\<bar>?diff (n + ?N) x\<bar>)\<bar> \<le> L (n + ?N)" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1111
    proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1112
      have "\<bar>?diff (n + ?N) x\<bar> \<le> L (n + ?N) * \<bar>(x0 + x) - x0\<bar> / \<bar>x\<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1113
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1114
        by (simp only: abs_divide)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1115
      with \<open>x \<noteq> 0\<close> show ?thesis by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1116
    qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1117
    note 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1118
    from 1 have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1119
      by (metis (lifting) abs_idempotent
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1120
          order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1121
    then have "\<bar>\<Sum>i. ?diff (i + ?N) x\<bar> \<le> r / 3" (is "?L_part \<le> r/3")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1122
      using L_estimate by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1123
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1124
    have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n\<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n\<bar>)" ..
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1125
    also have "\<dots> < (\<Sum>n<?N. ?r)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1126
    proof (rule sum_strict_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1127
      fix n
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1128
      assume "n \<in> {..< ?N}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1129
      have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> .
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1130
      also have "S \<le> S'" using \<open>S \<le> S'\<close> .
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1131
      also have "S' \<le> ?s n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1132
        unfolding S'_def
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1133
      proof (rule Min_le_iff[THEN iffD2])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1134
        have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1135
          using \<open>n \<in> {..< ?N}\<close> by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1136
        then show "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1137
          by blast
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1138
      qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1139
      finally have "\<bar>x\<bar> < ?s n" .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1140
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1141
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>,
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1142
          unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1143
      have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1144
      with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1145
        by blast
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1146
    qed auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1147
    also have "\<dots> = of_nat (card {..<?N}) * ?r"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1148
      by (rule sum_constant)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1149
    also have "\<dots> = real ?N * ?r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1150
      by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1151
    also have "\<dots> = r/3"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1152
      by (auto simp del: of_nat_Suc)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1153
    finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1154
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1155
    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1156
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1157
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1158
      unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1159
      using suminf_divide[OF diff_smbl, symmetric] by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1160
    also have "\<dots> \<le> ?diff_part + \<bar>(\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N))\<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1161
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1162
      unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1163
      apply (subst (5) add.commute)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1164
      apply (rule abs_triangle_ineq)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1165
      done
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1166
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1167
      using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1168
    also have "\<dots> < r /3 + r/3 + r/3"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1169
      using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close>
36842
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
  1170
      by (rule add_strict_mono [OF add_less_le_mono])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1171
    finally show ?thesis
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1172
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1173
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1174
  then show "\<exists>s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1175
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1176
    using \<open>0 < S\<close> by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1177
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1178
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1179
lemma DERIV_power_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1180
  fixes f :: "nat \<Rightarrow> real"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1181
  assumes converges: "\<And>x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda>n. f n * real (Suc n) * x^n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1182
    and x0_in_I: "x0 \<in> {-R <..< R}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1183
    and "0 < R"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1184
  shows "DERIV (\<lambda>x. (\<Sum>n. f n * x^(Suc n))) x0 :> (\<Sum>n. f n * real (Suc n) * x0^n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1185
    (is "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)")