src/HOL/Transcendental.thy
author wenzelm
Mon, 12 Apr 2021 14:14:47 +0200
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tuned;
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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 Series Deriv NthRoot
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begin
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text \<open>A theorem about the factcorial function on the 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> \<open>could be weakened to lim sup\<close>
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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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    83
proof -
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    84
  assume "odd n"
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  hence "Suc (n div 2) \<le> n" by presburger
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    86
  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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    89
  finally show ?thesis .
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    90
qed
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    91
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lemma binomial_less_binomial_Suc:
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  assumes k: "k < n div 2"
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    94
  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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    97
  from k' have "real (n choose k) = fact n / (fact k * fact (n - k))"
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    98
    by (simp add: binomial_fact)
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    99
  also from k' have "n - k = Suc (n - Suc k)" by simp
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   100
  also from k' have "fact \<dots> = (real n - real k) * fact (n - Suc k)"
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   101
    by (subst fact_Suc) (simp_all add: of_nat_diff)
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   102
  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))) =
695d60817cb1 Some facts about factorial and binomial coefficients
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   104
               (n choose (Suc k)) * ((real k + 1) / (real n - real k))"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
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   105
    using k by (simp add: field_split_simps binomial_fact)
63766
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   106
  also from assms have "(real k + 1) / (real n - real k) < 1" by simp
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   107
  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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parents: 63721
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   113
proof -
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   114
  from assms have "k \<le> k' - 1" by simp
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parents: 63721
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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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f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
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   118
    with assms binomial_less_binomial_Suc[of "k' - 1" n]
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   119
      show ?case by simp
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   120
  next
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   121
    case (step k)
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
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diff changeset
   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
695d60817cb1 Some facts about factorial and binomial coefficients
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   128
695d60817cb1 Some facts about factorial and binomial coefficients
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   129
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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   134
lemma binomial_strict_antimono:
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   135
  assumes "k < k'" "2 * k \<ge> n" "k' \<le> n"
695d60817cb1 Some facts about factorial and binomial coefficients
Manuel Eberl <eberlm@in.tum.de>
parents: 63721
diff changeset
   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
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65204
diff changeset
   153
    with not_eq assms binomial_strict_antimono[of k k' n]
63766
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))"
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65204
diff changeset
   158
    proof (cases "k' = Suc (n div 2)")
63766
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"]
68077
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   191
    have "4 ^ n = (\<Sum>k\<le>2*n. (2*n) choose k)"
63766
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)
68077
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   193
  also have "{..2*n} = {0<..<2*n} \<union> {0,2*n}" by auto
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65204
diff changeset
   194
  also have "(\<Sum>k\<in>\<dots>. (2*n) choose k) =
68077
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
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
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65204
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)"
63766
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)"
65680
378a2f11bec9 Simplification of some proofs. Also key lemmas using !! rather than ! in premises
paulson <lp15@cam.ac.uk>
parents: 65583
diff changeset
   200
    by (intro sum_mono2) 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>,
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
   236
  then it sums absolutely for \<open>z\<close> with \<^term>\<open>\<bar>z\<bar> < \<bar>x\<bar>\<close>.\<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"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   254
    by (auto simp: Bseq_def)
63558
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"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
   302
    using assms by (auto simp: field_split_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"]
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
   329
  by (simp add: norm_divide field_split_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
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   372
        by (auto simp: 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"
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
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))))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   615
  by (auto simp: 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
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   621
lemma lemma_termdiff2:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   622
  fixes h :: "'a::field"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   623
  assumes h: "h \<noteq> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   624
  shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   625
         h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   626
    (is "?lhs = ?rhs")
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   627
proof (cases n)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   628
  case (Suc m)
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   629
  have 0: "\<And>x k. (\<Sum>n<Suc k. h * (z ^ x * (z ^ (k - n) * (h + z) ^ n))) =
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   630
                 (\<Sum>j<Suc k.  h * ((h + z) ^ j * z ^ (x + k - j)))"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   631
    by (auto simp add: power_add [symmetric] mult.commute intro: sum.cong)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   632
  have *: "(\<Sum>i<m. z ^ i * ((z + h) ^ (m - i) - z ^ (m - i))) =
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   633
           (\<Sum>i<m. \<Sum>j<m - i. h * ((z + h) ^ j * z ^ (m - Suc j)))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   634
    by (force simp add: less_iff_Suc_add sum_distrib_left diff_power_eq_sum ac_simps 0
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   635
        simp del: sum.lessThan_Suc power_Suc intro: sum.cong)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   636
  have "h * ?lhs = (z + h) ^ n - z ^ n - h * of_nat n * z ^ (n - Suc 0)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   637
    by (simp add: right_diff_distrib diff_divide_distrib h mult.assoc [symmetric])
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   638
  also have "... = h * ((\<Sum>p<Suc m. (z + h) ^ p * z ^ (m - p)) - of_nat (Suc m) * z ^ m)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   639
    by (simp add: Suc diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc
70097
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69654
diff changeset
   640
        del: power_Suc sum.lessThan_Suc of_nat_Suc)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   641
  also have "... = h * ((\<Sum>p<Suc m. (z + h) ^ (m - p) * z ^ p) - of_nat (Suc m) * z ^ m)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   642
    by (subst sum.nat_diff_reindex[symmetric]) simp
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   643
  also have "... = h * (\<Sum>i<Suc m. (z + h) ^ (m - i) * z ^ i - z ^ m)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   644
    by (simp add: sum_subtractf)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   645
  also have "... = h * ?rhs"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   646
    by (simp add: lemma_termdiff1 sum_distrib_left Suc *)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   647
  finally have "h * ?lhs = h * ?rhs" .
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   648
  then show ?thesis
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   649
    by (simp add: h)
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   650
qed auto
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   651
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   652
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   653
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
   654
  fixes K :: "'a::linordered_semidom"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   655
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K" 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"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   657
proof -
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   658
  have "sum f {..<n-k} \<le> (\<Sum>i<n - k. K)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   659
    by (rule sum_mono [OF f]) auto
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   660
  also have "... \<le> of_nat n * K"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   661
    by (auto simp: mult_right_mono K)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   662
  finally show ?thesis .
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   663
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   664
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   665
lemma lemma_termdiff3:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   666
  fixes h z :: "'a::real_normed_field"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   667
  assumes 1: "h \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   668
    and 2: "norm z \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   669
    and 3: "norm (z + h) \<le> K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   670
  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
   671
    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
   672
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   673
  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
   674
    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
   675
    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
   676
  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
   677
  proof (rule mult_right_mono [OF _ norm_ge_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   678
    from norm_ge_zero 2 have K: "0 \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   679
      by (rule order_trans)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   680
    have le_Kn: "norm ((z + h) ^ i * z ^ j) \<le> K ^ n" if "i + j = n" for i j n
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   681
    proof -
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   682
      have "norm (z + h) ^ i * norm z ^ j \<le> K ^ i * K ^ j"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   683
        by (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   684
      also have "... = K^n"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   685
        by (metis power_add that)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   686
      finally show ?thesis
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   687
        by (simp add: norm_mult norm_power) 
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   688
    qed
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   689
    then have "\<And>p q.
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   690
       \<lbrakk>p < n; q < n - Suc 0\<rbrakk> \<Longrightarrow> norm ((z + h) ^ q * z ^ (n - 2 - q)) \<le> K ^ (n - 2)"
71959
ee2c7f0dd1be prefer single name
haftmann
parents: 71918
diff changeset
   691
      by (simp del: subst_all)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   692
    then
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   693
    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
   694
        of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   695
      by (intro order_trans [OF norm_sum]
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   696
          real_sum_nat_ivl_bounded2 mult_nonneg_nonneg of_nat_0_le_iff zero_le_power K)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   697
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   698
  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
   699
    by (simp only: mult.assoc)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   700
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   701
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   702
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   703
lemma lemma_termdiff4:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   704
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   705
    and k :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   706
  assumes k: "0 < k"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   707
    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
   708
  shows "f \<midarrow>0\<rightarrow> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   709
proof (rule tendsto_norm_zero_cancel)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   710
  show "(\<lambda>h. norm (f h)) \<midarrow>0\<rightarrow> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   711
  proof (rule real_tendsto_sandwich)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   712
    show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   713
      by simp
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   714
    show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   715
      using k by (auto simp: eventually_at dist_norm le)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   716
    show "(\<lambda>h. 0) \<midarrow>(0::'a)\<rightarrow> (0::real)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   717
      by (rule tendsto_const)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   718
    have "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> K * norm (0::'a)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   719
      by (intro tendsto_intros)
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   720
    then show "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   721
      by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   722
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   723
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   724
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   725
lemma lemma_termdiff5:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   726
  fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   727
    and k :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   728
  assumes k: "0 < k"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   729
    and f: "summable f"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   730
    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
   731
  shows "(\<lambda>h. suminf (g h)) \<midarrow>0\<rightarrow> 0"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   732
proof (rule lemma_termdiff4 [OF k])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   733
  fix h :: 'a
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   734
  assume "h \<noteq> 0" and "norm h < k"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   735
  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
   736
    by (simp add: le)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   737
  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
   738
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   739
  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
   740
    by (rule summable_mult2)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   741
  ultimately have 3: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   742
    by (rule summable_comparison_test)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   743
  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
   744
    by (rule summable_norm)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   745
  also from 1 3 2 have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 72211
diff changeset
   746
    by (simp add: suminf_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   747
  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
   748
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   749
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   750
qed
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
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   753
(* FIXME: Long proofs *)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   754
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   755
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   756
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   757
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   758
    and 2: "norm x < norm K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   759
  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
   760
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   761
  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
   762
    by fast
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   763
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   764
    by (rule order_le_less_trans)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   765
  then have r_neq_0: "r \<noteq> 0" by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   766
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   767
  proof (rule lemma_termdiff5)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   768
    show "0 < r - norm x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   769
      using r1 by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   770
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   771
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   772
    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
   773
      by (rule powser_insidea)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   774
    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
   775
      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
   776
    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
   777
      by (rule diffs_equiv [THEN sums_summable])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   778
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   779
               (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   780
      by (simp add: diffs_def r_neq_0 fun_eq_iff split: nat_diff_split)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   781
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   782
      (\<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
   783
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   784
    also have
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   785
      "(\<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
   786
       (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   787
      by (rule ext) (simp add: r_neq_0 split: nat_diff_split)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   788
    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
   789
  next
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   790
    fix h :: 'a and n
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   791
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   792
    assume "norm h < r - norm x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   793
    then have "norm x + norm h < r" by simp
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   794
    with norm_triangle_ineq 
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   795
    have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   796
      by (rule order_le_less_trans)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   797
    have "norm (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   798
    \<le> real n * (real (n - Suc 0) * (r ^ (n - 2) * norm h))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   799
      by (metis (mono_tags, lifting) h mult.assoc lemma_termdiff3 less_eq_real_def r1 xh)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   800
    then show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<le>
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   801
      norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   802
      by (simp only: norm_mult mult.assoc mult_left_mono [OF _ norm_ge_zero])
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   803
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   804
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   805
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   806
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   807
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   808
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   809
    and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   810
    and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   811
    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
   812
  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
   813
  unfolding DERIV_def
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   814
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
   815
  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
   816
            - 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
   817
  proof (rule LIM_equal2)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   818
    show "0 < norm K - norm x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   819
      using 4 by (simp add: less_diff_eq)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   820
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   821
    fix h :: 'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   822
    assume "norm (h - 0) < norm K - norm x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   823
    then have "norm x + norm h < norm K" by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   824
    then have 5: "norm (x + h) < norm K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   825
      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
   826
    have "summable (\<lambda>n. c n * x^n)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   827
      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
   828
      and "summable (\<lambda>n. diffs c n * x^n)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   829
      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
   830
    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
   831
          (\<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
   832
      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
   833
    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
   834
          (\<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
   835
      by (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   836
  next
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
   837
    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
   838
      by (rule termdiffs_aux [OF 3 4])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   839
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   840
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   841
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   842
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
   843
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   844
lemma termdiff_converges:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   845
  fixes x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   846
  assumes K: "norm x < K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   847
    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
   848
  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
   849
proof (cases "x = 0")
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   850
  case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   851
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   852
    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
   853
next
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   854
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   855
  then have "K > 0"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   856
    using K less_trans zero_less_norm_iff by blast
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   857
  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
   858
    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
   859
    by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   860
  have to0: "(\<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
   861
    using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   862
  obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   863
    using r LIMSEQ_D [OF to0, of 1]
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   864
    by (auto simp: norm_divide norm_mult norm_power field_simps)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   865
  have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   866
  proof (rule summable_comparison_test')
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   867
    show "summable (\<lambda>n. norm (c n * of_real r ^ n))"
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   868
      apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   869
      using N r norm_of_real [of "r + K", where 'a = 'a] by auto
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   870
    show "\<And>n. N \<le> n \<Longrightarrow> norm (of_nat n * c n * x ^ n) \<le> norm (c n * of_real r ^ n)"
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   871
      using N r by (fastforce simp add: norm_mult norm_power less_eq_real_def)
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   872
  qed
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   873
  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
   874
    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
   875
    by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   876
  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
   877
    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
   878
    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
   879
  then show ?thesis
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   880
    by (simp add: diffs_def)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   881
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   882
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   883
lemma termdiff_converges_all:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   884
  fixes x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   885
  assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   886
  shows "summable (\<lambda>n. diffs c n * x^n)"
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68527
diff changeset
   887
  by (rule termdiff_converges [where K = "1 + norm x"]) (use assms in auto)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   888
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   889
lemma termdiffs_strong:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   890
  fixes K x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   891
  assumes sm: "summable (\<lambda>n. c n * K ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   892
    and K: "norm x < norm K"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   893
  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
   894
proof -
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   895
  have "norm K + norm x < norm K + norm K"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   896
    using K by force
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   897
  then have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   898
    by (auto simp: norm_triangle_lt norm_divide field_simps)
60762
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
   899
  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
   900
    by simp
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   901
  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
   902
    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
   903
  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
   904
    by (blast intro: sm termdiff_converges powser_inside)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   905
  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
   906
    by (blast intro: sm termdiff_converges powser_inside)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   907
  ultimately show ?thesis
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   908
    by (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   909
       (use K in \<open>auto simp: field_simps simp flip: of_real_add\<close>)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   910
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   911
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   912
lemma termdiffs_strong_converges_everywhere:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   913
  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
   914
  assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   915
  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
   916
  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
   917
  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
   918
63721
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   919
lemma termdiffs_strong':
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   920
  fixes z :: "'a :: {real_normed_field,banach}"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   921
  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
   922
  assumes "norm z < K"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   923
  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
   924
proof (rule termdiffs_strong)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   925
  define L :: real where "L =  (norm z + K) / 2"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   926
  have "0 \<le> norm z" by simp
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   927
  also note \<open>norm z < K\<close>
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   928
  finally have K: "K \<ge> 0" by simp
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   929
  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
   930
  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
   931
  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
   932
qed
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   933
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   934
lemma termdiffs_sums_strong:
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   935
  fixes z :: "'a :: {banach,real_normed_field}"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   936
  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
   937
  assumes deriv: "(f has_field_derivative f') (at z)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   938
  assumes norm: "norm z < K"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   939
  shows   "(\<lambda>n. diffs c n * z ^ n) sums f'"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   940
proof -
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   941
  have summable: "summable (\<lambda>n. diffs c n * z^n)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   942
    by (intro termdiff_converges[OF norm] sums_summable[OF sums])
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   943
  from norm have "eventually (\<lambda>z. z \<in> norm -` {..<K}) (nhds z)"
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65204
diff changeset
   944
    by (intro eventually_nhds_in_open open_vimage)
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
   945
       (simp_all add: continuous_on_norm)
63721
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   946
  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
   947
    by eventually_elim (insert sums, simp add: sums_iff)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   948
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   949
  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
   950
    by (intro termdiffs_strong'[OF _ norm] sums_summable[OF sums])
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   951
  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
   952
    by (subst (asm) DERIV_cong_ev[OF refl eq refl])
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   953
  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
   954
  with summable show ?thesis by (simp add: sums_iff)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   955
qed
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
   956
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   957
lemma isCont_powser:
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   958
  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
   959
  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
   960
  assumes "norm x < norm K"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   961
  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
   962
  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
   963
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   964
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
   965
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   966
lemma isCont_powser_converges_everywhere:
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   967
  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
   968
  assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   969
  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
   970
  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
   971
  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
   972
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
   973
lemma powser_limit_0:
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   974
  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
   975
  assumes s: "0 < s"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
   976
    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
   977
  shows "(f \<longlongrightarrow> a 0) (at 0)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   978
proof -
68077
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   979
  have "norm (of_real s / 2 :: 'a) < s"
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   980
    using s  by (auto simp: norm_divide)
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   981
  then have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   982
    by (rule sums_summable [OF sm])
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   983
  then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)"
68077
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   984
    by (rule termdiffs_strong) (use s in \<open>auto simp: norm_divide\<close>)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   985
  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
   986
    by (blast intro: DERIV_continuous)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   987
  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
   988
    by (simp add: continuous_within)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   989
  moreover have "(\<lambda>x. f x - (\<Sum>n. a n * x ^ n)) \<midarrow>0\<rightarrow> 0"
68077
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   990
    apply (clarsimp simp: LIM_eq)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
   991
    apply (rule_tac x=s in exI)
68077
ee8c13ae81e9 Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents: 67727
diff changeset
   992
    using s sm sums_unique by fastforce
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   993
  ultimately show ?thesis
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   994
    by (rule Lim_transform)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   995
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   996
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
   997
lemma powser_limit_0_strong:
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   998
  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
   999
  assumes s: "0 < s"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1000
    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
  1001
  shows "(f \<longlongrightarrow> a 0) (at 0)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1002
proof -
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1003
  have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1004
    by (rule powser_limit_0 [OF s]) (auto simp: powser_sums_zero sm)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1005
  show ?thesis
72220
bb29e4eb938d but not the [cong] rule
paulson <lp15@cam.ac.uk>
parents: 72219
diff changeset
  1006
    using "*" by (auto cong: Lim_cong_within)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1007
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1008
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1009
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1010
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
  1011
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1012
lemma DERIV_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1013
  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
  1014
  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
  1015
    and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1016
    and x0_in_I: "x0 \<in> {a <..< b}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1017
    and "summable (f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1018
    and "summable L"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1019
    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
  1020
  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
  1021
  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
  1022
proof (rule LIM_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1023
  fix r :: real
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1024
  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
  1025
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1026
  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
  1027
    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
  1028
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1029
  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
  1030
    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
  1031
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1032
  let ?N = "Suc (max N_L N_f')"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1033
  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
  1034
    and L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1035
    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
  1036
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1037
  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
  1038
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1039
  let ?r = "r / (3 * real ?N)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1040
  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
  1041
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1042
  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
  1043
  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
  1044
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1045
  have "0 < S'"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1046
    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
  1047
  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
  1048
    show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1049
    proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1050
      fix x
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1051
      assume "x \<in> ?s ` {..<?N}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1052
      then obtain n where "x = ?s n" and "n \<in> {..<?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1053
        using image_iff[THEN iffD1] by blast
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1054
      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
  1055
      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
  1056
        by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1057
      have "0 < ?s n"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1058
        by (rule someI2[where a=s]) (auto simp: s_bound simp del: of_nat_Suc)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1059
      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
  1060
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1061
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1062
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62949
diff changeset
  1063
  define S where "S = min (min (x0 - a) (b - x0)) S'"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1064
  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
  1065
    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
  1066
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1067
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1068
  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
  1069
    if "x \<noteq> 0" and "\<bar>x\<bar> < S" for x
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1070
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1071
    from that have x_in_I: "x0 + x \<in> {a <..< b}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1072
      using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1073
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1074
    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
  1075
    note div_smbl = summable_divide[OF diff_smbl]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1076
    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
  1077
    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
  1078
    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
  1079
    note div_shft_smbl = summable_divide[OF diff_shft_smbl]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1080
    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
  1081
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1082
    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
  1083
    proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1084
      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
  1085
        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
  1086
        by (simp only: abs_divide)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1087
      with \<open>x \<noteq> 0\<close> show ?thesis by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1088
    qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1089
    note 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1090
    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
  1091
      by (metis (lifting) abs_idempotent
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1092
          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
  1093
    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
  1094
      using L_estimate by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1095
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1096
    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
  1097
    also have "\<dots> < (\<Sum>n<?N. ?r)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1098
    proof (rule sum_strict_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1099
      fix n
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1100
      assume "n \<in> {..< ?N}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1101
      have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> .
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1102
      also have "S \<le> S'" using \<open>S \<le> S'\<close> .
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1103
      also have "S' \<le> ?s n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1104
        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
  1105
      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
  1106
        have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1107
          using \<open>n \<in> {..< ?N}\<close> by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1108
        then show "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1109
          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
  1110
      qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1111
      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
  1112
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1113
      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
  1114
          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
  1115
      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
  1116
      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
  1117
        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
  1118
    qed auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1119
    also have "\<dots> = of_nat (card {..<?N}) * ?r"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1120
      by (rule sum_constant)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1121
    also have "\<dots> = real ?N * ?r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1122
      by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1123
    also have "\<dots> = r/3"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1124
      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
  1125
    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
  1126
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1127
    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
  1128
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1129
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1130
      unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1131
      using suminf_divide[OF diff_smbl, symmetric] by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1132
    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
  1133
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1134
      unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1135
      apply (simp only: add.commute)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1136
      using abs_triangle_ineq by blast
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1137
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1138
      using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1139
    also have "\<dots> < r /3 + r/3 + r/3"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1140
      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
  1141
      by (rule add_strict_mono [OF add_less_le_mono])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1142
    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
  1143
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1144
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1145
  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
  1146
      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
  1147
    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
  1148
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1149
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1150
lemma DERIV_power_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1151
  fixes f :: "nat \<Rightarrow> real"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1152
  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
  1153
    and x0_in_I: "x0 \<in> {-R <..< R}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1154
    and "0 < R"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1155
  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
  1156
    (is "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1157
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1158
  have for_subinterval: "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1159
    if "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" for R'
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1160
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1161
    from that have "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1162
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1163
    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
  1164
    proof (rule DERIV_series')
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1165
      show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1166
      proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1167
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
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
  1168
          using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1169
        then have in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1170
          using \<open>R' < R\<close> by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1171
        have "norm R' < norm ((R' + R) / 2)"
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
  1172
          using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1173
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1174
          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
  1175
      qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1176
    next
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1177
      fix n x y
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1178
      assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1179
      show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1180
      proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1181
        have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1182
          (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1183
          unfolding right_diff_distrib[symmetric] diff_power_eq_sum abs_mult
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1184
          by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1185
        also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1186
        proof (rule mult_left_mono)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1187
          have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1188
            by (rule sum_abs)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1189
          also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1190
          proof (rule sum_mono)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1191
            fix p
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1192
            assume "p \<in> {..<Suc n}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1193
            then have "p \<le> n" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1194
            have "\<bar>x^n\<bar> \<le> R'^n" if  "x \<in> {-R'<..<R'}" for n and x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1195
            proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1196
              from that have "\<bar>x\<bar> \<le> R'" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1197
              then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1198
                unfolding power_abs by (rule power_mono) auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  1199
            qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1200
            from mult_mono[OF this[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1201
              and \<open>0 < R'\<close>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1202
            have "\<bar>x^p * y^(n - p)\<bar> \<le> R'^p * R'^(n - p)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1203
              unfolding abs_mult by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1204
            then show "\<bar>x^p * y^(n - p)\<bar> \<le> R'^n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1205
              unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  1206
          qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1207
          also have "\<dots> = real (Suc n) * R' ^ n"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1208
            unfolding sum_constant card_atLeastLessThan by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1209
          finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1210
            unfolding abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1211
            by linarith
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1212
          show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1213
            unfolding abs_mult[symmetric] by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1214
        qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1215
        also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1216
          unfolding abs_mult mult.assoc[symmetric] by algebra
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1217
        finally show ?thesis .
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1218
      qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1219
    next
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1220
      show "DERIV (\<lambda>x. ?f x n) x0 :> ?f' x0 n" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1221
        by (auto intro!: derivative_eq_intros simp del: power_Suc)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1222
    next
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1223
      fix x
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1224
      assume "x \<in> {-R' <..< R'}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1225
      then have "R' \<in> {-R <..< R}" and "norm x < norm R'"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1226
        using assms \<open>R' < R\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1227
      have "summable (\<lambda>n. f n * x^n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1228
      proof (rule summable_comparison_test, intro exI allI impI)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1229
        fix n
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1230
        have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1231
          by (rule mult_left_mono) auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1232
        show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1233
          unfolding real_norm_def abs_mult
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1234
          using le mult_right_mono by fastforce
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1235
      qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1236
      from this[THEN summable_mult2[where c=x], simplified mult.assoc, simplified mult.commute]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1237
      show "summable (?f x)" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1238
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1239
      show "summable (?f' x0)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1240
        using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1241
      show "x0 \<in> {-R' <..< R'}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1242
        using \<open>x0 \<in> {-R' <..< R'}\<close> .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1243
    qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1244
  qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1245
  let ?R = "(R + \<bar>x0\<bar>) / 2"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1246
  have "\<bar>x0\<bar> < ?R"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1247
    using assms by (auto simp: field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1248
  then have "- ?R < x0"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1249
  proof (cases "x0 < 0")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1250
    case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1251
    then have "- x0 < ?R"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1252
      using \<open>\<bar>x0\<bar> < ?R\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1253
    then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1254
      unfolding neg_less_iff_less[symmetric, of "- x0"] 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
  1255
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1256
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1257
    have "- ?R < 0" using assms by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1258
    also have "\<dots> \<le> x0" using False 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
  1259
    finally show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1260
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1261
  then have "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
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
  1262
    using assms by (auto simp: field_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1263
  from for_subinterval[OF this] 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
  1264
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
  1265
63721
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1266
lemma geometric_deriv_sums:
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1267
  fixes z :: "'a :: {real_normed_field,banach}"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1268
  assumes "norm z < 1"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1269
  shows   "(\<lambda>n. of_nat (Suc n) * z ^ n) sums (1 / (1 - z)^2)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1270
proof -
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1271
  have "(\<lambda>n. diffs (\<lambda>n. 1) n * z^n) sums (1 / (1 - z)^2)"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1272
  proof (rule termdiffs_sums_strong)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1273
    fix z :: 'a assume "norm z < 1"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1274
    thus "(\<lambda>n. 1 * z^n) sums (1 / (1 - z))" by (simp add: geometric_sums)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1275
  qed (insert assms, auto intro!: derivative_eq_intros simp: power2_eq_square)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1276
  thus ?thesis unfolding diffs_def by simp
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63566
diff changeset
  1277
qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1278
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1279
lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z. pochhammer z n) z"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1280
  for z :: "'a::real_normed_field"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1281
  by (induct n) (auto simp: pochhammer_rec')
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1282
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1283
lemma continuous_on_pochhammer [continuous_intros]: "continuous_on A (\<lambda>z. pochhammer z n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1284
  for A :: "'a::real_normed_field set"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1285
  by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1286
66486
ffaaa83543b2 Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents: 66279
diff changeset
  1287
lemmas continuous_on_pochhammer' [continuous_intros] =
ffaaa83543b2 Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents: 66279
diff changeset
  1288
  continuous_on_compose2[OF continuous_on_pochhammer _ subset_UNIV]
ffaaa83543b2 Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents: 66279
diff changeset
  1289
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1290
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1291
subsection \<open>Exponential Function\<close>
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1292
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1293
definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1294
  where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1295
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1296
lemma summable_exp_generic:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1297
  fixes x :: "'a::{real_normed_algebra_1,banach}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1298
  defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1299
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1300
proof -
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1301
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30082
diff changeset
  1302
    unfolding S_def by (simp del: mult_Suc)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1303
  obtain r :: real where r0: "0 < r" and r1: "r < 1"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1304
    using dense [OF zero_less_one] by fast
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1305
  obtain N :: nat where N: "norm x < real N * r"
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
  1306
    using ex_less_of_nat_mult r0 by auto
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1307
  from r1 show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1308
  proof (rule summable_ratio_test [rule_format])
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1309
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1310
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1311
    have "norm x \<le> real N * r"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1312
      using N by (rule order_less_imp_le)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1313
    also have "real N * r \<le> real (Suc n) * r"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1314
      using r0 n by (simp add: mult_right_mono)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1315
    finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1316
      using norm_ge_zero by (rule mult_right_mono)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1317
    then have "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1318
      by (rule order_trans [OF norm_mult_ineq])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1319
    then have "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
  1320
      by (simp add: pos_divide_le_eq ac_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1321
    then show "norm (S (Suc n)) \<le> r * norm (S n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
  1322
      by (simp add: S_Suc inverse_eq_divide)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1323
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1324
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1325
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1326
lemma summable_norm_exp: "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1327
  for x :: "'a::{real_normed_algebra_1,banach}"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1328
proof (rule summable_norm_comparison_test [OF exI, rule_format])
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1329
  show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1330
    by (rule summable_exp_generic)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1331
  show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n" for n
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
  1332
    by (simp add: norm_power_ineq)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1333
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1334
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1335
lemma summable_exp: "summable (\<lambda>n. inverse (fact n) * x^n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1336
  for x :: "'a::{real_normed_field,banach}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1337
  using summable_exp_generic [where x=x]
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1338
  by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1339
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1340
lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1341
  unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1342
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1343
lemma exp_fdiffs:
60241
wenzelm
parents: 60036
diff changeset
  1344
  "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a::{real_normed_field,banach}))"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1345
  by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1346
      del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1347
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1348
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1349
  by (simp add: diffs_def)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1350
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1351
lemma DERIV_exp [simp]: "DERIV exp x :> exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1352
  unfolding exp_def scaleR_conv_of_real
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1353
proof (rule DERIV_cong)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1354
  have sinv: "summable (\<lambda>n. of_real (inverse (fact n)) * x ^ n)" for x::'a
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1355
    by (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1356
  note xx = exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real]
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1357
  show "((\<lambda>x. \<Sum>n. of_real (inverse (fact n)) * x ^ n) has_field_derivative
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1358
        (\<Sum>n. diffs (\<lambda>n. of_real (inverse (fact n))) n * x ^ n))  (at x)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1359
    by (rule termdiffs [where K="of_real (1 + norm x)"]) (simp_all only: diffs_of_real exp_fdiffs sinv norm_of_real)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1360
  show "(\<Sum>n. diffs (\<lambda>n. of_real (inverse (fact n))) n * x ^ n) = (\<Sum>n. of_real (inverse (fact n)) * x ^ n)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1361
    by (simp add: diffs_of_real exp_fdiffs)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1362
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1363
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
  1364
declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1365
  and DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  1366
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  1367
lemmas has_derivative_exp[derivative_intros] = DERIV_exp[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  1368
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1369
lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1370
proof -
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1371
  from summable_norm[OF summable_norm_exp, of x]
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1372
  have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1373
    by (simp add: exp_def)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1374
  also have "\<dots> \<le> exp (norm x)"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1375
    using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1376
    by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1377
  finally show ?thesis .
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1378
qed
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1379
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1380
lemma isCont_exp: "isCont exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1381
  for x :: "'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1382
  by (rule DERIV_exp [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1383
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1384
lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1385
  for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1386
  by (rule isCont_o2 [OF _ isCont_exp])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1387
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1388
lemma tendsto_exp [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1389
  for f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1390
  by (rule isCont_tendsto_compose [OF isCont_exp])
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  1391
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1392
lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1393
  for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1394
  unfolding continuous_def by (rule tendsto_exp)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1395
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1396
lemma continuous_on_exp [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1397
  for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1398
  unfolding continuous_on_def by (auto intro: tendsto_exp)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1399
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1400
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1401
subsubsection \<open>Properties of the Exponential Function\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1402
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1403
lemma exp_zero [simp]: "exp 0 = 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1404
  unfolding exp_def by (simp add: scaleR_conv_of_real)
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1405
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1406
lemma exp_series_add_commuting:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1407
  fixes x y :: "'a::{real_normed_algebra_1,banach}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1408
  defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1409
  assumes comm: "x * y = y * x"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1410
  shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1411
proof (induct n)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1412
  case 0
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1413
  show ?case
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1414
    unfolding S_def by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1415
next
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1416
  case (Suc n)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1417
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30082
diff changeset
  1418
    unfolding S_def by (simp del: mult_Suc)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1419
  then have times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1420
    by simp
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1421
  have S_comm: "\<And>n. S x n * y = y * S x n"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1422
    by (simp add: power_commuting_commutes comm S_def)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1423
72211
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1424
  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * (\<Sum>i\<le>n. S x i * S y (n - i))"
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1425
    by (metis Suc.hyps times_S)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1426
  also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n - i)) + y * (\<Sum>i\<le>n. S x i * S y (n - i))"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47489
diff changeset
  1427
    by (rule distrib_right)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1428
  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * y * S y (n - i))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1429
    by (simp add: sum_distrib_left ac_simps S_comm)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1430
  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * (y * S y (n - i)))"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1431
    by (simp add: ac_simps)
72211
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1432
  also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) 
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1433
                + (\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1434
    by (simp add: times_S Suc_diff_le)
72211
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1435
  also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i)))
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1436
           = (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))"
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  1437
    by (subst sum.atMost_Suc_shift) simp
72211
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1438
  also have "(\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1439
           = (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1440
    by simp
72211
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1441
  also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1442
           + (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) 
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1443
           = (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n - i)))"
a6cbf8ce979e tiny tidy-up of proofs
paulson <lp15@cam.ac.uk>
parents: 71959
diff changeset
  1444
    by (simp flip: sum.distrib scaleR_add_left of_nat_add) 
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1445
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  1446
    by (simp only: scaleR_right.sum)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1447
  finally show "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  1448
    by (simp del: sum.cl_ivl_Suc)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1449
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1450
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1451
lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1452
  by (simp only: exp_def Cauchy_product summable_norm_exp exp_series_add_commuting)
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1453
62949
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1454
lemma exp_times_arg_commute: "exp A * A = A * exp A"
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1455
  by (simp add: exp_def suminf_mult[symmetric] summable_exp_generic power_commutes suminf_mult2)
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1456
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1457
lemma exp_add: "exp (x + y) = exp x * exp y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1458
  for x y :: "'a::{real_normed_field,banach}"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1459
  by (rule exp_add_commuting) (simp add: ac_simps)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1460
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1461
lemma exp_double: "exp(2 * z) = exp z ^ 2"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1462
  by (simp add: exp_add_commuting mult_2 power2_eq_square)
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1463
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1464
lemmas mult_exp_exp = exp_add [symmetric]
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1465
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1466
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1467
  unfolding exp_def
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1468
  apply (subst suminf_of_real [OF summable_exp_generic])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1469
  apply (simp add: scaleR_conv_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1470
  done
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1471
65204
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  1472
lemmas of_real_exp = exp_of_real[symmetric]
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  1473
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  1474
corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  1475
  by (metis Reals_cases Reals_of_real exp_of_real)
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  1476
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1477
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1478
proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1479
  have "exp x * exp (- x) = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1480
    by (simp add: exp_add_commuting[symmetric])
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1481
  also assume "exp x = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1482
  finally show False by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1483
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1484
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1485
lemma exp_minus_inverse: "exp x * exp (- x) = 1"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1486
  by (simp add: exp_add_commuting[symmetric])
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1487
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1488
lemma exp_minus: "exp (- x) = inverse (exp x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1489
  for x :: "'a::{real_normed_field,banach}"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1490
  by (intro inverse_unique [symmetric] exp_minus_inverse)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1491
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1492
lemma exp_diff: "exp (x - y) = exp x / exp y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1493
  for x :: "'a::{real_normed_field,banach}"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
  1494
  using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1495
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1496
lemma exp_of_nat_mult: "exp (of_nat n * x) = exp x ^ n"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1497
  for x :: "'a::{real_normed_field,banach}"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1498
  by (induct n) (auto simp: distrib_left exp_add mult.commute)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1499
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1500
corollary exp_of_nat2_mult: "exp (x * of_nat n) = exp x ^ n"
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  1501
  for x :: "'a::{real_normed_field,banach}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  1502
  by (metis exp_of_nat_mult mult_of_nat_commute)
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1503
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  1504
lemma exp_sum: "finite I \<Longrightarrow> exp (sum f I) = prod (\<lambda>x. exp (f x)) I"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1505
  by (induct I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1506
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1507
lemma exp_divide_power_eq:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1508
  fixes x :: "'a::{real_normed_field,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1509
  assumes "n > 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1510
  shows "exp (x / of_nat n) ^ n = exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1511
  using assms
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1512
proof (induction n arbitrary: x)
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1513
  case (Suc n)
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1514
  show ?case
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1515
  proof (cases "n = 0")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1516
    case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1517
    then show ?thesis by simp
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1518
  next
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1519
    case False
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  1520
    have [simp]: "1 + (of_nat n * of_nat n + of_nat n * 2) \<noteq> (0::'a)"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  1521
      using of_nat_eq_iff [of "1 + n * n + n * 2" "0"]
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  1522
      by simp
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  1523
    from False have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)"
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1524
      by simp
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1525
    have [simp]: "x / (1 + of_nat n) + x * of_nat n / (1 + of_nat n) = x"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  1526
      using of_nat_neq_0
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  1527
      by (auto simp add: field_split_simps)
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1528
    show ?thesis
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1529
      using Suc.IH [of "x * of_nat n / (1 + of_nat n)"] False
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1530
      by (simp add: exp_add [symmetric])
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1531
  qed
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1532
qed simp
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62347
diff changeset
  1533
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1534
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1535
subsubsection \<open>Properties of the Exponential Function on Reals\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1536
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  1537
text \<open>Comparisons of \<^term>\<open>exp x\<close> with zero.\<close>
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1538
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1539
text \<open>Proof: because every exponential can be seen as a square.\<close>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1540
lemma exp_ge_zero [simp]: "0 \<le> exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1541
  for x :: real
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1542
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1543
  have "0 \<le> exp (x/2) * exp (x/2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1544
    by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1545
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1546
    by (simp add: exp_add [symmetric])
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1547
qed
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1548
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1549
lemma exp_gt_zero [simp]: "0 < exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1550
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1551
  by (simp add: order_less_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1552
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1553
lemma not_exp_less_zero [simp]: "\<not> exp x < 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1554
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1555
  by (simp add: not_less)
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1556
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1557
lemma not_exp_le_zero [simp]: "\<not> exp x \<le> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1558
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1559
  by (simp add: not_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1560
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1561
lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1562
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1563
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1564
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1565
text \<open>Strict monotonicity of exponential.\<close>
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1566
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1567
lemma exp_ge_add_one_self_aux:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1568
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1569
  assumes "0 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1570
  shows "1 + x \<le> exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1571
  using order_le_imp_less_or_eq [OF assms]
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1572
proof
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1573
  assume "0 < x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1574
  have "1 + x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1575
    by (auto simp: numeral_2_eq_2)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1576
  also have "\<dots> \<le> (\<Sum>n. inverse (fact n) * x^n)"
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 72211
diff changeset
  1577
    using \<open>0 < x\<close> by (auto  simp add: zero_le_mult_iff intro: sum_le_suminf [OF summable_exp])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1578
  finally show "1 + x \<le> exp x"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1579
    by (simp add: exp_def)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1580
qed auto
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1581
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1582
lemma exp_gt_one: "0 < x \<Longrightarrow> 1 < exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1583
  for x :: real
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1584
proof -
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1585
  assume x: "0 < x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1586
  then have "1 < 1 + x" by simp
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1587
  also from x have "1 + x \<le> exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1588
    by (simp add: exp_ge_add_one_self_aux)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1589
  finally show ?thesis .
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1590
qed
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1591
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1592
lemma exp_less_mono:
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1593
  fixes x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1594
  assumes "x < y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1595
  shows "exp x < exp y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1596
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1597
  from \<open>x < y\<close> have "0 < y - x" by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1598
  then have "1 < exp (y - x)" by (rule exp_gt_one)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1599
  then have "1 < exp y / exp x" by (simp only: exp_diff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1600
  then show "exp x < exp y" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1601
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1602
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1603
lemma exp_less_cancel: "exp x < exp y \<Longrightarrow> x < y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1604
  for x y :: real
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1605
  unfolding linorder_not_le [symmetric]
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1606
  by (auto simp: order_le_less exp_less_mono)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1607
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1608
lemma exp_less_cancel_iff [iff]: "exp x < exp y \<longleftrightarrow> x < y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1609
  for x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1610
  by (auto intro: exp_less_mono exp_less_cancel)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1611
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1612
lemma exp_le_cancel_iff [iff]: "exp x \<le> exp y \<longleftrightarrow> x \<le> y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1613
  for x y :: real
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1614
  by (auto simp: linorder_not_less [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1615
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1616
lemma exp_inj_iff [iff]: "exp x = exp y \<longleftrightarrow> x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1617
  for x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1618
  by (simp add: order_eq_iff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1619
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  1620
text \<open>Comparisons of \<^term>\<open>exp x\<close> with one.\<close>
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1621
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1622
lemma one_less_exp_iff [simp]: "1 < exp x \<longleftrightarrow> 0 < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1623
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1624
  using exp_less_cancel_iff [where x = 0 and y = x] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1625
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1626
lemma exp_less_one_iff [simp]: "exp x < 1 \<longleftrightarrow> x < 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1627
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1628
  using exp_less_cancel_iff [where x = x and y = 0] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1629
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1630
lemma one_le_exp_iff [simp]: "1 \<le> exp x \<longleftrightarrow> 0 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1631
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1632
  using exp_le_cancel_iff [where x = 0 and y = x] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1633
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1634
lemma exp_le_one_iff [simp]: "exp x \<le> 1 \<longleftrightarrow> x \<le> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1635
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1636
  using exp_le_cancel_iff [where x = x and y = 0] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1637
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1638
lemma exp_eq_one_iff [simp]: "exp x = 1 \<longleftrightarrow> x = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1639
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1640
  using exp_inj_iff [where x = x and y = 0] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1641
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1642
lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x \<le> y - 1 \<and> exp x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1643
  for y :: real
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1644
proof (rule IVT)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1645
  assume "1 \<le> y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1646
  then have "0 \<le> y - 1" by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1647
  then have "1 + (y - 1) \<le> exp (y - 1)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1648
    by (rule exp_ge_add_one_self_aux)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1649
  then show "y \<le> exp (y - 1)" by simp
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1650
qed (simp_all add: le_diff_eq)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1651
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1652
lemma exp_total: "0 < y \<Longrightarrow> \<exists>x. exp x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1653
  for y :: real
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1654
proof (rule linorder_le_cases [of 1 y])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1655
  assume "1 \<le> y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1656
  then show "\<exists>x. exp x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1657
    by (fast dest: lemma_exp_total)
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1658
next
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1659
  assume "0 < y" and "y \<le> 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1660
  then have "1 \<le> inverse y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1661
    by (simp add: one_le_inverse_iff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1662
  then obtain x where "exp x = inverse y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1663
    by (fast dest: lemma_exp_total)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1664
  then have "exp (- x) = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1665
    by (simp add: exp_minus)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1666
  then show "\<exists>x. exp x = y" ..
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1667
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1668
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1669
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1670
subsection \<open>Natural Logarithm\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1671
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1672
class ln = real_normed_algebra_1 + banach +
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1673
  fixes ln :: "'a \<Rightarrow> 'a"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1674
  assumes ln_one [simp]: "ln 1 = 0"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1675
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1676
definition powr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a::ln"  (infixr "powr" 80)
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
  1677
  \<comment> \<open>exponentation via ln and exp\<close>
68774
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  1678
  where "x powr a \<equiv> if x = 0 then 0 else exp (a * ln x)"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1679
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1680
lemma powr_0 [simp]: "0 powr z = 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1681
  by (simp add: powr_def)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  1682
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1683
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1684
instantiation real :: ln
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1685
begin
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1686
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1687
definition ln_real :: "real \<Rightarrow> real"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1688
  where "ln_real x = (THE u. exp u = x)"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1689
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
  1690
instance
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1691
  by intro_classes (simp add: ln_real_def)
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1692
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1693
end
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1694
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1695
lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1696
  by (simp add: powr_def)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1697
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1698
lemma ln_exp [simp]: "ln (exp x) = x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1699
  for x :: real
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1700
  by (simp add: ln_real_def)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1701
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1702
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1703
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1704
  by (auto dest: exp_total)
22654
c2b6b5a9e136 new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents: 22653
diff changeset
  1705
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1706
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1707
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1708
  by (metis exp_gt_zero exp_ln)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1709
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1710
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1711
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1712
  by (erule subst) (rule ln_exp)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1713
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1714
lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1715
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1716
  by (rule ln_unique) (simp add: exp_add)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1717
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  1718
lemma ln_prod: "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i > 0) \<Longrightarrow> ln (prod f I) = sum (\<lambda>x. ln(f x)) I"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1719
  for f :: "'a \<Rightarrow> real"
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  1720
  by (induct I rule: finite_induct) (auto simp: ln_mult prod_pos)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1721
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1722
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1723
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1724
  by (rule ln_unique) (simp add: exp_minus)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1725
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1726
lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1727
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1728
  by (rule ln_unique) (simp add: exp_diff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1729
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1730
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  1731
  by (rule ln_unique) (simp add: exp_of_nat_mult)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1732
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1733
lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1734
  for x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1735
  by (subst exp_less_cancel_iff [symmetric]) simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1736
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1737
lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1738
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1739
  by (simp add: linorder_not_less [symmetric])
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1740
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1741
lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1742
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1743
  by (simp add: order_eq_iff)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1744
65680
378a2f11bec9 Simplification of some proofs. Also key lemmas using !! rather than ! in premises
paulson <lp15@cam.ac.uk>
parents: 65583
diff changeset
  1745
lemma ln_add_one_self_le_self: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1746
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1747
  by (rule exp_le_cancel_iff [THEN iffD1]) (simp add: exp_ge_add_one_self_aux)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1748
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1749
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1750
  for x :: real
65680
378a2f11bec9 Simplification of some proofs. Also key lemmas using !! rather than ! in premises
paulson <lp15@cam.ac.uk>
parents: 65583
diff changeset
  1751
  by (rule order_less_le_trans [where y = "ln (1 + x)"]) (simp_all add: ln_add_one_self_le_self)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1752
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  1753
lemma ln_ge_iff: "\<And>x::real. 0 < x \<Longrightarrow> y \<le> ln x \<longleftrightarrow> exp y \<le> x"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  1754
  using exp_le_cancel_iff exp_total by force
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  1755
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1756
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1757
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1758
  using ln_le_cancel_iff [of 1 x] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1759
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1760
lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1761
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1762
  using ln_le_cancel_iff [of 1 x] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1763
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1764
lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1765
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1766
  using ln_le_cancel_iff [of 1 x] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1767
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1768
lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1769
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1770
  using ln_less_cancel_iff [of x 1] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1771
65204
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  1772
lemma ln_le_zero_iff [simp]: "0 < x \<Longrightarrow> ln x \<le> 0 \<longleftrightarrow> x \<le> 1"
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  1773
  for x :: real
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  1774
  by (metis less_numeral_extra(1) ln_le_cancel_iff ln_one)
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  1775
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1776
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1777
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1778
  using ln_less_cancel_iff [of 1 x] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1779
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1780
lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1781
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1782
  using ln_less_cancel_iff [of 1 x] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1783
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1784
lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1785
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1786
  using ln_less_cancel_iff [of 1 x] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1787
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1788
lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1789
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1790
  using ln_inj_iff [of x 1] by simp
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1791
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1792
lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1793
  for x :: real
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1794
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1795
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1796
lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1797
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1798
  by (auto simp: ln_real_def intro!: arg_cong[where f = The])
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1799
70350
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  1800
lemma powr_eq_one_iff [simp]:
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  1801
  "a powr x = 1 \<longleftrightarrow> x = 0" if "a > 1" for a x :: real
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  1802
  using that by (auto simp: powr_def split: if_splits)
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  1803
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
  1804
lemma isCont_ln:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1805
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1806
  assumes "x \<noteq> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1807
  shows "isCont ln x"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  1808
proof (cases "0 < x")
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  1809
  case True
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  1810
  then have "isCont ln (exp (ln x))"
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  1811
    by (intro isCont_inverse_function[where d = "\<bar>x\<bar>" and f = exp]) auto
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  1812
  with True show ?thesis
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  1813
    by simp
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  1814
next
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  1815
  case False
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  1816
  with \<open>x \<noteq> 0\<close> show "isCont ln x"
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  1817
    unfolding isCont_def
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  1818
    by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  1819
       (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1820
         intro!: exI[of _ "\<bar>x\<bar>"])
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  1821
qed
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  1822
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1823
lemma tendsto_ln [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1824
  for a :: real
45915
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45309
diff changeset
  1825
  by (rule isCont_tendsto_compose [OF isCont_ln])
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45309
diff changeset
  1826
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1827
lemma continuous_ln:
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1828
  "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1829
  unfolding continuous_def by (rule tendsto_ln)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1830
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1831
lemma isCont_ln' [continuous_intros]:
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1832
  "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1833
  unfolding continuous_at by (rule tendsto_ln)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1834
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1835
lemma continuous_within_ln [continuous_intros]:
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1836
  "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1837
  unfolding continuous_within by (rule tendsto_ln)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1838
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  1839
lemma continuous_on_ln [continuous_intros]:
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1840
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1841
  unfolding continuous_on_def by (auto intro: tendsto_ln)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  1842
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1843
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1844
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1845
  by (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1846
    (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1847
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1848
lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1849
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1850
  by (rule DERIV_ln[THEN DERIV_cong]) (simp_all add: divide_inverse)
33667
958dc9f03611 A little rationalisation
paulson
parents: 33549
diff changeset
  1851
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  1852
declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1853
  and DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  1854
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  1855
lemmas has_derivative_ln[derivative_intros] = DERIV_ln[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  1856
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1857
lemma ln_series:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1858
  assumes "0 < x" and "x < 2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1859
  shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1860
    (is "ln x = suminf (?f (x - 1))")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1861
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1862
  let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1863
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1864
  have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1865
  proof (rule DERIV_isconst3 [where x = x])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1866
    fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1867
    assume "x \<in> {0 <..< 2}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1868
    then have "0 < x" and "x < 2" by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1869
    have "norm (1 - x) < 1"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1870
      using \<open>0 < x\<close> and \<open>x < 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
  1871
    have "1 / x = 1 / (1 - (1 - x))" by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1872
    also have "\<dots> = (\<Sum> n. (1 - x)^n)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1873
      using geometric_sums[OF \<open>norm (1 - x) < 1\<close>] by (rule sums_unique)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1874
    also have "\<dots> = suminf (?f' x)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1875
      unfolding power_mult_distrib[symmetric]
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67268
diff changeset
  1876
      by (rule arg_cong[where f=suminf], rule arg_cong[where f="(^)"], auto)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1877
    finally have "DERIV ln x :> suminf (?f' x)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1878
      using DERIV_ln[OF \<open>0 < x\<close>] unfolding divide_inverse 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
  1879
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1880
    have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1881
    have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1882
      (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1883
    proof (rule DERIV_power_series')
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1884
      show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1885
        using \<open>0 < x\<close> \<open>x < 2\<close> by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1886
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1887
      fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1888
      assume "x \<in> {- 1<..<1}"
72980
4fc3dc37f406 default simprule for geometric series
paulson <lp15@cam.ac.uk>
parents: 72220
diff changeset
  1889
      then show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)"
4fc3dc37f406 default simprule for geometric series
paulson <lp15@cam.ac.uk>
parents: 72220
diff changeset
  1890
        by (simp add: abs_if flip: power_mult_distrib)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1891
    qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1892
    then have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1893
      unfolding One_nat_def by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1894
    then have "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  1895
      unfolding DERIV_def repos .
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1896
    ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> suminf (?f' x) - suminf (?f' x)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1897
      by (rule DERIV_diff)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1898
    then show "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1899
  qed (auto simp: assms)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1900
  then show ?thesis 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
  1901
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1902
62949
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1903
lemma exp_first_terms:
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1904
  fixes x :: "'a::{real_normed_algebra_1,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1905
  shows "exp x = (\<Sum>n<k. inverse(fact n) *\<^sub>R (x ^ n)) + (\<Sum>n. inverse(fact (n + k)) *\<^sub>R (x ^ (n + k)))"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1906
proof -
62949
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1907
  have "exp x = suminf (\<lambda>n. inverse(fact n) *\<^sub>R (x^n))"
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1908
    by (simp add: exp_def)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1909
  also from summable_exp_generic have "\<dots> = (\<Sum> n. inverse(fact(n+k)) *\<^sub>R (x ^ (n + k))) +
62949
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1910
    (\<Sum> n::nat<k. inverse(fact n) *\<^sub>R (x^n))" (is "_ = _ + ?a")
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1911
    by (rule suminf_split_initial_segment)
62949
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1912
  finally show ?thesis by simp
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1913
qed
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1914
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1915
lemma exp_first_term: "exp x = 1 + (\<Sum>n. inverse (fact (Suc n)) *\<^sub>R (x ^ Suc n))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1916
  for x :: "'a::{real_normed_algebra_1,banach}"
62949
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1917
  using exp_first_terms[of x 1] by simp
f36a54da47a4 added derivative of scaling in exponential function
immler
parents: 62948
diff changeset
  1918
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1919
lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum>n. inverse (fact (n + 2)) *\<^sub>R (x ^ (n + 2)))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1920
  for x :: "'a::{real_normed_algebra_1,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1921
  using exp_first_terms[of x 2] by (simp add: eval_nat_numeral)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1922
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1923
lemma exp_bound:
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1924
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1925
  assumes a: "0 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1926
    and b: "x \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1927
  shows "exp x \<le> 1 + x + x\<^sup>2"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1928
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1929
  have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> x\<^sup>2"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1930
  proof -
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1931
    have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1932
      by (intro sums_mult geometric_sums) simp
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1933
    then have sumsx: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1934
      by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1935
    have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1936
    proof (intro suminf_le allI)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1937
      show "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" for n :: nat
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1938
      proof -
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1939
        have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1940
          by (induct n) simp_all
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1941
        then have "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1942
          by (simp only: of_nat_le_iff)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1943
        then have "((2::real) * 2 ^ n) \<le> fact (n + 2)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1944
          unfolding of_nat_fact by simp
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1945
        then have "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1946
          by (rule le_imp_inverse_le) simp
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1947
        then have "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1948
          by (simp add: power_inverse [symmetric])
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1949
        then have "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1950
          by (rule mult_mono) (rule mult_mono, simp_all add: power_le_one a b)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1951
        then show ?thesis
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1952
          unfolding power_add by (simp add: ac_simps del: fact_Suc)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1953
      qed
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1954
      show "summable (\<lambda>n. inverse (fact (n + 2)) * x ^ (n + 2))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1955
        by (rule summable_exp [THEN summable_ignore_initial_segment])
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1956
      show "summable (\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1957
        by (rule sums_summable [OF sumsx])
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1958
    qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1959
    also have "\<dots> = x\<^sup>2"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1960
      by (rule sums_unique [THEN sym]) (rule sumsx)
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1961
    finally show ?thesis .
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1962
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1963
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1964
    unfolding exp_first_two_terms by auto
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1965
qed
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  1966
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1967
corollary exp_half_le2: "exp(1/2) \<le> (2::real)"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1968
  using exp_bound [of "1/2"]
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1969
  by (simp add: field_simps)
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1970
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  1971
corollary exp_le: "exp 1 \<le> (3::real)"
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  1972
  using exp_bound [of 1]
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  1973
  by (simp add: field_simps)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  1974
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1975
lemma exp_bound_half: "norm z \<le> 1/2 \<Longrightarrow> norm (exp z) \<le> 2"
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1976
  by (blast intro: order_trans intro!: exp_half_le2 norm_exp)
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1977
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1978
lemma exp_bound_lemma:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1979
  assumes "norm z \<le> 1/2"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1980
  shows "norm (exp z) \<le> 1 + 2 * norm z"
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1981
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1982
  have *: "(norm z)\<^sup>2 \<le> norm z * 1"
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1983
    unfolding power2_eq_square
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1984
    by (rule mult_left_mono) (use assms in auto)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1985
  have "norm (exp z) \<le> exp (norm z)"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1986
    by (rule norm_exp)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1987
  also have "\<dots> \<le> 1 + (norm z) + (norm z)\<^sup>2"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1988
    using assms exp_bound by auto
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1989
  also have "\<dots> \<le> 1 + 2 * norm z"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1990
    using * by auto
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1991
  finally show ?thesis .
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1992
qed
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1993
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1994
lemma real_exp_bound_lemma: "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp x \<le> 1 + 2 * x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1995
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1996
  using exp_bound_lemma [of x] by simp
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  1997
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  1998
lemma ln_one_minus_pos_upper_bound:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  1999
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2000
  assumes a: "0 \<le> x" and b: "x < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2001
  shows "ln (1 - x) \<le> - x"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2002
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2003
  have "(1 - x) * (1 + x + x\<^sup>2) = 1 - x^3"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2004
    by (simp add: algebra_simps power2_eq_square power3_eq_cube)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2005
  also have "\<dots> \<le> 1"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2006
    by (auto simp: a)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2007
  finally have "(1 - x) * (1 + x + x\<^sup>2) \<le> 1" .
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  2008
  moreover have c: "0 < 1 + x + x\<^sup>2"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2009
    by (simp add: add_pos_nonneg a)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2010
  ultimately have "1 - x \<le> 1 / (1 + x + x\<^sup>2)"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2011
    by (elim mult_imp_le_div_pos)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2012
  also have "\<dots> \<le> 1 / exp x"
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  2013
    by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2014
        real_sqrt_pow2_iff real_sqrt_power)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2015
  also have "\<dots> = exp (- x)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2016
    by (auto simp: exp_minus divide_inverse)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2017
  finally have "1 - x \<le> exp (- x)" .
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2018
  also have "1 - x = exp (ln (1 - x))"
54576
e877eec2b698 tidied more proofs
paulson
parents: 54575
diff changeset
  2019
    by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2020
  finally have "exp (ln (1 - x)) \<le> exp (- x)" .
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2021
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2022
    by (auto simp only: exp_le_cancel_iff)
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2023
qed
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2024
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2025
lemma exp_ge_add_one_self [simp]: "1 + x \<le> exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2026
  for x :: real
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2027
proof (cases "0 \<le> x \<or> x \<le> -1")
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2028
  case True
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2029
  then show ?thesis
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2030
    by (meson exp_ge_add_one_self_aux exp_ge_zero order.trans real_add_le_0_iff)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2031
next
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2032
  case False
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2033
  then have ln1: "ln (1 + x) \<le> x"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2034
    using ln_one_minus_pos_upper_bound [of "-x"] by simp
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2035
  have "1 + x = exp (ln (1 + x))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2036
    using False by auto
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2037
  also have "\<dots> \<le> exp x"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2038
    by (simp add: ln1)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2039
  finally show ?thesis .
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2040
qed
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2041
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2042
lemma ln_one_plus_pos_lower_bound:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2043
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2044
  assumes a: "0 \<le> x" and b: "x \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2045
  shows "x - x\<^sup>2 \<le> ln (1 + x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2046
proof -
53076
47c9aff07725 more symbols;
wenzelm
parents: 53015
diff changeset
  2047
  have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2048
    by (rule exp_diff)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2049
  also have "\<dots> \<le> (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
54576
e877eec2b698 tidied more proofs
paulson
parents: 54575
diff changeset
  2050
    by (metis a b divide_right_mono exp_bound exp_ge_zero)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2051
  also have "\<dots> \<le> (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56541
diff changeset
  2052
    by (simp add: a divide_left_mono add_pos_nonneg)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2053
  also from a have "\<dots> \<le> 1 + x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2054
    by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2055
  finally have "exp (x - x\<^sup>2) \<le> 1 + x" .
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2056
  also have "\<dots> = exp (ln (1 + x))"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2057
  proof -
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2058
    from a have "0 < 1 + x" by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2059
    then show ?thesis
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2060
      by (auto simp only: exp_ln_iff [THEN sym])
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2061
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2062
  finally have "exp (x - x\<^sup>2) \<le> exp (ln (1 + x))" .
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2063
  then show ?thesis
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  2064
    by (metis exp_le_cancel_iff)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2065
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2066
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2067
lemma ln_one_minus_pos_lower_bound:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2068
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2069
  assumes a: "0 \<le> x" and b: "x \<le> 1 / 2"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2070
  shows "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2071
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2072
  from b have c: "x < 1" by auto
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2073
  then have "ln (1 - x) = - ln (1 + x / (1 - x))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2074
    by (auto simp: ln_inverse [symmetric] field_simps intro: arg_cong [where f=ln])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2075
  also have "- (x / (1 - x)) \<le> \<dots>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2076
  proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2077
    have "ln (1 + x / (1 - x)) \<le> x / (1 - x)"
56571
f4635657d66f added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents: 56544
diff changeset
  2078
      using a c by (intro ln_add_one_self_le_self) auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2079
    then show ?thesis
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2080
      by auto
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2081
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2082
  also have "- (x / (1 - x)) = - x / (1 - x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2083
    by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2084
  finally have d: "- x / (1 - x) \<le> ln (1 - x)" .
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2085
  have "0 < 1 - x" using a b by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2086
  then have e: "- x - 2 * x\<^sup>2 \<le> - x / (1 - x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2087
    using mult_right_le_one_le[of "x * x" "2 * x"] a b
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2088
    by (simp add: field_simps power2_eq_square)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2089
  from e d show "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2090
    by (rule order_trans)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2091
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2092
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2093
lemma ln_add_one_self_le_self2:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2094
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2095
  shows "-1 < x \<Longrightarrow> ln (1 + x) \<le> x"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2096
  by (metis diff_gt_0_iff_gt diff_minus_eq_add exp_ge_add_one_self exp_le_cancel_iff exp_ln minus_less_iff)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2097
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2098
lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2099
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2100
  assumes x: "0 \<le> x" and x1: "x \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2101
  shows "\<bar>ln (1 + x) - x\<bar> \<le> x\<^sup>2"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2102
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2103
  from x have "ln (1 + x) \<le> x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2104
    by (rule ln_add_one_self_le_self)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2105
  then have "ln (1 + x) - x \<le> 0"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2106
    by simp
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61942
diff changeset
  2107
  then have "\<bar>ln(1 + x) - x\<bar> = - (ln(1 + x) - x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2108
    by (rule abs_of_nonpos)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2109
  also have "\<dots> = x - ln (1 + x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2110
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2111
  also have "\<dots> \<le> x\<^sup>2"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2112
  proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2113
    from x x1 have "x - x\<^sup>2 \<le> ln (1 + x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2114
      by (intro ln_one_plus_pos_lower_bound)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2115
    then show ?thesis
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2116
      by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2117
  qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2118
  finally show ?thesis .
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2119
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2120
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2121
lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2122
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2123
  assumes a: "-(1 / 2) \<le> x" and b: "x \<le> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2124
  shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2125
proof -
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2126
  have *: "- (-x) - 2 * (-x)\<^sup>2 \<le> ln (1 - (- x))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2127
    by (metis a b diff_zero ln_one_minus_pos_lower_bound minus_diff_eq neg_le_iff_le) 
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2128
  have "\<bar>ln (1 + x) - x\<bar> = x - ln (1 - (- x))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2129
    using a ln_add_one_self_le_self2 [of x] by (simp add: abs_if)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2130
  also have "\<dots> \<le> 2 * x\<^sup>2"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2131
    using * by (simp add: algebra_simps)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2132
  finally show ?thesis .
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2133
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2134
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2135
lemma abs_ln_one_plus_x_minus_x_bound:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2136
  fixes x :: real
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2137
  assumes "\<bar>x\<bar> \<le> 1 / 2"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2138
  shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2139
proof (cases "0 \<le> x")
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2140
  case True
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2141
  then show ?thesis
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2142
    using abs_ln_one_plus_x_minus_x_bound_nonneg assms by fastforce
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2143
next
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2144
  case False
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2145
  then show ?thesis
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2146
    using abs_ln_one_plus_x_minus_x_bound_nonpos assms by auto
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2147
qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2148
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2149
lemma ln_x_over_x_mono:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2150
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2151
  assumes x: "exp 1 \<le> x" "x \<le> y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2152
  shows "ln y / y \<le> ln x / x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2153
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2154
  note x
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2155
  moreover have "0 < exp (1::real)" by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2156
  ultimately have a: "0 < x" and b: "0 < y"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2157
    by (fast intro: less_le_trans order_trans)+
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2158
  have "x * ln y - x * ln x = x * (ln y - ln x)"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2159
    by (simp add: algebra_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2160
  also have "\<dots> = x * ln (y / x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2161
    by (simp only: ln_div a b)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2162
  also have "y / x = (x + (y - x)) / x"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2163
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2164
  also have "\<dots> = 1 + (y - x) / x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2165
    using x a by (simp add: field_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2166
  also have "x * ln (1 + (y - x) / x) \<le> x * ((y - x) / x)"
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  2167
    using x a
56571
f4635657d66f added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents: 56544
diff changeset
  2168
    by (intro mult_left_mono ln_add_one_self_le_self) simp_all
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2169
  also have "\<dots> = y - x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2170
    using a by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2171
  also have "\<dots> = (y - x) * ln (exp 1)" by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2172
  also have "\<dots> \<le> (y - x) * ln x"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2173
    using a x exp_total of_nat_1 x(1)  by (fastforce intro: mult_left_mono)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2174
  also have "\<dots> = y * ln x - x * ln x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2175
    by (rule left_diff_distrib)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2176
  finally have "x * ln y \<le> y * ln x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2177
    by arith
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2178
  then have "ln y \<le> (y * ln x) / x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2179
    using a by (simp add: field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2180
  also have "\<dots> = y * (ln x / x)" by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2181
  finally show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2182
    using b by (simp add: field_simps)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2183
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2184
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2185
lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2186
  for x :: real
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2187
  using exp_ge_add_one_self[of "ln x"] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2188
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2189
corollary ln_diff_le: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2190
  for x :: real
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  2191
  by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  2192
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2193
lemma ln_eq_minus_one:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2194
  fixes x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2195
  assumes "0 < x" "ln x = x - 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2196
  shows "x = 1"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2197
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2198
  let ?l = "\<lambda>y. ln y - y + 1"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2199
  have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  2200
    by (auto intro!: derivative_eq_intros)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2201
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2202
  show ?thesis
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2203
  proof (cases rule: linorder_cases)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2204
    assume "x < 1"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2205
    from dense[OF \<open>x < 1\<close>] obtain a where "x < a" "a < 1" by blast
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2206
    from \<open>x < a\<close> have "?l x < ?l a"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  2207
    proof (rule DERIV_pos_imp_increasing)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2208
      fix y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2209
      assume "x \<le> y" "y \<le> a"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2210
      with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2211
        by (auto simp: field_simps)
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  2212
      with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" by blast
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2213
    qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2214
    also have "\<dots> \<le> 0"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2215
      using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2216
    finally show "x = 1" using assms by auto
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2217
  next
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2218
    assume "1 < x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2219
    from dense[OF this] obtain a where "1 < a" "a < x" by blast
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2220
    from \<open>a < x\<close> have "?l x < ?l a"
68638
87d1bff264df de-applying and meta-quantifying
paulson <lp15@cam.ac.uk>
parents: 68635
diff changeset
  2221
    proof (rule DERIV_neg_imp_decreasing)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2222
      fix y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2223
      assume "a \<le> y" "y \<le> x"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2224
      with \<open>1 < a\<close> have "1 / y - 1 < 0" "0 < y"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2225
        by (auto simp: field_simps)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2226
      with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2227
        by blast
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2228
    qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2229
    also have "\<dots> \<le> 0"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2230
      using ln_le_minus_one \<open>1 < a\<close> by (auto simp: field_simps)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2231
    finally show "x = 1" using assms by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2232
  next
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2233
    assume "x = 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2234
    then show ?thesis by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2235
  qed
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2236
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2237
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2238
lemma ln_x_over_x_tendsto_0: "((\<lambda>x::real. ln x / x) \<longlongrightarrow> 0) at_top"
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2239
proof (rule lhospital_at_top_at_top[where f' = inverse and g' = "\<lambda>_. 1"])
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2240
  from eventually_gt_at_top[of "0::real"]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2241
  show "\<forall>\<^sub>F x in at_top. (ln has_real_derivative inverse x) (at x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2242
    by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2243
qed (use tendsto_inverse_0 in
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2244
      \<open>auto simp: filterlim_ident dest!: tendsto_mono[OF at_top_le_at_infinity]\<close>)
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2245
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2246
lemma exp_ge_one_plus_x_over_n_power_n:
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2247
  assumes "x \<ge> - real n" "n > 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2248
  shows "(1 + x / of_nat n) ^ n \<le> exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2249
proof (cases "x = - of_nat n")
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2250
  case False
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2251
  from assms False have "(1 + x / of_nat n) ^ n = exp (of_nat n * ln (1 + x / of_nat n))"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2252
    by (subst exp_of_nat_mult, subst exp_ln) (simp_all add: field_simps)
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2253
  also from assms False have "ln (1 + x / real n) \<le> x / real n"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2254
    by (intro ln_add_one_self_le_self2) (simp_all add: field_simps)
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2255
  with assms have "exp (of_nat n * ln (1 + x / of_nat n)) \<le> exp x"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2256
    by (simp add: field_simps)
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2257
  finally show ?thesis .
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2258
next
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2259
  case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2260
  then show ?thesis by (simp add: zero_power)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2261
qed
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2262
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2263
lemma exp_ge_one_minus_x_over_n_power_n:
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2264
  assumes "x \<le> real n" "n > 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2265
  shows "(1 - x / of_nat n) ^ n \<le> exp (-x)"
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2266
  using exp_ge_one_plus_x_over_n_power_n[of n "-x"] assms by simp
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2267
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2268
lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot"
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2269
  unfolding tendsto_Zfun_iff
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2270
proof (rule ZfunI, simp add: eventually_at_bot_dense)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2271
  fix r :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2272
  assume "0 < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2273
  have "exp x < r" if "x < ln r" for x
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2274
    by (metis \<open>0 < r\<close> exp_less_mono exp_ln that)
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2275
  then show "\<exists>k. \<forall>n<k. exp n < r" by auto
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2276
qed
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2277
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2278
lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2279
  by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g=ln])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2280
    (auto intro: eventually_gt_at_top)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2281
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2282
lemma lim_exp_minus_1: "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2283
  for x :: "'a::{real_normed_field,banach}"
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  2284
proof -
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  2285
  have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  2286
    by (intro derivative_eq_intros | simp)+
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  2287
  then show ?thesis
68634
db0980691ef4 more de-applying and a fix
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2288
    by (simp add: Deriv.has_field_derivative_iff)
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  2289
qed
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59587
diff changeset
  2290
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2291
lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2292
  by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g=exp])
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51527
diff changeset
  2293
     (auto simp: eventually_at_filter)
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2294
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2295
lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2296
  by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g=exp])
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  2297
     (auto intro: eventually_gt_at_top)
50326
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 49962
diff changeset
  2298
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60688
diff changeset
  2299
lemma filtermap_ln_at_top: "filtermap (ln::real \<Rightarrow> real) at_top = at_top"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60688
diff changeset
  2300
  by (intro filtermap_fun_inverse[of exp] exp_at_top ln_at_top) auto
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60688
diff changeset
  2301
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60688
diff changeset
  2302
lemma filtermap_exp_at_top: "filtermap (exp::real \<Rightarrow> real) at_top = at_top"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60688
diff changeset
  2303
  by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60688
diff changeset
  2304
     (auto simp: eventually_at_top_dense)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60688
diff changeset
  2305
65204
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  2306
lemma filtermap_ln_at_right: "filtermap ln (at_right (0::real)) = at_bot"
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  2307
  by (auto intro!: filtermap_fun_inverse[where g="\<lambda>x. exp x"] ln_at_0
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  2308
      simp: filterlim_at exp_at_bot)
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65109
diff changeset
  2309
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2310
lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) \<longlongrightarrow> (0::real)) at_top"
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2311
proof (induct k)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2312
  case 0
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2313
  show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top"
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2314
    by (simp add: inverse_eq_divide[symmetric])
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2315
       (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2316
         at_top_le_at_infinity order_refl)
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2317
next
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2318
  case (Suc k)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2319
  show ?case
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2320
  proof (rule lhospital_at_top_at_top)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2321
    show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  2322
      by eventually_elim (intro derivative_eq_intros, auto)
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2323
    show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  2324
      by eventually_elim auto
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2325
    show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2326
      by auto
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2327
    from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2328
    show "((\<lambda>x. real (Suc k) * x ^ k / exp x) \<longlongrightarrow> 0) at_top"
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2329
      by simp
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2330
  qed (rule exp_at_top)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2331
qed
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  2332
64758
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2333
subsubsection\<open> A couple of simple bounds\<close>
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2334
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2335
lemma exp_plus_inverse_exp:
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2336
  fixes x::real
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2337
  shows "2 \<le> exp x + inverse (exp x)"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2338
proof -
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2339
  have "2 \<le> exp x + exp (-x)"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2340
    using exp_ge_add_one_self [of x] exp_ge_add_one_self [of "-x"]
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2341
    by linarith
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2342
  then show ?thesis
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2343
    by (simp add: exp_minus)
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2344
qed
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2345
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2346
lemma real_le_x_sinh:
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2347
  fixes x::real
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2348
  assumes "0 \<le> x"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2349
  shows "x \<le> (exp x - inverse(exp x)) / 2"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2350
proof -
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2351
  have *: "exp a - inverse(exp a) - 2*a \<le> exp b - inverse(exp b) - 2*b" if "a \<le> b" for a b::real
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2352
    using exp_plus_inverse_exp
68638
87d1bff264df de-applying and meta-quantifying
paulson <lp15@cam.ac.uk>
parents: 68635
diff changeset
  2353
    by (fastforce intro: derivative_eq_intros DERIV_nonneg_imp_nondecreasing [OF that])
64758
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2354
  show ?thesis
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2355
    using*[OF assms] by simp
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2356
qed
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2357
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2358
lemma real_le_abs_sinh:
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2359
  fixes x::real
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2360
  shows "abs x \<le> abs((exp x - inverse(exp x)) / 2)"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2361
proof (cases "0 \<le> x")
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2362
  case True
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2363
  show ?thesis
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2364
    using real_le_x_sinh [OF True] True by (simp add: abs_if)
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2365
next
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2366
  case False
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2367
  have "-x \<le> (exp(-x) - inverse(exp(-x))) / 2"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2368
    by (meson False linear neg_le_0_iff_le real_le_x_sinh)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2369
  also have "\<dots> \<le> \<bar>(exp x - inverse (exp x)) / 2\<bar>"
64758
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2370
    by (metis (no_types, hide_lams) abs_divide abs_le_iff abs_minus_cancel
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2371
       add.inverse_inverse exp_minus minus_diff_eq order_refl)
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2372
  finally show ?thesis
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2373
    using False by linarith
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2374
qed
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2375
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2376
subsection\<open>The general logarithm\<close>
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64446
diff changeset
  2377
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2378
definition log :: "real \<Rightarrow> real \<Rightarrow> real"
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  2379
  \<comment> \<open>logarithm of \<^term>\<open>x\<close> to base \<^term>\<open>a\<close>\<close>
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2380
  where "log a x = ln x / ln a"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2381
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2382
lemma tendsto_log [tendsto_intros]:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2383
  "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2384
    ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2385
  unfolding log_def by (intro tendsto_intros) auto
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2386
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2387
lemma continuous_log:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2388
  assumes "continuous F f"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2389
    and "continuous F g"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2390
    and "0 < f (Lim F (\<lambda>x. x))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2391
    and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2392
    and "0 < g (Lim F (\<lambda>x. x))"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2393
  shows "continuous F (\<lambda>x. log (f x) (g x))"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2394
  using assms unfolding continuous_def by (rule tendsto_log)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2395
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2396
lemma continuous_at_within_log[continuous_intros]:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2397
  assumes "continuous (at a within s) f"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2398
    and "continuous (at a within s) g"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2399
    and "0 < f a"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2400
    and "f a \<noteq> 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2401
    and "0 < g a"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2402
  shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2403
  using assms unfolding continuous_within by (rule tendsto_log)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2404
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2405
lemma isCont_log[continuous_intros, simp]:
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2406
  assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2407
  shows "isCont (\<lambda>x. log (f x) (g x)) a"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2408
  using assms unfolding continuous_at by (rule tendsto_log)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2409
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  2410
lemma continuous_on_log[continuous_intros]:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2411
  assumes "continuous_on s f" "continuous_on s g"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2412
    and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2413
  shows "continuous_on s (\<lambda>x. log (f x) (g x))"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2414
  using assms unfolding continuous_on_def by (fast intro: tendsto_log)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2415
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2416
lemma powr_one_eq_one [simp]: "1 powr a = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2417
  by (simp add: powr_def)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2418
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2419
lemma powr_zero_eq_one [simp]: "x powr 0 = (if x = 0 then 0 else 1)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2420
  by (simp add: powr_def)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2421
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2422
lemma powr_one_gt_zero_iff [simp]: "x powr 1 = x \<longleftrightarrow> 0 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2423
  for x :: real
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2424
  by (auto simp: powr_def)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2425
declare powr_one_gt_zero_iff [THEN iffD2, simp]
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2426
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  2427
lemma powr_diff:
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  2428
  fixes w:: "'a::{ln,real_normed_field}" shows  "w powr (z1 - z2) = w powr z1 / w powr z2"
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  2429
  by (simp add: powr_def algebra_simps exp_diff)
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  2430
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2431
lemma powr_mult: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2432
  for a x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2433
  by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2434
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2435
lemma powr_ge_pzero [simp]: "0 \<le> x powr y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2436
  for x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2437
  by (simp add: powr_def)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2438
67573
ed0a7090167d added lemmas, avoid 'float_of 0'
immler
parents: 67443
diff changeset
  2439
lemma powr_non_neg[simp]: "\<not>a powr x < 0" for a x::real
ed0a7090167d added lemmas, avoid 'float_of 0'
immler
parents: 67443
diff changeset
  2440
  using powr_ge_pzero[of a x] by arith
ed0a7090167d added lemmas, avoid 'float_of 0'
immler
parents: 67443
diff changeset
  2441
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2442
lemma inverse_powr: "\<And>y::real. 0 \<le> y \<Longrightarrow> inverse y powr a = inverse (y powr a)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2443
    by (simp add: exp_minus ln_inverse powr_def)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2444
70723
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70722
diff changeset
  2445
lemma powr_divide: "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2446
  for a b x :: real
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2447
    by (simp add: divide_inverse powr_mult inverse_powr)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2448
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2449
lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  2450
  for a b x :: "'a::{ln,real_normed_field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2451
  by (simp add: powr_def exp_add [symmetric] distrib_right)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2452
70723
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70722
diff changeset
  2453
lemma powr_mult_base: "0 \<le> x \<Longrightarrow>x * x powr y = x powr (1 + y)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2454
  for x :: real
63092
a949b2a5f51d eliminated use of empty "assms";
wenzelm
parents: 63040
diff changeset
  2455
  by (auto simp: powr_add)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2456
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2457
lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2458
  for a b x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2459
  by (simp add: powr_def)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2460
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2461
lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2462
  for a b x :: real
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  2463
  by (simp add: powr_powr mult.commute)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2464
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2465
lemma powr_minus: "x powr (- a) = inverse (x powr a)"
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  2466
      for a x :: "'a::{ln,real_normed_field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2467
  by (simp add: powr_def exp_minus [symmetric])
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2468
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2469
lemma powr_minus_divide: "x powr (- a) = 1/(x powr a)"
67268
bdf25939a550 new/improved theories involving convergence; better pretty-printing for bounded quantifiers and sum/product
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  2470
      for a x :: "'a::{ln,real_normed_field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2471
  by (simp add: divide_inverse powr_minus)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2472
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2473
lemma divide_powr_uminus: "a / b powr c = a * b powr (- c)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2474
  for a b c :: real
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2475
  by (simp add: powr_minus_divide)
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2476
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2477
lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2478
  for a b x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2479
  by (simp add: powr_def)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2480
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2481
lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2482
  for a b x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2483
  by (simp add: powr_def)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2484
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2485
lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a < x powr b \<longleftrightarrow> a < b"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2486
  for a b x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2487
  by (blast intro: powr_less_cancel powr_less_mono)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2488
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2489
lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a \<le> x powr b \<longleftrightarrow> a \<le> b"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2490
  for a b x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2491
  by (simp add: linorder_not_less [symmetric])
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2492
66511
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2493
lemma powr_realpow: "0 < x \<Longrightarrow> x powr (real n) = x^n"
71837
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2494
  by (induction n) (simp_all add: ac_simps powr_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2495
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2496
lemma powr_real_of_int':
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2497
  assumes "x \<ge> 0" "x \<noteq> 0 \<or> n > 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2498
  shows   "x powr real_of_int n = power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2499
proof (cases "x = 0")
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2500
  case False
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2501
  with assms have "x > 0" by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2502
  show ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2503
  proof (cases n rule: int_cases4)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2504
    case (nonneg n)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2505
    thus ?thesis using \<open>x > 0\<close>
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2506
      by (auto simp add: powr_realpow)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2507
  next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2508
    case (neg n)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2509
    thus ?thesis using \<open>x > 0\<close>
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2510
      by (auto simp add: powr_realpow powr_minus power_int_minus)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2511
  qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71585
diff changeset
  2512
qed (use assms in auto)
66511
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2513
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2514
lemma log_ln: "ln x = log (exp(1)) x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2515
  by (simp add: log_def)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2516
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2517
lemma DERIV_log:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2518
  assumes "x > 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2519
  shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2520
proof -
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62949
diff changeset
  2521
  define lb where "lb = 1 / ln b"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2522
  moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2523
    using \<open>x > 0\<close> by (auto intro!: derivative_eq_intros)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2524
  ultimately show ?thesis
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2525
    by (simp add: log_def)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2526
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2527
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  2528
lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2529
  and DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2530
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2531
lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2532
  by (simp add: powr_def log_def)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2533
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2534
lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2535
  by (simp add: log_def powr_def)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2536
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2537
lemma log_mult:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2538
  "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2539
    log a (x * y) = log a x + log a y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2540
  by (simp add: log_def ln_mult divide_inverse distrib_right)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2541
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2542
lemma log_eq_div_ln_mult_log:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2543
  "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2544
    log a x = (ln b/ln a) * log b x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2545
  by (simp add: log_def divide_inverse)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2546
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2547
text\<open>Base 10 logarithms\<close>
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2548
lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2549
  by (simp add: log_def)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2550
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2551
lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2552
  by (simp add: log_def)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2553
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2554
lemma log_one [simp]: "log a 1 = 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2555
  by (simp add: log_def)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2556
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2557
lemma log_eq_one [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a a = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2558
  by (simp add: log_def)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2559
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2560
lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
68638
87d1bff264df de-applying and meta-quantifying
paulson <lp15@cam.ac.uk>
parents: 68635
diff changeset
  2561
  using ln_inverse log_def by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2562
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2563
lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2564
  by (simp add: log_mult divide_inverse log_inverse)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2565
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2566
lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> x \<noteq> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2567
  for a x :: real
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2568
  by (simp add: powr_def)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2569
67573
ed0a7090167d added lemmas, avoid 'float_of 0'
immler
parents: 67443
diff changeset
  2570
lemma powr_nonneg_iff[simp]: "a powr x \<le> 0 \<longleftrightarrow> a = 0"
ed0a7090167d added lemmas, avoid 'float_of 0'
immler
parents: 67443
diff changeset
  2571
  for a x::real
ed0a7090167d added lemmas, avoid 'float_of 0'
immler
parents: 67443
diff changeset
  2572
  by (meson not_less powr_gt_zero)
ed0a7090167d added lemmas, avoid 'float_of 0'
immler
parents: 67443
diff changeset
  2573
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2574
lemma log_add_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x + y = log b (x * b powr y)"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2575
  and add_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y + log b x = log b (b powr y * x)"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2576
  and log_minus_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x - y = log b (x * b powr -y)"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2577
  and minus_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y - log b x = log b (b powr y / x)"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2578
  by (simp_all add: log_mult log_divide)
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2579
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2580
lemma log_less_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  2581
  using powr_less_cancel_iff [of a] powr_log_cancel [of a x] powr_log_cancel [of a y]
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  2582
  by (metis less_eq_real_def less_trans not_le zero_less_one)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2583
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2584
lemma log_inj:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2585
  assumes "1 < b"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2586
  shows "inj_on (log b) {0 <..}"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2587
proof (rule inj_onI, simp)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2588
  fix x y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2589
  assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2590
  show "x = y"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2591
  proof (cases rule: linorder_cases)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2592
    assume "x = y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2593
    then show ?thesis by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2594
  next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2595
    assume "x < y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2596
    then have "log b x < log b y"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2597
      using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2598
    then show ?thesis using * by simp
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2599
  next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2600
    assume "y < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2601
    then have "log b y < log b x"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2602
      using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2603
    then show ?thesis using * by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2604
  qed
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2605
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2606
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2607
lemma log_le_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x \<le> log a y \<longleftrightarrow> x \<le> y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2608
  by (simp add: linorder_not_less [symmetric])
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2609
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2610
lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2611
  using log_less_cancel_iff[of a 1 x] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2612
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2613
lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2614
  using log_le_cancel_iff[of a 1 x] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2615
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2616
lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2617
  using log_less_cancel_iff[of a x 1] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2618
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2619
lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2620
  using log_le_cancel_iff[of a x 1] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2621
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2622
lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2623
  using log_less_cancel_iff[of a a x] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2624
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2625
lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2626
  using log_le_cancel_iff[of a a x] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2627
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2628
lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2629
  using log_less_cancel_iff[of a x a] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2630
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2631
lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2632
  using log_le_cancel_iff[of a x a] by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2633
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2634
lemma le_log_iff:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2635
  fixes b x y :: real
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2636
  assumes "1 < b" "x > 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2637
  shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> x"
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
  2638
  using assms
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  2639
  by (metis less_irrefl less_trans powr_le_cancel_iff powr_log_cancel zero_less_one)
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2640
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2641
lemma less_log_iff:
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2642
  assumes "1 < b" "x > 0"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2643
  shows "y < log b x \<longleftrightarrow> b powr y < x"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2644
  by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2645
    powr_log_cancel zero_less_one)
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2646
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2647
lemma
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2648
  assumes "1 < b" "x > 0"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2649
  shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2650
    and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y"
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2651
  using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y]
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2652
  by auto
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2653
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2654
lemmas powr_le_iff = le_log_iff[symmetric]
66515
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2655
  and powr_less_iff = less_log_iff[symmetric]
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2656
  and less_powr_iff = log_less_iff[symmetric]
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2657
  and le_powr_iff = log_le_iff[symmetric]
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2658
66511
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2659
lemma le_log_of_power:
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2660
  assumes "b ^ n \<le> m" "1 < b"
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2661
  shows "n \<le> log b m"
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2662
proof -
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2663
  from assms have "0 < m" by (metis less_trans zero_less_power less_le_trans zero_less_one)
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2664
  thus ?thesis using assms by (simp add: le_log_iff powr_realpow)
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2665
qed
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2666
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2667
lemma le_log2_of_power: "2 ^ n \<le> m \<Longrightarrow> n \<le> log 2 m" for m n :: nat
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2668
using le_log_of_power[of 2] by simp
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2669
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2670
lemma log_of_power_le: "\<lbrakk> m \<le> b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) \<le> n"
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2671
by (simp add: log_le_iff powr_realpow)
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2672
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2673
lemma log2_of_power_le: "\<lbrakk> m \<le> 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m \<le> n" for m n :: nat
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2674
using log_of_power_le[of _ 2] by simp
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2675
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2676
lemma log_of_power_less: "\<lbrakk> m < b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) < n"
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2677
by (simp add: log_less_iff powr_realpow)
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2678
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2679
lemma log2_of_power_less: "\<lbrakk> m < 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m < n" for m n :: nat
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2680
using log_of_power_less[of _ 2] by simp
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2681
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2682
lemma less_log_of_power:
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2683
  assumes "b ^ n < m" "1 < b"
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2684
  shows "n < log b m"
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2685
proof -
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2686
  have "0 < m" by (metis assms less_trans zero_less_power zero_less_one)
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2687
  thus ?thesis using assms by (simp add: less_log_iff powr_realpow)
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2688
qed
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2689
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2690
lemma less_log2_of_power: "2 ^ n < m \<Longrightarrow> n < log 2 m" for m n :: nat
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2691
using less_log_of_power[of 2] by simp
9756684f4d74 tuned proofs
nipkow
parents: 66510
diff changeset
  2692
64446
ec766f7b887e added simp rule
nipkow
parents: 64272
diff changeset
  2693
lemma gr_one_powr[simp]:
ec766f7b887e added simp rule
nipkow
parents: 64272
diff changeset
  2694
  fixes x y :: real shows "\<lbrakk> x > 1; y > 0 \<rbrakk> \<Longrightarrow> 1 < x powr y"
ec766f7b887e added simp rule
nipkow
parents: 64272
diff changeset
  2695
by(simp add: less_powr_iff)
ec766f7b887e added simp rule
nipkow
parents: 64272
diff changeset
  2696
70350
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  2697
lemma log_pow_cancel [simp]:
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  2698
  "a > 0 \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a ^ b) = b"
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  2699
  by (simp add: ln_realpow log_def)
571ae57313a4 moved some theorems into HOL main corpus
haftmann
parents: 70270
diff changeset
  2700
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2701
lemma floor_log_eq_powr_iff: "x > 0 \<Longrightarrow> b > 1 \<Longrightarrow> \<lfloor>log b x\<rfloor> = k \<longleftrightarrow> b powr k \<le> x \<and> x < b powr (k + 1)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2702
  by (auto simp: floor_eq_iff powr_le_iff less_powr_iff)
58984
ae0c56c485ae added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents: 58981
diff changeset
  2703
66515
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2704
lemma floor_log_nat_eq_powr_iff: fixes b n k :: nat
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2705
  shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow>
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2706
  floor (log b (real k)) = n \<longleftrightarrow> b^n \<le> k \<and> k < b^(n+1)"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2707
by (auto simp: floor_log_eq_powr_iff powr_add powr_realpow
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2708
               of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2709
         simp del: of_nat_power of_nat_mult)
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2710
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2711
lemma floor_log_nat_eq_if: fixes b n k :: nat
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2712
  assumes "b^n \<le> k" "k < b^(n+1)" "b \<ge> 2"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2713
  shows "floor (log b (real k)) = n"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2714
proof -
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2715
  have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2716
  with assms show ?thesis by(simp add: floor_log_nat_eq_powr_iff)
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2717
qed
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2718
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2719
lemma ceiling_log_eq_powr_iff: "\<lbrakk> x > 0; b > 1 \<rbrakk>
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2720
  \<Longrightarrow> \<lceil>log b x\<rceil> = int k + 1 \<longleftrightarrow> b powr k < x \<and> x \<le> b powr (k + 1)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2721
by (auto simp: ceiling_eq_iff powr_less_iff le_powr_iff)
66515
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2722
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2723
lemma ceiling_log_nat_eq_powr_iff: fixes b n k :: nat
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2724
  shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow>
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2725
  ceiling (log b (real k)) = int n + 1 \<longleftrightarrow> (b^n < k \<and> k \<le> b^(n+1))"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2726
using ceiling_log_eq_powr_iff
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2727
by (auto simp: powr_add powr_realpow of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2728
         simp del: of_nat_power of_nat_mult)
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2729
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2730
lemma ceiling_log_nat_eq_if: fixes b n k :: nat
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2731
  assumes "b^n < k" "k \<le> b^(n+1)" "b \<ge> 2"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2732
  shows "ceiling (log b (real k)) = int n + 1"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2733
proof -
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2734
  have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2735
  with assms show ?thesis by(simp add: ceiling_log_nat_eq_powr_iff)
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2736
qed
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2737
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2738
lemma floor_log2_div2: fixes n :: nat assumes "n \<ge> 2"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2739
shows "floor(log 2 n) = floor(log 2 (n div 2)) + 1"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2740
proof cases
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2741
  assume "n=2" thus ?thesis by simp
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2742
next
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2743
  let ?m = "n div 2"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2744
  assume "n\<noteq>2"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2745
  hence "1 \<le> ?m" using assms by arith
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2746
  then obtain i where i: "2 ^ i \<le> ?m" "?m < 2 ^ (i + 1)"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2747
    using ex_power_ivl1[of 2 ?m] by auto
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2748
  have "2^(i+1) \<le> 2*?m" using i(1) by simp
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2749
  also have "2*?m \<le> n" by arith
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2750
  finally have *: "2^(i+1) \<le> \<dots>" .
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2751
  have "n < 2^(i+1+1)" using i(2) by simp
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2752
  from floor_log_nat_eq_if[OF * this] floor_log_nat_eq_if[OF i]
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2753
  show ?thesis by simp
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2754
qed
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2755
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2756
lemma ceiling_log2_div2: assumes "n \<ge> 2"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2757
shows "ceiling(log 2 (real n)) = ceiling(log 2 ((n-1) div 2 + 1)) + 1"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2758
proof cases
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2759
  assume "n=2" thus ?thesis by simp
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2760
next
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2761
  let ?m = "(n-1) div 2 + 1"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2762
  assume "n\<noteq>2"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2763
  hence "2 \<le> ?m" using assms by arith
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2764
  then obtain i where i: "2 ^ i < ?m" "?m \<le> 2 ^ (i + 1)"
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2765
    using ex_power_ivl2[of 2 ?m] by auto
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2766
  have "n \<le> 2*?m" by arith
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2767
  also have "2*?m \<le> 2 ^ ((i+1)+1)" using i(2) by simp
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2768
  finally have *: "n \<le> \<dots>" .
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2769
  have "2^(i+1) < n" using i(1) by (auto simp: less_Suc_eq_0_disj)
66515
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2770
  from ceiling_log_nat_eq_if[OF this *] ceiling_log_nat_eq_if[OF i]
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2771
  show ?thesis by simp
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2772
qed
85c505c98332 reorganized and added log-related lemmas
nipkow
parents: 66511
diff changeset
  2773
62679
092cb9c96c99 add le_log_of_power and le_log2_of_power by Tobias Nipkow
hoelzl
parents: 62393
diff changeset
  2774
lemma powr_real_of_int:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2775
  "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (- n)))"
62049
b0f941e207cf Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents: 61976
diff changeset
  2776
  using powr_realpow[of x "nat n"] powr_realpow[of x "nat (-n)"]
62679
092cb9c96c99 add le_log_of_power and le_log2_of_power by Tobias Nipkow
hoelzl
parents: 62393
diff changeset
  2777
  by (auto simp: field_simps powr_minus)
62049
b0f941e207cf Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents: 61976
diff changeset
  2778
70270
4065e3b0e5bf Generalisations involving numerals; comparisons should now work for ennreal
paulson <lp15@cam.ac.uk>
parents: 70113
diff changeset
  2779
lemma powr_numeral [simp]: "0 \<le> x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
4065e3b0e5bf Generalisations involving numerals; comparisons should now work for ennreal
paulson <lp15@cam.ac.uk>
parents: 70113
diff changeset
  2780
  by (metis less_le power_zero_numeral powr_0 of_nat_numeral powr_realpow)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2781
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2782
lemma powr_int:
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2783
  assumes "x > 0"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2784
  shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2785
proof (cases "i < 0")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2786
  case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2787
  have r: "x powr i = 1 / x powr (- i)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2788
    by (simp add: powr_minus field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2789
  show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2790
    by (simp add: r field_simps powr_realpow[symmetric])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2791
next
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2792
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2793
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2794
    by (simp add: assms powr_realpow[symmetric])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2795
qed
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2796
68774
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2797
definition powr_real :: "real \<Rightarrow> real \<Rightarrow> real"
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2798
  where [code_abbrev, simp]: "powr_real = Transcendental.powr"
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2799
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2800
lemma compute_powr_real [code]:
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2801
  "powr_real b i =
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2802
    (if b \<le> 0 then Code.abort (STR ''powr_real with nonpositive base'') (\<lambda>_. powr_real b i)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2803
     else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>)
68774
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2804
     else Code.abort (STR ''powr_real with non-integer exponent'') (\<lambda>_. powr_real b i))"
9fc50a3e07f6 proper code abbreviation for power on real
haftmann
parents: 68642
diff changeset
  2805
    for b i :: real
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 58984
diff changeset
  2806
  by (auto simp: powr_int)
58981
11b6c099f5f3 code equation for powr
immler
parents: 58889
diff changeset
  2807
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2808
lemma powr_one: "0 \<le> x \<Longrightarrow> x powr 1 = x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2809
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2810
  using powr_realpow [of x 1] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2811
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2812
lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2813
  for x :: real
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54230
diff changeset
  2814
  using powr_int [of x "- 1"] by simp
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54230
diff changeset
  2815
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2816
lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2817
  for x :: real
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54230
diff changeset
  2818
  using powr_int [of x "- numeral n"] by simp
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2819
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2820
lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2821
  by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2822
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2823
lemma ln_powr: "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2824
  for x :: real
56483
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2825
  by (simp add: powr_def)
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2826
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2827
lemma ln_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> ln (root n b) =  ln b / n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2828
  by (simp add: root_powr_inverse ln_powr)
56952
efa2a83d548b added lemmas
nipkow
parents: 56571
diff changeset
  2829
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  2830
lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
65109
a79c1080f1e9 added numeral_powr_numeral
nipkow
parents: 65057
diff changeset
  2831
  by (simp add: ln_powr ln_powr[symmetric] mult.commute)
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  2832
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2833
lemma log_root: "n > 0 \<Longrightarrow> a > 0 \<Longrightarrow> log b (root n a) =  log b a / n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2834
  by (simp add: log_def ln_root)
56952
efa2a83d548b added lemmas
nipkow
parents: 56571
diff changeset
  2835
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2836
lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x"
56483
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2837
  by (simp add: log_def ln_powr)
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2838
64446
ec766f7b887e added simp rule
nipkow
parents: 64272
diff changeset
  2839
(* [simp] is not worth it, interferes with some proofs *)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  2840
lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x"
56483
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2841
  by (simp add: log_powr powr_realpow [symmetric])
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2842
66510
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2843
lemma log_of_power_eq:
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2844
  assumes "m = b ^ n" "b > 1"
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2845
  shows "n = log b (real m)"
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2846
proof -
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2847
  have "n = log b (b ^ n)" using assms(2) by (simp add: log_nat_power)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2848
  also have "\<dots> = log b m" using assms by simp
66510
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2849
  finally show ?thesis .
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2850
qed
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2851
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2852
lemma log2_of_power_eq: "m = 2 ^ n \<Longrightarrow> n = log 2 m" for m n :: nat
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2853
using log_of_power_eq[of _ 2] by simp
ca7a369301f6 reorganization of tree lemmas; new lemmas
nipkow
parents: 66486
diff changeset
  2854
56483
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2855
lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2856
  by (simp add: log_def)
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2857
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2858
lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n"
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2859
  by (simp add: log_def ln_realpow)
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2860
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2861
lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b"
56483
5b82c58b665c generalize ln/log_powr; add log_base_powr/pow
hoelzl
parents: 56479
diff changeset
  2862
  by (simp add: log_def ln_powr)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2863
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2864
lemma log_base_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> log (root n b) x = n * (log b x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2865
  by (simp add: log_def ln_root)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2866
67727
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  2867
lemma ln_bound: "0 < x \<Longrightarrow> ln x \<le> x" for x :: real
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  2868
  using ln_le_minus_one by force
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2869
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2870
lemma powr_mono:
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2871
  fixes x :: real
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2872
  assumes "a \<le> b" and "1 \<le> x" shows "x powr a \<le> x powr b"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2873
  using assms less_eq_real_def by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2874
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2875
lemma ge_one_powr_ge_zero: "1 \<le> x \<Longrightarrow> 0 \<le> a \<Longrightarrow> 1 \<le> x powr a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2876
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2877
  using powr_mono by fastforce
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2878
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2879
lemma powr_less_mono2: "0 < a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> x powr a < y powr a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2880
  for x :: real
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2881
  by (simp add: powr_def)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2882
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2883
lemma powr_less_mono2_neg: "a < 0 \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> y powr a < x powr a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2884
  for x :: real
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2885
  by (simp add: powr_def)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2886
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2887
lemma powr_mono2: "x powr a \<le> y powr a" if "0 \<le> a" "0 \<le> x" "x \<le> y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2888
  for x :: real
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2889
  using less_eq_real_def powr_less_mono2 that by auto
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2890
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2891
lemma powr_le1: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> x powr a \<le> 1"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2892
  for x :: real
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2893
  using powr_mono2 by fastforce
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  2894
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  2895
lemma powr_mono2':
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2896
  fixes a x y :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2897
  assumes "a \<le> 0" "x > 0" "x \<le> y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2898
  shows "x powr a \<ge> y powr a"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  2899
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2900
  from assms have "x powr - a \<le> y powr - a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2901
    by (intro powr_mono2) simp_all
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2902
  with assms show ?thesis
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2903
    by (auto simp: powr_minus field_simps)
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  2904
qed
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  2905
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2906
lemma powr_mono_both:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2907
  fixes x :: real
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2908
  assumes "0 \<le> a" "a \<le> b" "1 \<le> x" "x \<le> y"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2909
    shows "x powr a \<le> y powr b"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2910
  by (meson assms order.trans powr_mono powr_mono2 zero_le_one)
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65552
diff changeset
  2911
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2912
lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2913
  for x :: real
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2914
  unfolding powr_def exp_inj_iff by simp
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2915
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  2916
lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  2917
  by (simp add: powr_def root_powr_inverse sqrt_def)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
  2918
70365
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70350
diff changeset
  2919
lemma square_powr_half [simp]:
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70350
diff changeset
  2920
  fixes x::real shows "x\<^sup>2 powr (1/2) = \<bar>x\<bar>"
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70350
diff changeset
  2921
  by (simp add: powr_half_sqrt)
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70350
diff changeset
  2922
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2923
lemma ln_powr_bound: "1 \<le> x \<Longrightarrow> 0 < a \<Longrightarrow> ln x \<le> (x powr a) / a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2924
  for x :: real
62679
092cb9c96c99 add le_log_of_power and le_log2_of_power by Tobias Nipkow
hoelzl
parents: 62393
diff changeset
  2925
  by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2926
      mult_imp_le_div_pos not_less powr_gt_zero)
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2927
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2928
lemma ln_powr_bound2:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2929
  fixes x :: real
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2930
  assumes "1 < x" and "0 < a"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2931
  shows "(ln x) powr a \<le> (a powr a) * x"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2932
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2933
  from assms have "ln x \<le> (x powr (1 / a)) / (1 / a)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  2934
    by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2935
  also have "\<dots> = a * (x powr (1 / a))"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2936
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2937
  finally have "(ln x) powr a \<le> (a * (x powr (1 / a))) powr a"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  2938
    by (metis assms less_imp_le ln_gt_zero powr_mono2)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2939
  also have "\<dots> = (a powr a) * ((x powr (1 / a)) powr a)"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  2940
    using assms powr_mult by auto
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2941
  also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2942
    by (rule powr_powr)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2943
  also have "\<dots> = x" using assms
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  2944
    by auto
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2945
  finally show ?thesis .
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2946
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2947
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2948
lemma tendsto_powr:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2949
  fixes a b :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2950
  assumes f: "(f \<longlongrightarrow> a) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2951
    and g: "(g \<longlongrightarrow> b) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2952
    and a: "a \<noteq> 0"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2953
  shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  2954
  unfolding powr_def
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  2955
proof (rule filterlim_If)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2956
  from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
  2957
    by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2958
  from f g a show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a)))
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2959
      (inf F (principal {x. f x \<noteq> 0}))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2960
    by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2961
qed
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2962
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2963
lemma tendsto_powr'[tendsto_intros]:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2964
  fixes a :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2965
  assumes f: "(f \<longlongrightarrow> a) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2966
    and g: "(g \<longlongrightarrow> b) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2967
    and a: "a \<noteq> 0 \<or> (b > 0 \<and> eventually (\<lambda>x. f x \<ge> 0) F)"
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2968
  shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2969
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2970
  from a consider "a \<noteq> 0" | "a = 0" "b > 0" "eventually (\<lambda>x. f x \<ge> 0) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2971
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2972
  then show ?thesis
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2973
  proof cases
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2974
    case 1
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2975
    with f g show ?thesis by (rule tendsto_powr)
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2976
  next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2977
    case 2
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2978
    have "((\<lambda>x. if f x = 0 then 0 else exp (g x * ln (f x))) \<longlongrightarrow> 0) F"
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2979
    proof (intro filterlim_If)
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2980
      have "filterlim f (principal {0<..}) (inf F (principal {z. f z \<noteq> 0}))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2981
        using \<open>eventually (\<lambda>x. f x \<ge> 0) F\<close>
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2982
        by (auto simp: filterlim_iff eventually_inf_principal
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2983
            eventually_principal elim: eventually_mono)
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2984
      moreover have "filterlim f (nhds a) (inf F (principal {z. f z \<noteq> 0}))"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2985
        by (rule tendsto_mono[OF _ f]) simp_all
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2986
      ultimately have f: "filterlim f (at_right 0) (inf F (principal {x. f x \<noteq> 0}))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2987
        by (simp add: at_within_def filterlim_inf \<open>a = 0\<close>)
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2988
      have g: "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2989
        by (rule tendsto_mono[OF _ g]) simp_all
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2990
      show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2991
        by (rule filterlim_compose[OF exp_at_bot] filterlim_tendsto_pos_mult_at_bot
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2992
                 filterlim_compose[OF ln_at_0] f g \<open>b > 0\<close>)+
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2993
    qed simp_all
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2994
    with \<open>a = 0\<close> show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  2995
      by (simp add: powr_def)
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2996
  qed
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2997
qed
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  2998
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  2999
lemma continuous_powr:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3000
  assumes "continuous F f"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3001
    and "continuous F g"
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3002
    and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  3003
  shows "continuous F (\<lambda>x. (f x) powr (g x :: real))"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3004
  using assms unfolding continuous_def by (rule tendsto_powr)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3005
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3006
lemma continuous_at_within_powr[continuous_intros]:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3007
  fixes f g :: "_ \<Rightarrow> real"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3008
  assumes "continuous (at a within s) f"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3009
    and "continuous (at a within s) g"
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3010
    and "f a \<noteq> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3011
  shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3012
  using assms unfolding continuous_within by (rule tendsto_powr)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3013
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3014
lemma isCont_powr[continuous_intros, simp]:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3015
  fixes f g :: "_ \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3016
  assumes "isCont f a" "isCont g a" "f a \<noteq> 0"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3017
  shows "isCont (\<lambda>x. (f x) powr g x) a"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3018
  using assms unfolding continuous_at by (rule tendsto_powr)
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3019
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  3020
lemma continuous_on_powr[continuous_intros]:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3021
  fixes f g :: "_ \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3022
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0"
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3023
  shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3024
  using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3025
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3026
lemma tendsto_powr2:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3027
  fixes a :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3028
  assumes f: "(f \<longlongrightarrow> a) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3029
    and g: "(g \<longlongrightarrow> b) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3030
    and "\<forall>\<^sub>F x in F. 0 \<le> f x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3031
    and b: "0 < b"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3032
  shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3033
  using tendsto_powr'[of f a F g b] assms by auto
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3034
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3035
lemma has_derivative_powr[derivative_intros]:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3036
  assumes g[derivative_intros]: "(g has_derivative g') (at x within X)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3037
    and f[derivative_intros]:"(f has_derivative f') (at x within X)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3038
  assumes pos: "0 < g x" and "x \<in> X"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3039
  shows "((\<lambda>x. g x powr f x::real) has_derivative (\<lambda>h. (g x powr f x) * (f' h * ln (g x) + g' h * f x / g x))) (at x within X)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3040
proof -
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3041
  have "\<forall>\<^sub>F x in at x within X. g x > 0"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3042
    by (rule order_tendstoD[OF _ pos])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3043
      (rule has_derivative_continuous[OF g, unfolded continuous_within])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3044
  then obtain d where "d > 0" and pos': "\<And>x'. x' \<in> X \<Longrightarrow> dist x' x < d \<Longrightarrow> 0 < g x'"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3045
    using pos unfolding eventually_at by force
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3046
  have "((\<lambda>x. exp (f x * ln (g x))) has_derivative
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3047
    (\<lambda>h. (g x powr f x) * (f' h * ln (g x) + g' h * f x / g x))) (at x within X)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3048
    using pos
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  3049
    by (auto intro!: derivative_eq_intros simp: field_split_simps powr_def)
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3050
  then show ?thesis
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3051
    by (rule has_derivative_transform_within[OF _ \<open>d > 0\<close> \<open>x \<in> X\<close>]) (auto simp: powr_def dest: pos')
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3052
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3053
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3054
lemma DERIV_powr:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3055
  fixes r :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3056
  assumes g: "DERIV g x :> m"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3057
    and pos: "g x > 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3058
    and f: "DERIV f x :> r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3059
  shows "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3060
  using assms
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3061
  by (auto intro!: derivative_eq_intros ext simp: has_field_derivative_def algebra_simps)
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3062
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3063
lemma DERIV_fun_powr:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3064
  fixes r :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3065
  assumes g: "DERIV g x :> m"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3066
    and pos: "g x > 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3067
  shows "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m"
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3068
  using DERIV_powr[OF g pos DERIV_const, of r] pos
65583
8d53b3bebab4 Further new material. The simprule status of some exp and ln identities was reverted.
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
  3069
  by (simp add: powr_diff field_simps)
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3070
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3071
lemma has_real_derivative_powr:
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3072
  assumes "z > 0"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3073
  shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3074
proof (subst DERIV_cong_ev[OF refl _ refl])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3075
  from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3076
    by (intro t1_space_nhds) auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3077
  then show "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3078
    unfolding powr_def by eventually_elim simp
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3079
  from assms show "((\<lambda>z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3080
    by (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3081
qed
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3082
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3083
declare has_real_derivative_powr[THEN DERIV_chain2, derivative_intros]
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61518
diff changeset
  3084
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3085
lemma tendsto_zero_powrI:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3086
  assumes "(f \<longlongrightarrow> (0::real)) F" "(g \<longlongrightarrow> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b"
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3087
  shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F"
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3088
  using tendsto_powr2[OF assms] by simp
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3089
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3090
lemma continuous_on_powr':
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3091
  fixes f g :: "_ \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3092
  assumes "continuous_on s f" "continuous_on s g"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3093
    and "\<forall>x\<in>s. f x \<ge> 0 \<and> (f x = 0 \<longrightarrow> g x > 0)"
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3094
  shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3095
  unfolding continuous_on_def
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3096
proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3097
  fix x
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3098
  assume x: "x \<in> s"
63295
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3099
  from assms x show "((\<lambda>x. f x powr g x) \<longlongrightarrow> f x powr g x) (at x within s)"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3100
  proof (cases "f x = 0")
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3101
    case True
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3102
    from assms(3) have "eventually (\<lambda>x. f x \<ge> 0) (at x within s)"
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3103
      by (auto simp: at_within_def eventually_inf_principal)
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3104
    with True x assms show ?thesis
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3105
      by (auto intro!: tendsto_zero_powrI[of f _ g "g x"] simp: continuous_on_def)
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3106
  next
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3107
    case False
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3108
    with assms x show ?thesis
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3109
      by (auto intro!: tendsto_powr' simp: continuous_on_def)
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3110
  qed
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3111
qed
52792bb9126e Facts about HK integration, complex powers, Gamma function
eberlm
parents: 63170
diff changeset
  3112
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3113
lemma tendsto_neg_powr:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3114
  assumes "s < 0"
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3115
    and f: "LIM x F. f x :> at_top"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3116
  shows "((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3117
proof -
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3118
  have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X")
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3119
    by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3120
        filterlim_tendsto_neg_mult_at_bot assms)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3121
  also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3122
    using f filterlim_at_top_dense[of f F]
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
  3123
    by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono)
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60162
diff changeset
  3124
  finally show ?thesis .
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3125
qed
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  3126
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3127
lemma tendsto_exp_limit_at_right: "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3128
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3129
proof (cases "x = 0")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3130
  case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3131
  then show ?thesis by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3132
next
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3133
  case False
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3134
  have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3135
    by (auto intro!: derivative_eq_intros)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3136
  then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3137
    by (auto simp: has_field_derivative_def field_has_derivative_at)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  3138
  then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) \<longlongrightarrow> exp x) (at 0)"
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3139
    by (rule tendsto_intros)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3140
  then show ?thesis
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3141
  proof (rule filterlim_mono_eventually)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3142
    show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3143
      unfolding eventually_at_right[OF zero_less_one]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3144
      using False
68638
87d1bff264df de-applying and meta-quantifying
paulson <lp15@cam.ac.uk>
parents: 68635
diff changeset
  3145
      by (intro exI[of _ "1 / \<bar>x\<bar>"]) (auto simp: field_simps powr_def abs_if add_nonneg_eq_0_iff)
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3146
  qed (simp_all add: at_eq_sup_left_right)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3147
qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3148
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3149
lemma tendsto_exp_limit_at_top: "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3150
  for x :: real
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3151
  by (simp add: filterlim_at_top_to_right inverse_eq_divide tendsto_exp_limit_at_right)
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3152
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3153
lemma tendsto_exp_limit_sequentially: "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3154
  for x :: real
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3155
proof (rule filterlim_mono_eventually)
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61942
diff changeset
  3156
  from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" ..
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3157
  then have "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  3158
    by (intro eventually_sequentiallyI [of n]) (auto simp: field_split_simps)
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3159
  then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
  3160
    by (rule eventually_mono) (erule powr_realpow)
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
  3161
  show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x"
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3162
    by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3163
qed auto
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57180
diff changeset
  3164
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3165
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  3166
subsection \<open>Sine and Cosine\<close>
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  3167
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3168
definition sin_coeff :: "nat \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3169
  where "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3170
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3171
definition cos_coeff :: "nat \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3172
  where "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
31271
0237e5e40b71 add constants sin_coeff, cos_coeff
huffman
parents: 31148
diff changeset
  3173
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3174
definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3175
  where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3176
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3177
definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3178
  where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)"
31271
0237e5e40b71 add constants sin_coeff, cos_coeff
huffman
parents: 31148
diff changeset
  3179
44319
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3180
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3181
  unfolding sin_coeff_def by simp
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3182
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3183
lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3184
  unfolding cos_coeff_def by simp
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3185
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3186
lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3187
  unfolding cos_coeff_def sin_coeff_def
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3188
  by (simp del: mult_Suc)
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3189
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3190
lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3191
  unfolding cos_coeff_def sin_coeff_def
58709
efdc6c533bd3 prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents: 58656
diff changeset
  3192
  by (simp del: mult_Suc) (auto elim: oddE)
44319
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3193
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3194
lemma summable_norm_sin: "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3195
  for x :: "'a::{real_normed_algebra_1,banach}"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3196
proof (rule summable_comparison_test [OF _ summable_norm_exp])
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3197
  show "\<exists>N. \<forall>n\<ge>N. norm (norm (sin_coeff n *\<^sub>R x ^ n)) \<le> norm (x ^ n /\<^sub>R fact n)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3198
    unfolding sin_coeff_def
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3199
    by (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3200
qed
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  3201
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3202
lemma summable_norm_cos: "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3203
  for x :: "'a::{real_normed_algebra_1,banach}"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3204
proof (rule summable_comparison_test [OF _ summable_norm_exp])
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3205
  show "\<exists>N. \<forall>n\<ge>N. norm (norm (cos_coeff n *\<^sub>R x ^ n)) \<le> norm (x ^ n /\<^sub>R fact n)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3206
    unfolding cos_coeff_def
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3207
    by (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3208
qed
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3209
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  3210
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3211
lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3212
  unfolding sin_def
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3213
  by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3214
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3215
lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3216
  unfolding cos_def
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3217
  by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3218
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3219
lemma sin_of_real: "sin (of_real x) = of_real (sin x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3220
  for x :: real
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3221
proof -
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3222
  have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R  x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R  (of_real x)^n)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3223
  proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3224
    show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n" for n
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3225
      by (simp add: scaleR_conv_of_real)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3226
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3227
  also have "\<dots> sums (sin (of_real x))"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3228
    by (rule sin_converges)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3229
  finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" .
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3230
  then show ?thesis
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3231
    using sums_unique2 sums_of_real [OF sin_converges] by blast
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3232
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3233
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  3234
corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  3235
  by (metis Reals_cases Reals_of_real sin_of_real)
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  3236
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3237
lemma cos_of_real: "cos (of_real x) = of_real (cos x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3238
  for x :: real
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3239
proof -
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3240
  have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R  x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R  (of_real x)^n)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3241
  proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3242
    show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n" for n
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3243
      by (simp add: scaleR_conv_of_real)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3244
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3245
  also have "\<dots> sums (cos (of_real x))"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3246
    by (rule cos_converges)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3247
  finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" .
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3248
  then show ?thesis
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3249
    using sums_unique2 sums_of_real [OF cos_converges]
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3250
    by blast
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3251
qed
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  3252
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  3253
corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  3254
  by (metis Reals_cases Reals_of_real cos_of_real)
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  3255
44319
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3256
lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
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
  3257
  by (simp add: diffs_def sin_coeff_Suc del: of_nat_Suc)
44319
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3258
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3259
lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
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
  3260
  by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc)
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  3261
65036
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  3262
lemma sin_int_times_real: "sin (of_int m * of_real x) = of_real (sin (of_int m * x))"
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  3263
  by (metis sin_of_real of_real_mult of_real_of_int_eq)
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  3264
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  3265
lemma cos_int_times_real: "cos (of_int m * of_real x) = of_real (cos (of_int m * x))"
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  3266
  by (metis cos_of_real of_real_mult of_real_of_int_eq)
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  3267
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3268
text \<open>Now at last we can get the derivatives of exp, sin and cos.\<close>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3269
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3270
lemma DERIV_sin [simp]: "DERIV sin x :> cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3271
  for x :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3272
  unfolding sin_def cos_def scaleR_conv_of_real
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3273
  apply (rule DERIV_cong)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3274
   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3275
      apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3276
              summable_minus_iff scaleR_conv_of_real [symmetric]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3277
              summable_norm_sin [THEN summable_norm_cancel]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3278
              summable_norm_cos [THEN summable_norm_cancel])
44319
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3279
  done
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3280
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  3281
declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3282
  and DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3283
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3284
lemmas has_derivative_sin[derivative_intros] = DERIV_sin[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3285
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3286
lemma DERIV_cos [simp]: "DERIV cos x :> - sin x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3287
  for x :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3288
  unfolding sin_def cos_def scaleR_conv_of_real
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3289
  apply (rule DERIV_cong)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3290
   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3291
      apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3292
              diffs_sin_coeff diffs_cos_coeff
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3293
              summable_minus_iff scaleR_conv_of_real [symmetric]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3294
              summable_norm_sin [THEN summable_norm_cancel]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3295
              summable_norm_cos [THEN summable_norm_cancel])
44319
806e0390de53 move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents: 44318
diff changeset
  3296
  done
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  3297
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  3298
declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3299
  and DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3300
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3301
lemmas has_derivative_cos[derivative_intros] = DERIV_cos[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  3302
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3303
lemma isCont_sin: "isCont sin x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3304
  for x :: "'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3305
  by (rule DERIV_sin [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3306
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3307
lemma continuous_on_sin_real: "continuous_on {a..b} sin" for a::real
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3308
  using continuous_at_imp_continuous_on isCont_sin by blast
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3309
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3310
lemma isCont_cos: "isCont cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3311
  for x :: "'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3312
  by (rule DERIV_cos [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3313
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3314
lemma continuous_on_cos_real: "continuous_on {a..b} cos" for a::real
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3315
  using continuous_at_imp_continuous_on isCont_cos by blast
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3316
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3317
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3318
context
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3319
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{real_normed_field,banach}"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3320
begin
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3321
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3322
lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3323
  by (rule isCont_o2 [OF _ isCont_sin])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3324
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3325
lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3326
  by (rule isCont_o2 [OF _ isCont_cos])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3327
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3328
lemma tendsto_sin [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3329
  by (rule isCont_tendsto_compose [OF isCont_sin])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3330
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3331
lemma tendsto_cos [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  3332
  by (rule isCont_tendsto_compose [OF isCont_cos])
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  3333
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3334
lemma continuous_sin [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  3335
  unfolding continuous_def by (rule tendsto_sin)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  3336
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3337
lemma continuous_on_sin [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  3338
  unfolding continuous_on_def by (auto intro: tendsto_sin)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  3339
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3340
lemma continuous_cos [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3341
  unfolding continuous_def by (rule tendsto_cos)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3342
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3343
lemma continuous_on_cos [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3344
  unfolding continuous_on_def by (auto intro: tendsto_cos)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3345
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3346
end
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3347
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3348
lemma continuous_within_sin: "continuous (at z within s) sin"     
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3349
  for z :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3350
  by (simp add: continuous_within tendsto_sin)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3351
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3352
lemma continuous_within_cos: "continuous (at z within s) cos"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3353
  for z :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3354
  by (simp add: continuous_within tendsto_cos)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3355
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3356
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  3357
subsection \<open>Properties of Sine and Cosine\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3358
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3359
lemma sin_zero [simp]: "sin 0 = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3360
  by (simp add: sin_def sin_coeff_def scaleR_conv_of_real)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3361
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3362
lemma cos_zero [simp]: "cos 0 = 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3363
  by (simp add: cos_def cos_coeff_def scaleR_conv_of_real)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3364
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3365
lemma DERIV_fun_sin: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin (g x)) x :> cos (g x) * m"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3366
  by (fact derivative_intros)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3367
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3368
lemma DERIV_fun_cos: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> - sin (g x) * m"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3369
  by (fact derivative_intros)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3370
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3371
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  3372
subsection \<open>Deriving the Addition Formulas\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  3373
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3374
text \<open>The product of two cosine series.\<close>
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3375
lemma cos_x_cos_y:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3376
  fixes x :: "'a::{real_normed_field,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3377
  shows
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3378
    "(\<lambda>p. \<Sum>n\<le>p.
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3379
        if even p \<and> even n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3380
        then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3381
      sums (cos x * cos y)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3382
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3383
  have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p - n)) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3384
    (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p - n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3385
     else 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3386
    if "n \<le> p" for n p :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3387
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3388
    from that have *: "even n \<Longrightarrow> even p \<Longrightarrow>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3389
        (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3390
      by (metis div_add power_add le_add_diff_inverse odd_add)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3391
    with that show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3392
      by (auto simp: algebra_simps cos_coeff_def binomial_fact)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3393
  qed
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3394
  then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3395
                  then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3396
             (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3397
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3398
  also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3399
    by (simp add: algebra_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3400
  also have "\<dots> sums (cos x * cos y)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3401
    using summable_norm_cos
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3402
    by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3403
  finally show ?thesis .
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3404
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3405
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3406
text \<open>The product of two sine series.\<close>
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3407
lemma sin_x_sin_y:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3408
  fixes x :: "'a::{real_normed_field,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3409
  shows
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3410
    "(\<lambda>p. \<Sum>n\<le>p.
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3411
        if even p \<and> odd n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3412
        then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3413
        else 0)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3414
      sums (sin x * sin y)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3415
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3416
  have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3417
    (if even p \<and> odd n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3418
     then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3419
     else 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3420
    if "n \<le> p" for n p :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3421
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3422
    have "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3423
      if np: "odd n" "even p"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3424
    proof -
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3425
      have "p > 0"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3426
        using \<open>n \<le> p\<close> neq0_conv that(1) by blast
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3427
      then have \<section>: "(- 1::real) ^ (p div 2 - Suc 0) = - ((- 1) ^ (p div 2))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3428
        using \<open>even p\<close> by (auto simp add: dvd_def power_eq_if)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3429
      from \<open>n \<le> p\<close> np have *: "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3430
        by arith+
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3431
      have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3432
        by simp
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3433
      with \<open>n \<le> p\<close> np  \<section> * show ?thesis
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3434
        by (simp add: flip: div_add power_add)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3435
    qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3436
    then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3437
      using \<open>n\<le>p\<close> by (auto simp: algebra_simps sin_coeff_def binomial_fact)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3438
  qed
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3439
  then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3440
               then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3441
             (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3442
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3443
  also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3444
    by (simp add: algebra_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3445
  also have "\<dots> sums (sin x * sin y)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3446
    using summable_norm_sin
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3447
    by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3448
  finally show ?thesis .
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3449
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3450
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3451
lemma sums_cos_x_plus_y:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3452
  fixes x :: "'a::{real_normed_field,banach}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3453
  shows
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3454
    "(\<lambda>p. \<Sum>n\<le>p.
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3455
        if even p
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3456
        then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3457
        else 0)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3458
      sums cos (x + y)"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  3459
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3460
  have
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3461
    "(\<Sum>n\<le>p.
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3462
      if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3463
      else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3464
    for p :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3465
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3466
    have
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3467
      "(\<Sum>n\<le>p. if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3468
       (if even p then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3469
      by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3470
    also have "\<dots> =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3471
       (if even p
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3472
        then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3473
        else 0)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  3474
      by (auto simp: sum_distrib_left field_simps scaleR_conv_of_real nonzero_of_real_divide)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3475
    also have "\<dots> = cos_coeff p *\<^sub>R ((x + y) ^ p)"
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
  3476
      by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real atLeast0AtMost)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3477
    finally show ?thesis .
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3478
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3479
  then have
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3480
    "(\<lambda>p. \<Sum>n\<le>p.
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3481
        if even p
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3482
        then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3483
        else 0) = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3484
    by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3485
   also have "\<dots> sums cos (x + y)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3486
    by (rule cos_converges)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3487
   finally show ?thesis .
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3488
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3489
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3490
theorem cos_add:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3491
  fixes x :: "'a::{real_normed_field,banach}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3492
  shows "cos (x + y) = cos x * cos y - sin x * sin y"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3493
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3494
  have
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3495
    "(if even p \<and> even n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3496
      then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3497
     (if even p \<and> odd n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3498
      then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3499
     (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3500
    if "n \<le> p" for n p :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3501
    by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3502
  then have
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3503
    "(\<lambda>p. \<Sum>n\<le>p. (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3504
      sums (cos x * cos y - sin x * sin y)"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3505
    using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]]
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  3506
    by (simp add: sum_subtractf [symmetric])
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3507
  then show ?thesis
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3508
    by (blast intro: sums_cos_x_plus_y sums_unique2)
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  3509
qed
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  3510
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3511
lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3512
proof -
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3513
  have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3514
    by (auto simp: sin_coeff_def elim!: oddE)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3515
  show ?thesis
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3516
    by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums])
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3517
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3518
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3519
lemma sin_minus [simp]: "sin (- x) = - sin x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3520
  for x :: "'a::{real_normed_algebra_1,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3521
  using sin_minus_converges [of x]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3522
  by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3523
      suminf_minus sums_iff equation_minus_iff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3524
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3525
lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3526
proof -
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3527
  have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3528
    by (auto simp: Transcendental.cos_coeff_def elim!: evenE)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3529
  show ?thesis
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3530
    by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums])
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3531
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3532
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3533
lemma cos_minus [simp]: "cos (-x) = cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3534
  for x :: "'a::{real_normed_algebra_1,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3535
  using cos_minus_converges [of x]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3536
  by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3537
      suminf_minus sums_iff equation_minus_iff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3538
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3539
lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3540
  for x :: "'a::{real_normed_field,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3541
  using cos_add [of x "-x"]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3542
  by (simp add: power2_eq_square algebra_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3543
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3544
lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3545
  for x :: "'a::{real_normed_field,banach}"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  3546
  by (subst add.commute, rule sin_cos_squared_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3547
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3548
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3549
  for x :: "'a::{real_normed_field,banach}"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  3550
  using sin_cos_squared_add2 [unfolded power2_eq_square] .
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3551
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3552
lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3553
  for x :: "'a::{real_normed_field,banach}"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  3554
  unfolding eq_diff_eq by (rule sin_cos_squared_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3555
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3556
lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3557
  for x :: "'a::{real_normed_field,banach}"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  3558
  unfolding eq_diff_eq by (rule sin_cos_squared_add2)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3559
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3560
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3561
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3562
  by (rule power2_le_imp_le) (simp_all add: sin_squared_eq)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3563
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3564
lemma sin_ge_minus_one [simp]: "- 1 \<le> sin x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3565
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3566
  using abs_sin_le_one [of x] by (simp add: abs_le_iff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3567
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3568
lemma sin_le_one [simp]: "sin x \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3569
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3570
  using abs_sin_le_one [of x] by (simp add: abs_le_iff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3571
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3572
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3573
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3574
  by (rule power2_le_imp_le) (simp_all add: cos_squared_eq)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3575
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3576
lemma cos_ge_minus_one [simp]: "- 1 \<le> cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3577
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3578
  using abs_cos_le_one [of x] by (simp add: abs_le_iff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3579
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3580
lemma cos_le_one [simp]: "cos x \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3581
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3582
  using abs_cos_le_one [of x] by (simp add: abs_le_iff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3583
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3584
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3585
  for x :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3586
  using cos_add [of x "- y"] by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3587
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3588
lemma cos_double: "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3589
  for x :: "'a::{real_normed_field,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3590
  using cos_add [where x=x and y=x] by (simp add: power2_eq_square)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3591
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3592
lemma sin_cos_le1: "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3593
  for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3594
  using cos_diff [of x y] by (metis abs_cos_le_one add.commute)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3595
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3596
lemma DERIV_fun_pow: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
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
  3597
  by (auto intro!: derivative_eq_intros simp:)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3598
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3599
lemma DERIV_fun_exp: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. exp (g x)) x :> exp (g x) * m"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  3600
  by (auto intro!: derivative_intros)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3601
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3602
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  3603
subsection \<open>The Constant Pi\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3604
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3605
definition pi :: real
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3606
  where "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3607
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  3608
text \<open>Show that there's a least positive \<^term>\<open>x\<close> with \<^term>\<open>cos x = 0\<close>;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  3609
   hence define pi.\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3610
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3611
lemma sin_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3612
  for x :: real
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3613
proof -
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3614
  have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3615
    by (rule sums_group) (use sin_converges [of x, unfolded scaleR_conv_of_real] in auto)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3616
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3617
    by (simp add: sin_coeff_def ac_simps)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3618
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3619
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3620
lemma sin_gt_zero_02:
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3621
  fixes x :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3622
  assumes "0 < x" and "x < 2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3623
  shows "0 < sin x"
44728
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3624
proof -
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3625
  let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)"
44728
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3626
  have pos: "\<forall>n. 0 < ?f n"
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3627
  proof
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3628
    fix n :: nat
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3629
    let ?k2 = "real (Suc (Suc (4 * n)))"
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3630
    let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3631
    have "x * x < ?k2 * ?k3"
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3632
      using assms by (intro mult_strict_mono', simp_all)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3633
    then have "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  3634
      by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3635
    then show "0 < ?f n"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  3636
      by (simp add: ac_simps divide_less_eq)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3637
qed
44728
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3638
  have sums: "?f sums sin x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3639
    by (rule sin_paired [THEN sums_group]) simp
44728
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3640
  show "0 < sin x"
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 72211
diff changeset
  3641
    unfolding sums_unique [OF sums] using sums_summable [OF sums] pos by (simp add: suminf_pos)
44728
86f43cca4886 convert lemma sin_gt_zero to Isar style;
huffman
parents: 44727
diff changeset
  3642
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3643
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3644
lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3645
  for x :: real
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3646
  using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3647
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3648
lemma cos_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3649
  for x :: real
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3650
proof -
31271
0237e5e40b71 add constants sin_coeff, cos_coeff
huffman
parents: 31148
diff changeset
  3651
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3652
    by (rule sums_group) (use cos_converges [of x, unfolded scaleR_conv_of_real] in auto)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3653
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3654
    by (simp add: cos_coeff_def ac_simps)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3655
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3656
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3657
lemma sum_pos_lt_pair:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  3658
  fixes f :: "nat \<Rightarrow> real"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3659
  assumes f: "summable f" and fplus: "\<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc (Suc 0) * d) + 1))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3660
  shows "sum f {..<k} < suminf f"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3661
proof -
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3662
  have "(\<lambda>n. \<Sum>n = n * Suc (Suc 0)..<n * Suc (Suc 0) +  Suc (Suc 0). f (n + k)) 
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3663
             sums (\<Sum>n. f (n + k))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3664
  proof (rule sums_group)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3665
    show "(\<lambda>n. f (n + k)) sums (\<Sum>n. f (n + k))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3666
      by (simp add: f summable_iff_shift summable_sums)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3667
  qed auto
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3668
  with fplus have "0 < (\<Sum>n. f (n + k))"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3669
    apply (simp add: add.commute)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3670
    apply (metis (no_types, lifting) suminf_pos summable_def sums_unique)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3671
    done
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3672
  then show ?thesis
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3673
    by (simp add: f suminf_minus_initial_segment)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3674
qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3675
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3676
lemma cos_two_less_zero [simp]: "cos 2 < (0::real)"
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3677
proof -
63367
6c731c8b7f03 simplified definitions of combinatorial functions
haftmann
parents: 63365
diff changeset
  3678
  note fact_Suc [simp del]
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3679
  from sums_minus [OF cos_paired]
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3680
  have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)"
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3681
    by simp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
  3682
  then have sm: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3683
    by (rule sums_summable)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3684
  have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3685
    by (simp add: fact_num_eq_if power_eq_if)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3686
  moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n)))) <
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3687
    (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3688
  proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3689
    {
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3690
      fix d
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
  3691
      let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))"
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
  3692
      have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3693
        unfolding of_nat_mult by (rule mult_strict_mono) (simp_all add: fact_less_mono)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
  3694
      then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
63367
6c731c8b7f03 simplified definitions of combinatorial functions
haftmann
parents: 63365
diff changeset
  3695
        by (simp only: fact_Suc [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
  3696
      then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3697
        by (simp add: inverse_eq_divide less_divide_eq)
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
  3698
    }
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60155
diff changeset
  3699
    then show ?thesis
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3700
      by (force intro!: sum_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps)
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3701
  qed
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3702
  ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3703
    by (rule order_less_trans)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  3704
  moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3705
    by (rule sums_unique)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3706
  ultimately have "(0::real) < - cos 2" by simp
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3707
  then show ?thesis by simp
0ae3db699a3e tuned proofs
haftmann
parents: 53599
diff changeset
  3708
qed
23053
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3709
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3710
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3711
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3712
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3713
lemma cos_is_zero: "\<exists>!x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
44730
11a1290fd0ac convert lemma cos_is_zero to Isar-style
huffman
parents: 44728
diff changeset
  3714
proof (rule ex_ex1I)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3715
  show "\<exists>x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3716
    by (rule IVT2) simp_all
44730
11a1290fd0ac convert lemma cos_is_zero to Isar-style
huffman
parents: 44728
diff changeset
  3717
next
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3718
  fix a b :: real
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3719
  assume ab: "0 \<le> a \<and> a \<le> 2 \<and> cos a = 0" "0 \<le> b \<and> b \<le> 2 \<and> cos b = 0"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3720
  have cosd: "\<And>x::real. cos differentiable (at x)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 56167
diff changeset
  3721
    unfolding real_differentiable_def by (auto intro: DERIV_cos)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3722
  show "a = b"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3723
  proof (cases a b rule: linorder_cases)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3724
    case less
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3725
    then obtain z where "a < z" "z < b" "(cos has_real_derivative 0) (at z)"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3726
      using Rolle by (metis cosd continuous_on_cos_real ab)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3727
    then have "sin z = 0"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3728
      using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3729
    then show ?thesis
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3730
      by (metis \<open>a < z\<close> \<open>z < b\<close> ab order_less_le_trans less_le sin_gt_zero_02)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3731
  next
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3732
    case greater
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3733
    then obtain z where "b < z" "z < a" "(cos has_real_derivative 0) (at z)"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  3734
      using Rolle by (metis cosd continuous_on_cos_real ab)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3735
    then have "sin z = 0"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3736
      using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3737
    then show ?thesis
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3738
      by (metis \<open>b < z\<close> \<open>z < a\<close> ab order_less_le_trans less_le sin_gt_zero_02)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3739
  qed auto
44730
11a1290fd0ac convert lemma cos_is_zero to Isar-style
huffman
parents: 44728
diff changeset
  3740
qed
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31790
diff changeset
  3741
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3742
lemma pi_half: "pi/2 = (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3743
  by (simp add: pi_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3744
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3745
lemma cos_pi_half [simp]: "cos (pi/2) = 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3746
  by (simp add: pi_half cos_is_zero [THEN theI'])
23053
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3747
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3748
lemma cos_of_real_pi_half [simp]: "cos ((of_real pi/2) :: 'a) = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3749
  if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3750
  by (metis cos_pi_half cos_of_real eq_numeral_simps(4)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3751
      nonzero_of_real_divide of_real_0 of_real_numeral)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3752
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3753
lemma pi_half_gt_zero [simp]: "0 < pi/2"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3754
proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3755
  have "0 \<le> pi/2"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3756
    by (simp add: pi_half cos_is_zero [THEN theI'])
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3757
  then show ?thesis
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3758
    by (metis cos_pi_half cos_zero less_eq_real_def one_neq_zero)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3759
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3760
23053
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3761
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3762
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3763
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3764
lemma pi_half_less_two [simp]: "pi/2 < 2"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3765
proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3766
  have "pi/2 \<le> 2"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3767
    by (simp add: pi_half cos_is_zero [THEN theI'])
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3768
  then show ?thesis
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3769
    by (metis cos_pi_half cos_two_neq_zero le_less)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3770
qed
23053
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3771
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3772
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3773
lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3774
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3775
lemma pi_gt_zero [simp]: "0 < pi"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3776
  using pi_half_gt_zero by simp
23053
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3777
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3778
lemma pi_ge_zero [simp]: "0 \<le> pi"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3779
  by (rule pi_gt_zero [THEN order_less_imp_le])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3780
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3781
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3782
  by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3783
23053
03fe1dafa418 define pi with THE instead of SOME; cleaned up
huffman
parents: 23052
diff changeset
  3784
lemma pi_not_less_zero [simp]: "\<not> pi < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3785
  by (simp add: linorder_not_less)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3786
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  3787
lemma minus_pi_half_less_zero: "-(pi/2) < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3788
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3789
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3790
lemma m2pi_less_pi: "- (2*pi) < pi"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3791
  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
  3792
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3793
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3794
  using sin_cos_squared_add2 [where x = "pi/2"]
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3795
  using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3796
  by (simp add: power2_eq_1_iff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3797
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3798
lemma sin_of_real_pi_half [simp]: "sin ((of_real pi/2) :: 'a) = 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3799
  if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3800
  using sin_pi_half
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3801
  by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3802
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3803
lemma sin_cos_eq: "sin x = cos (of_real pi/2 - x)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3804
  for x :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3805
  by (simp add: cos_diff)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3806
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3807
lemma minus_sin_cos_eq: "- sin x = cos (x + of_real pi/2)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3808
  for x :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3809
  by (simp add: cos_add nonzero_of_real_divide)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3810
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3811
lemma cos_sin_eq: "cos x = sin (of_real pi/2 - x)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3812
  for x :: "'a::{real_normed_field,banach}"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3813
  using sin_cos_eq [of "of_real pi/2 - x"] by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3814
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3815
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3816
  for x :: "'a::{real_normed_field,banach}"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3817
  using cos_add [of "of_real pi/2 - x" "-y"]
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3818
  by (simp add: cos_sin_eq) (simp add: sin_cos_eq)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3819
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3820
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3821
  for x :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3822
  using sin_add [of x "- y"] by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3823
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3824
lemma sin_double: "sin(2 * x) = 2 * sin x * cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3825
  for x :: "'a::{real_normed_field,banach}"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3826
  using sin_add [where x=x and y=x] by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3827
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3828
lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3829
  using cos_add [where x = "pi/2" and y = "pi/2"]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3830
  by (simp add: cos_of_real)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3831
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3832
lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0"
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3833
  using sin_add [where x = "pi/2" and y = "pi/2"]
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3834
  by (simp add: sin_of_real)
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  3835
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3836
lemma cos_pi [simp]: "cos pi = -1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3837
  using cos_add [where x = "pi/2" and y = "pi/2"] by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3838
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3839
lemma sin_pi [simp]: "sin pi = 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3840
  using sin_add [where x = "pi/2" and y = "pi/2"] by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3841
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3842
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3843
  by (simp add: sin_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3844
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3845
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3846
  by (simp add: sin_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3847
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3848
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3849
  by (simp add: cos_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3850
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
  3851
lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x"
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
  3852
  by (simp add: cos_add)
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
  3853
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3854
lemma sin_periodic [simp]: "sin (x + 2 * pi) = sin x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3855
  by (simp add: sin_add sin_double cos_double)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3856
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3857
lemma cos_periodic [simp]: "cos (x + 2 * pi) = cos x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3858
  by (simp add: cos_add sin_double cos_double)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3859
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
  3860
lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
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
  3861
  by (induct n) (auto simp: distrib_right)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3862
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
  3863
lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  3864
  by (metis cos_npi mult.commute)
15383
c49e4225ef4f made proofs more robust
paulson
parents: 15251
diff changeset
  3865
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3866
lemma sin_npi [simp]: "sin (real n * pi) = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3867
  for n :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3868
  by (induct n) (auto simp: distrib_right)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3869
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3870
lemma sin_npi2 [simp]: "sin (pi * real n) = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3871
  for n :: nat
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  3872
  by (simp add: mult.commute [of pi])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3873
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3874
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3875
  by (simp add: cos_double)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3876
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3877
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3878
  by (simp add: sin_double)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3879
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3880
context
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3881
  fixes w :: "'a::{real_normed_field,banach}"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3882
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3883
begin
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3884
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3885
lemma sin_times_sin: "sin w * sin z = (cos (w - z) - cos (w + z)) / 2"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3886
  by (simp add: cos_diff cos_add)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3887
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3888
lemma sin_times_cos: "sin w * cos z = (sin (w + z) + sin (w - z)) / 2"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3889
  by (simp add: sin_diff sin_add)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3890
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3891
lemma cos_times_sin: "cos w * sin z = (sin (w + z) - sin (w - z)) / 2"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3892
  by (simp add: sin_diff sin_add)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3893
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3894
lemma cos_times_cos: "cos w * cos z = (cos (w - z) + cos (w + z)) / 2"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3895
  by (simp add: cos_diff cos_add)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3896
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3897
lemma cos_double_cos: "cos (2 * w) = 2 * cos w ^ 2 - 1"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3898
  by (simp add: cos_double sin_squared_eq)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3899
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3900
lemma cos_double_sin: "cos (2 * w) = 1 - 2 * sin w ^ 2"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3901
  by (simp add: cos_double sin_squared_eq)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3902
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3903
end
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  3904
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3905
lemma sin_plus_sin: "sin w + sin z = 2 * sin ((w + z) / 2) * cos ((w - z) / 2)"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3906
  for w :: "'a::{real_normed_field,banach}" 
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3907
  apply (simp add: mult.assoc sin_times_cos)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3908
  apply (simp add: field_simps)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3909
  done
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3910
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3911
lemma sin_diff_sin: "sin w - sin z = 2 * sin ((w - z) / 2) * cos ((w + z) / 2)"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  3912
  for w :: "'a::{real_normed_field,banach}"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3913
  apply (simp add: mult.assoc sin_times_cos)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3914
  apply (simp add: field_simps)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3915
  done
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3916
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3917
lemma cos_plus_cos: "cos w + cos z = 2 * cos ((w + z) / 2) * cos ((w - z) / 2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3918
  for w :: "'a::{real_normed_field,banach,field}"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3919
  apply (simp add: mult.assoc cos_times_cos)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3920
  apply (simp add: field_simps)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3921
  done
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3922
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3923
lemma cos_diff_cos: "cos w - cos z = 2 * sin ((w + z) / 2) * sin ((z - w) / 2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3924
  for w :: "'a::{real_normed_field,banach,field}"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3925
  apply (simp add: mult.assoc sin_times_sin)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3926
  apply (simp add: field_simps)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3927
  done
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3928
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3929
lemma sin_pi_minus [simp]: "sin (pi - x) = sin x"
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3930
  by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3931
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3932
lemma cos_pi_minus [simp]: "cos (pi - x) = - (cos x)"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3933
  by (metis cos_minus cos_periodic_pi uminus_add_conv_diff)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3934
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3935
lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)"
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3936
  by (simp add: sin_diff)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3937
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3938
lemma cos_minus_pi [simp]: "cos (x - pi) = - (cos x)"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3939
  by (simp add: cos_diff)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3940
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3941
lemma sin_2pi_minus [simp]: "sin (2 * pi - x) = - (sin x)"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3942
  by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59731
diff changeset
  3943
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3944
lemma cos_2pi_minus [simp]: "cos (2 * pi - x) = cos x"
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
  3945
  by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3946
      diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3947
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3948
lemma sin_gt_zero2: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < sin x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3949
  by (metis sin_gt_zero_02 order_less_trans pi_half_less_two)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3950
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  3951
lemma sin_less_zero:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3952
  assumes "- pi/2 < x" and "x < 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3953
  shows "sin x < 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3954
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3955
  have "0 < sin (- x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3956
    using assms by (simp only: sin_gt_zero2)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3957
  then show ?thesis by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3958
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3959
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3960
lemma pi_less_4: "pi < 4"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3961
  using pi_half_less_two by auto
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  3962
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3963
lemma cos_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3964
  by (simp add: cos_sin_eq sin_gt_zero2)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3965
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3966
lemma cos_gt_zero_pi: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3967
  using cos_gt_zero [of x] cos_gt_zero [of "-x"]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3968
  by (cases rule: linorder_cases [of x 0]) auto
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3969
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3970
lemma cos_ge_zero: "-(pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> 0 \<le> cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3971
  by (auto simp: order_le_less cos_gt_zero_pi)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3972
    (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3973
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3974
lemma sin_gt_zero: "0 < x \<Longrightarrow> x < pi \<Longrightarrow> 0 < sin x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  3975
  by (simp add: sin_cos_eq cos_gt_zero_pi)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  3976
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3977
lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x < 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3978
  using sin_gt_zero [of "x - pi"]
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3979
  by (simp add: sin_diff)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  3980
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  3981
lemma pi_ge_two: "2 \<le> pi"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  3982
proof (rule ccontr)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3983
  assume "\<not> ?thesis"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3984
  then have "pi < 2" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3985
  have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3986
  proof (cases "2 < 2 * pi")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3987
    case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3988
    with dense[OF \<open>pi < 2\<close>] show ?thesis 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
  3989
  next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3990
    case False
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3991
    have "pi < 2 * pi" 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
  3992
    from dense[OF this] and False 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
  3993
  qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3994
  then obtain y where "pi < y" and "y < 2" and "y < 2 * pi"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3995
    by blast
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3996
  then have "0 < sin y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3997
    using sin_gt_zero_02 by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3998
  moreover have "sin y < 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  3999
    using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2 * pi\<close> sin_periodic_pi[of "y - pi"]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4000
    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
  4001
  ultimately show False by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4002
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4003
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4004
lemma sin_ge_zero: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> sin x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4005
  by (auto simp: order_le_less sin_gt_zero)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4006
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4007
lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x \<le> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4008
  using sin_ge_zero [of "x - pi"] by (simp add: sin_diff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4009
62948
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62679
diff changeset
  4010
lemma sin_pi_divide_n_ge_0 [simp]:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4011
  assumes "n \<noteq> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4012
  shows "0 \<le> sin (pi / real n)"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  4013
  by (rule sin_ge_zero) (use assms in \<open>simp_all add: field_split_simps\<close>)
62948
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62679
diff changeset
  4014
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62679
diff changeset
  4015
lemma sin_pi_divide_n_gt_0:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4016
  assumes "2 \<le> n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4017
  shows "0 < sin (pi / real n)"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  4018
  by (rule sin_gt_zero) (use assms in \<open>simp_all add: field_split_simps\<close>)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4019
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  4020
text\<open>Proof resembles that of \<open>cos_is_zero\<close> but with \<^term>\<open>pi\<close> for the upper bound\<close>
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4021
lemma cos_total:
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4022
  assumes y: "-1 \<le> y" "y \<le> 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4023
  shows "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
44745
b068207a7400 convert lemma cos_total to Isar-style proof
huffman
parents: 44730
diff changeset
  4024
proof (rule ex_ex1I)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4025
  show "\<exists>x::real. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4026
    by (rule IVT2) (simp_all add: y)
44745
b068207a7400 convert lemma cos_total to Isar-style proof
huffman
parents: 44730
diff changeset
  4027
next
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4028
  fix a b :: real
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4029
  assume ab: "0 \<le> a \<and> a \<le> pi \<and> cos a = y" "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4030
  have cosd: "\<And>x::real. cos differentiable (at x)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 56167
diff changeset
  4031
    unfolding real_differentiable_def by (auto intro: DERIV_cos)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4032
  show "a = b"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4033
  proof (cases a b rule: linorder_cases)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4034
    case less
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4035
    then obtain z where "a < z" "z < b" "(cos has_real_derivative 0) (at z)"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4036
      using Rolle by (metis cosd continuous_on_cos_real ab)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4037
    then have "sin z = 0"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4038
      using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4039
    then show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4040
      by (metis \<open>a < z\<close> \<open>z < b\<close> ab order_less_le_trans less_le sin_gt_zero)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4041
  next
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4042
    case greater
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4043
    then obtain z where "b < z" "z < a" "(cos has_real_derivative 0) (at z)"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4044
      using Rolle by (metis cosd continuous_on_cos_real ab)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4045
    then have "sin z = 0"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4046
      using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4047
    then show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4048
      by (metis \<open>b < z\<close> \<open>z < a\<close> ab order_less_le_trans less_le sin_gt_zero)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4049
  qed auto
44745
b068207a7400 convert lemma cos_total to Isar-style proof
huffman
parents: 44730
diff changeset
  4050
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4051
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4052
lemma sin_total:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4053
  assumes y: "-1 \<le> y" "y \<le> 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4054
  shows "\<exists>!x. - (pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4055
proof -
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4056
  from cos_total [OF y]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4057
  obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4058
    and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4059
    by blast
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4060
  show ?thesis
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4061
    unfolding sin_cos_eq
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4062
  proof (rule ex1I [where a="pi/2 - x"])
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4063
    show "- (pi/2) \<le> z \<and> z \<le> pi/2 \<and> cos (of_real pi/2 - z) = y \<Longrightarrow>
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4064
          z = pi/2 - x" for z
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4065
      using uniq [of "pi/2 -z"] by auto
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4066
  qed (use x in auto)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4067
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4068
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  4069
lemma cos_zero_lemma:
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4070
  assumes "0 \<le> x" "cos x = 0"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4071
  shows "\<exists>n. odd n \<and> x = of_nat n * (pi/2)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4072
proof -
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4073
  have xle: "x < (1 + real_of_int \<lfloor>x/pi\<rfloor>) * pi"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4074
    using floor_correct [of "x/pi"]
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4075
    by (simp add: add.commute divide_less_eq)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4076
  obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4077
  proof 
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4078
    show "real (nat \<lfloor>x / pi\<rfloor>) * pi \<le> x"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4079
      using assms floor_divide_lower [of pi x] by auto
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4080
    show "x < real (Suc (nat \<lfloor>x / pi\<rfloor>)) * pi"
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4081
      using assms floor_divide_upper [of pi x]  by (simp add: xle)
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  4082
  qed
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4083
  then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4084
    by (auto simp: algebra_simps cos_diff assms)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4085
  then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4086
    by (auto simp: intro!: cos_total)
62679
092cb9c96c99 add le_log_of_power and le_log2_of_power by Tobias Nipkow
hoelzl
parents: 62393
diff changeset
  4087
  then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4088
    and uniq: "\<And>\<phi>. 0 \<le> \<phi> \<Longrightarrow> \<phi> \<le> pi \<Longrightarrow> cos \<phi> = 0 \<Longrightarrow> \<phi> = \<theta>"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4089
    by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4090
  then have "x - real n * pi = \<theta>"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4091
    using x by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4092
  moreover have "pi/2 = \<theta>"
62679
092cb9c96c99 add le_log_of_power and le_log2_of_power by Tobias Nipkow
hoelzl
parents: 62393
diff changeset
  4093
    using pi_half_ge_zero uniq by fastforce
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4094
  ultimately show ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4095
    by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4096
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4097
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4098
lemma sin_zero_lemma:
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4099
  assumes "0 \<le> x" "sin x = 0"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4100
  shows "\<exists>n::nat. even n \<and> x = real n * (pi/2)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4101
proof -
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4102
  obtain n where "odd n" and n: "x + pi/2 = of_nat n * (pi/2)" "n > 0"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4103
    using cos_zero_lemma [of "x + pi/2"] assms by (auto simp add: cos_add)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4104
  then have "x = real (n - 1) * (pi / 2)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4105
    by (simp add: algebra_simps of_nat_diff)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4106
  then show ?thesis
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4107
    by (simp add: \<open>odd n\<close>)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4108
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4109
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  4110
lemma cos_zero_iff:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4111
  "cos x = 0 \<longleftrightarrow> ((\<exists>n. odd n \<and> x = real n * (pi/2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi/2))))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4112
  (is "?lhs = ?rhs")
58709
efdc6c533bd3 prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents: 58656
diff changeset
  4113
proof -
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4114
  have *: "cos (real n * pi/2) = 0" if "odd n" for n :: nat
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4115
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4116
    from that obtain m where "n = 2 * m + 1" ..
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4117
    then show ?thesis
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4118
      by (simp add: field_simps) (simp add: cos_add add_divide_distrib)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4119
  qed
58709
efdc6c533bd3 prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents: 58656
diff changeset
  4120
  show ?thesis
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4121
  proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4122
    show ?rhs if ?lhs
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4123
      using that cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4124
    show ?lhs if ?rhs
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4125
      using that by (auto dest: * simp del: eq_divide_eq_numeral1)
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4126
  qed
58709
efdc6c533bd3 prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents: 58656
diff changeset
  4127
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4128
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  4129
lemma sin_zero_iff:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4130
  "sin x = 0 \<longleftrightarrow> ((\<exists>n. even n \<and> x = real n * (pi/2)) \<or> (\<exists>n. even n \<and> x = - (real n * (pi/2))))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4131
  (is "?lhs = ?rhs")
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4132
proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4133
  show ?rhs if ?lhs
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4134
    using that sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4135
  show ?lhs if ?rhs
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4136
    using that by (auto elim: evenE)
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  4137
qed
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4138
70532
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4139
lemma sin_zero_pi_iff:
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4140
  fixes x::real
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4141
  assumes "\<bar>x\<bar> < pi"
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4142
  shows "sin x = 0 \<longleftrightarrow> x = 0"
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4143
proof
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4144
  show "x = 0" if "sin x = 0"
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4145
    using that assms by (auto simp: sin_zero_iff)
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4146
qed auto
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  4147
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4148
lemma cos_zero_iff_int: "cos x = 0 \<longleftrightarrow> (\<exists>i. odd i \<and> x = of_int i * (pi/2))"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4149
proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4150
  have 1: "\<And>n. odd n \<Longrightarrow> \<exists>i. odd i \<and> real n = real_of_int i"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4151
    by (metis even_of_nat of_int_of_nat_eq)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4152
  have 2: "\<And>n. odd n \<Longrightarrow> \<exists>i. odd i \<and> - (real n * pi) = real_of_int i * pi"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4153
    by (metis even_minus even_of_nat mult.commute mult_minus_right of_int_minus of_int_of_nat_eq)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4154
  have 3: "\<lbrakk>odd i;  \<forall>n. even n \<or> real_of_int i \<noteq> - (real n)\<rbrakk>
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4155
         \<Longrightarrow> \<exists>n. odd n \<and> real_of_int i = real n" for i
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4156
    by (cases i rule: int_cases2) auto
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4157
  show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4158
    by (force simp: cos_zero_iff intro!: 1 2 3)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4159
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4160
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4161
lemma sin_zero_iff_int: "sin x = 0 \<longleftrightarrow> (\<exists>i. even i \<and> x = of_int i * (pi/2))" (is "?lhs = ?rhs")
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4162
proof safe
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4163
  assume ?lhs
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4164
  then consider (plus) n where "even n" "x = real n * (pi/2)" | (minus) n where "even n"  "x = - (real n * (pi/2))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4165
    using sin_zero_iff by auto
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4166
  then show "\<exists>n. even n \<and> x = of_int n * (pi/2)"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4167
  proof cases
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4168
    case plus
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4169
    then show ?rhs
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4170
      by (metis even_of_nat of_int_of_nat_eq)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4171
  next
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4172
    case minus
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4173
    then show ?thesis
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4174
      by (rule_tac x="- (int n)" in exI) simp
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4175
  qed
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4176
next
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4177
  fix i :: int
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4178
  assume "even i"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4179
  then show "sin (of_int i * (pi/2)) = 0"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4180
    by (cases i rule: int_cases2, simp_all add: sin_zero_iff)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4181
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4182
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4183
lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>i::int. x = of_int i * pi)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4184
proof -
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4185
  have "sin x = 0 \<longleftrightarrow> (\<exists>i. even i \<and> x = real_of_int i * (pi / 2))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4186
    by (auto simp: sin_zero_iff_int)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4187
  also have "... = (\<exists>j. x = real_of_int (2*j) * (pi / 2))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4188
    using dvd_triv_left by blast
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4189
  also have "... = (\<exists>i::int. x = of_int i * pi)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4190
    by auto
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4191
  finally show ?thesis .
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4192
qed
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4193
65036
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4194
lemma sin_npi_int [simp]: "sin (pi * of_int n) = 0"
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4195
  by (simp add: sin_zero_iff_int2)
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4196
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4197
lemma cos_monotone_0_pi:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4198
  assumes "0 \<le> y" and "y < x" and "x \<le> pi"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4199
  shows "cos x < cos y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4200
proof -
33549
39f2855ce41b tuned proofs;
wenzelm
parents: 32960
diff changeset
  4201
  have "- (x - y) < 0" using assms by auto
68635
8094b853a92f fixes and more de-applying
paulson <lp15@cam.ac.uk>
parents: 68634
diff changeset
  4202
  from MVT2[OF \<open>y < x\<close> DERIV_cos]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4203
  obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4204
    by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4205
  then have "0 < z" and "z < pi"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4206
    using assms by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4207
  then have "0 < sin z"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4208
    using sin_gt_zero by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4209
  then have "cos x - cos y < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4210
    unfolding cos_diff minus_mult_commute[symmetric]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4211
    using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4212
  then show ?thesis 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
  4213
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4214
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4215
lemma cos_monotone_0_pi_le:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4216
  assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4217
  shows "cos x \<le> cos y"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4218
proof (cases "y < x")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4219
  case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4220
  show ?thesis
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4221
    using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<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
  4222
next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4223
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4224
  then have "y = x" using \<open>y \<le> x\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4225
  then show ?thesis 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
  4226
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4227
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4228
lemma cos_monotone_minus_pi_0:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4229
  assumes "- pi \<le> y" and "y < x" and "x \<le> 0"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4230
  shows "cos y < cos x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4231
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4232
  have "0 \<le> - x" and "- x < - y" and "- y \<le> pi"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4233
    using assms by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4234
  from cos_monotone_0_pi[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4235
    unfolding cos_minus .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4236
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4237
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4238
lemma cos_monotone_minus_pi_0':
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4239
  assumes "- pi \<le> y" and "y \<le> x" and "x \<le> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4240
  shows "cos y \<le> cos x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4241
proof (cases "y < x")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4242
  case True
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4243
  show ?thesis using cos_monotone_minus_pi_0[OF \<open>-pi \<le> y\<close> True \<open>x \<le> 0\<close>]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4244
    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
  4245
next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4246
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4247
  then have "y = x" using \<open>y \<le> x\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4248
  then show ?thesis 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
  4249
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4250
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4251
lemma sin_monotone_2pi:
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4252
  assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4253
  shows "sin y < sin x"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4254
  unfolding sin_cos_eq
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4255
  using assms by (auto intro: cos_monotone_0_pi)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4256
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4257
lemma sin_monotone_2pi_le:
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4258
  assumes "- (pi/2) \<le> y" and "y \<le> x" and "x \<le> pi/2"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4259
  shows "sin y \<le> sin x"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4260
  by (metis assms le_less sin_monotone_2pi)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4261
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4262
lemma sin_x_le_x:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4263
  fixes x :: real
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4264
  assumes "x \<ge> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4265
  shows "sin x \<le> x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4266
proof -
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4267
  let ?f = "\<lambda>x. x - sin x"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4268
  have "\<And>u. \<lbrakk>0 \<le> u; u \<le> x\<rbrakk> \<Longrightarrow> \<exists>y. (?f has_real_derivative 1 - cos u) (at u)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4269
    by (auto intro!: derivative_eq_intros simp: field_simps)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4270
  then have "?f x \<ge> ?f 0"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4271
    by (metis cos_le_one diff_ge_0_iff_ge DERIV_nonneg_imp_nondecreasing [OF assms])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4272
  then show "sin x \<le> x" 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
  4273
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4274
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4275
lemma sin_x_ge_neg_x:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4276
  fixes x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4277
  assumes x: "x \<ge> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4278
  shows "sin x \<ge> - x"
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4279
proof -
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4280
  let ?f = "\<lambda>x. x + sin x"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4281
  have \<section>: "\<And>u. \<lbrakk>0 \<le> u; u \<le> x\<rbrakk> \<Longrightarrow> \<exists>y. (?f has_real_derivative 1 + cos u) (at u)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4282
    by (auto intro!: derivative_eq_intros simp: field_simps)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4283
  have "?f x \<ge> ?f 0"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4284
    by (rule DERIV_nonneg_imp_nondecreasing [OF assms]) (use \<section> real_0_le_add_iff in force)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4285
  then show "sin x \<ge> -x" by simp
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4286
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4287
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4288
lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4289
  for x :: real
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4290
  using sin_x_ge_neg_x [of x] sin_x_le_x [of x] sin_x_ge_neg_x [of "-x"] sin_x_le_x [of "-x"]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4291
  by (auto simp: abs_real_def)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4292
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4293
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4294
subsection \<open>More Corollaries about Sine and Cosine\<close>
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4295
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4296
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi/2) = (-1) ^ n"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4297
proof -
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4298
  have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4299
    by (auto simp: algebra_simps sin_add)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4300
  then show ?thesis
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
  4301
    by (simp add: distrib_right add_divide_distrib add.commute mult.commute [of pi])
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4302
qed
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4303
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4304
lemma cos_2npi [simp]: "cos (2 * real n * pi) = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4305
  for n :: nat
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4306
  by (cases "even n") (simp_all add: cos_double mult.assoc)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4307
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4308
lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4309
proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4310
  have "cos (3/2*pi) = cos (pi + pi/2)"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4311
    by simp
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4312
  also have "... = 0"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4313
    by (subst cos_add, simp)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4314
  finally show ?thesis .
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4315
qed
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4316
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4317
lemma sin_2npi [simp]: "sin (2 * real n * pi) = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4318
  for n :: nat
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4319
  by (auto simp: mult.assoc sin_double)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4320
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4321
lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4322
proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4323
  have "sin (3/2*pi) = sin (pi + pi/2)"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4324
    by simp
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4325
  also have "... = -1"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4326
    by (subst sin_add, simp)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4327
  finally show ?thesis .
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4328
qed
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4329
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4330
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4331
  by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4332
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4333
lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4334
  by (auto intro!: derivative_eq_intros)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4335
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4336
lemma sin_zero_norm_cos_one:
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4337
  fixes x :: "'a::{real_normed_field,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4338
  assumes "sin x = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4339
  shows "norm (cos x) = 1"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4340
  using sin_cos_squared_add [of x, unfolded assms]
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4341
  by (simp add: square_norm_one)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4342
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4343
lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4344
  using sin_zero_norm_cos_one by fastforce
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4345
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4346
lemma cos_one_sin_zero:
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4347
  fixes x :: "'a::{real_normed_field,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4348
  assumes "cos x = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4349
  shows "sin x = 0"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4350
  using sin_cos_squared_add [of x, unfolded assms]
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4351
  by simp
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4352
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4353
lemma sin_times_pi_eq_0: "sin (x * pi) = 0 \<longleftrightarrow> x \<in> \<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
  4354
  by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int)
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
  4355
67091
1393c2340eec more symbols;
wenzelm
parents: 66827
diff changeset
  4356
lemma cos_one_2pi: "cos x = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2 * pi) \<or> (\<exists>n::nat. x = - (n * 2 * pi))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4357
  (is "?lhs = ?rhs")
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4358
proof
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4359
  assume ?lhs
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4360
  then have "sin x = 0"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4361
    by (simp add: cos_one_sin_zero)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4362
  then show ?rhs
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4363
  proof (simp only: sin_zero_iff, elim exE disjE conjE)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4364
    fix n :: nat
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4365
    assume n: "even n" "x = real n * (pi/2)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4366
    then obtain m where m: "n = 2 * m"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4367
      using dvdE by blast
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4368
    then have me: "even m" using \<open>?lhs\<close> n
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4369
      by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4370
    show ?rhs
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4371
      using m me n
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4372
      by (auto simp: field_simps elim!: evenE)
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
  4373
  next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4374
    fix n :: nat
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4375
    assume n: "even n" "x = - (real n * (pi/2))"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4376
    then obtain m where m: "n = 2 * m"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4377
      using dvdE by blast
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4378
    then have me: "even m" using \<open>?lhs\<close> n
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4379
      by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4380
    show ?rhs
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4381
      using m me n
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4382
      by (auto simp: field_simps elim!: evenE)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4383
  qed
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4384
next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4385
  assume ?rhs
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4386
  then show "cos x = 1"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4387
    by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4388
qed
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4389
65036
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4390
lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)" (is "?lhs = ?rhs")
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4391
proof
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4392
  assume "cos x = 1"
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4393
  then show ?rhs
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4394
    by (metis cos_one_2pi mult.commute mult_minus_right of_int_minus of_int_of_nat_eq)
65036
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4395
next
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4396
  assume ?rhs
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4397
  then show "cos x = 1"
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4398
    by (clarsimp simp add: cos_one_2pi) (metis mult_minus_right of_int_of_nat)
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4399
qed
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4400
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4401
lemma cos_npi_int [simp]:
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4402
  fixes n::int shows "cos (pi * of_int n) = (if even n then 1 else -1)"
ab7e11730ad8 Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents: 64758
diff changeset
  4403
    by (auto simp: algebra_simps cos_one_2pi_int elim!: oddE evenE)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4404
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4405
lemma sin_cos_sqrt: "0 \<le> sin x \<Longrightarrow> sin x = sqrt (1 - (cos(x) ^ 2))"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4406
  using sin_squared_eq real_sqrt_unique by fastforce
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4407
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4408
lemma sin_eq_0_pi: "- pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin x = 0 \<Longrightarrow> x = 0"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4409
  by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4410
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4411
lemma cos_treble_cos: "cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4412
  for x :: "'a::{real_normed_field,banach}"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4413
proof -
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4414
  have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4415
    by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4416
  have "cos(3 * x) = cos(2*x + x)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4417
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4418
  also have "\<dots> = 4 * cos x ^ 3 - 3 * cos x"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4419
    unfolding cos_add cos_double sin_double
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4420
    by (simp add: * field_simps power2_eq_square power3_eq_cube)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4421
  finally show ?thesis .
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4422
qed
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4423
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4424
lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4425
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4426
  let ?c = "cos (pi / 4)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4427
  let ?s = "sin (pi / 4)"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4428
  have nonneg: "0 \<le> ?c"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4429
    by (simp add: cos_ge_zero)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4430
  have "0 = cos (pi / 4 + pi / 4)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4431
    by simp
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4432
  also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4433
    by (simp only: cos_add power2_eq_square)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4434
  also have "\<dots> = 2 * ?c\<^sup>2 - 1"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4435
    by (simp add: sin_squared_eq)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4436
  finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4437
    by (simp add: power_divide)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4438
  then show ?thesis
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4439
    using nonneg by (rule power2_eq_imp_eq) simp
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4440
qed
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4441
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4442
lemma cos_30: "cos (pi / 6) = sqrt 3/2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4443
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4444
  let ?c = "cos (pi / 6)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4445
  let ?s = "sin (pi / 6)"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4446
  have pos_c: "0 < ?c"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4447
    by (rule cos_gt_zero) simp_all
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4448
  have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4449
    by simp
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4450
  also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4451
    by (simp only: cos_add sin_add)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4452
  also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4453
    by (simp add: algebra_simps power2_eq_square)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4454
  finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4455
    using pos_c by (simp add: sin_squared_eq power_divide)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4456
  then show ?thesis
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4457
    using pos_c [THEN order_less_imp_le]
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4458
    by (rule power2_eq_imp_eq) simp
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4459
qed
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4460
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4461
lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4462
  by (simp add: sin_cos_eq cos_45)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4463
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4464
lemma sin_60: "sin (pi / 3) = sqrt 3/2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4465
  by (simp add: sin_cos_eq cos_30)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4466
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4467
lemma cos_60: "cos (pi / 3) = 1 / 2"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4468
proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4469
  have "0 \<le> cos (pi / 3)"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4470
    by (rule cos_ge_zero) (use pi_half_ge_zero in \<open>linarith+\<close>)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4471
  then show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4472
    by (simp add: cos_squared_eq sin_60 power_divide power2_eq_imp_eq)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4473
qed
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4474
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4475
lemma sin_30: "sin (pi / 6) = 1 / 2"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4476
  by (simp add: sin_cos_eq cos_60)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4477
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4478
lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2 * pi * n) = 1"
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
  4479
  by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4480
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4481
lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4482
  by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4483
68499
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  4484
lemma cos_int_2pin [simp]: "cos ((2 * pi) * of_int n) = 1"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4485
  by (simp add: cos_one_2pi_int)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4486
68499
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  4487
lemma sin_int_2pin [simp]: "sin ((2 * pi) * of_int n) = 0"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  4488
  by (metis Ints_of_int sin_integer_2pi)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4489
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4490
lemma sincos_principal_value: "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> (sin y = sin x \<and> cos y = cos x)"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4491
proof -
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4492
  define y where "y \<equiv> pi - (2 * pi) * frac ((pi - x) / (2 * pi))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4493
  have "-pi < y"" y \<le> pi"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4494
    by (auto simp: field_simps frac_lt_1 y_def)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4495
  moreover
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4496
  have "sin y = sin x" "cos y = cos x"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4497
    unfolding y_def
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4498
     apply (simp_all add: frac_def divide_simps sin_add cos_add)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4499
    by (metis sin_int_2pin cos_int_2pin diff_zero add.right_neutral mult.commute mult.left_neutral mult_zero_left)+
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4500
  ultimately
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4501
  show ?thesis by metis
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  4502
qed
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4503
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4504
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4505
subsection \<open>Tangent\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4506
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4507
definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4508
  where "tan = (\<lambda>x. sin x / cos x)"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  4509
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4510
lemma tan_of_real: "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})"
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  4511
  by (simp add: tan_def sin_of_real cos_of_real)
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  4512
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4513
lemma tan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4514
  for z :: "'a::{real_normed_field,banach}"
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  4515
  by (simp add: tan_def)
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59751
diff changeset
  4516
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4517
lemma tan_zero [simp]: "tan 0 = 0"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4518
  by (simp add: tan_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4519
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4520
lemma tan_pi [simp]: "tan pi = 0"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4521
  by (simp add: tan_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4522
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4523
lemma tan_npi [simp]: "tan (real n * pi) = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4524
  for n :: nat
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4525
  by (simp add: tan_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4526
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4527
lemma tan_minus [simp]: "tan (- x) = - tan x"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4528
  by (simp add: tan_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4529
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4530
lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4531
  by (simp add: tan_def)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4532
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4533
lemma lemma_tan_add1: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4534
  by (simp add: tan_def cos_add field_simps)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4535
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4536
lemma add_tan_eq: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4537
  for x :: "'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4538
  by (simp add: tan_def sin_add field_simps)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4539
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  4540
lemma tan_add:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4541
  "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x + y) \<noteq> 0 \<Longrightarrow> tan (x + y) = (tan x + tan y)/(1 - tan x * tan y)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4542
  for x :: "'a::{real_normed_field,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4543
  by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4544
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4545
lemma tan_double: "cos x \<noteq> 0 \<Longrightarrow> cos (2 * x) \<noteq> 0 \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4546
  for x :: "'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4547
  using tan_add [of x x] by (simp add: power2_eq_square)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4548
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4549
lemma tan_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < tan x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4550
  by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  4551
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  4552
lemma tan_less_zero:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4553
  assumes "- pi/2 < x" and "x < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4554
  shows "tan x < 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4555
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4556
  have "0 < tan (- x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4557
    using assms by (simp only: tan_gt_zero)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4558
  then show ?thesis by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4559
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4560
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4561
lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4562
  for x :: "'a::{real_normed_field,banach,field}"
44756
efcd71fbaeec simplify proof of tan_half, removing unused assumptions
huffman
parents: 44755
diff changeset
  4563
  unfolding tan_def sin_double cos_double sin_squared_eq
efcd71fbaeec simplify proof of tan_half, removing unused assumptions
huffman
parents: 44755
diff changeset
  4564
  by (simp add: power2_eq_square)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4565
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4566
lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4567
  unfolding tan_def by (simp add: sin_30 cos_30)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4568
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4569
lemma tan_45: "tan (pi / 4) = 1"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4570
  unfolding tan_def by (simp add: sin_45 cos_45)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4571
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4572
lemma tan_60: "tan (pi / 3) = sqrt 3"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4573
  unfolding tan_def by (simp add: sin_60 cos_60)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4574
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4575
lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4576
  for x :: "'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4577
  unfolding tan_def
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  4578
  by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4579
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4580
declare DERIV_tan[THEN DERIV_chain2, derivative_intros]
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4581
  and DERIV_tan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4582
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  4583
lemmas has_derivative_tan[derivative_intros] = DERIV_tan[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  4584
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4585
lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4586
  for x :: "'a::{real_normed_field,banach}"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4587
  by (rule DERIV_tan [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4588
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4589
lemma isCont_tan' [simp,continuous_intros]:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4590
  fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4591
  shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4592
  by (rule isCont_o2 [OF _ isCont_tan])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4593
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4594
lemma tendsto_tan [tendsto_intros]:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4595
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4596
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cos a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4597
  by (rule isCont_tendsto_compose [OF isCont_tan])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  4598
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  4599
lemma continuous_tan:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4600
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4601
  shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  4602
  unfolding continuous_def by (rule tendsto_tan)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  4603
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4604
lemma continuous_on_tan [continuous_intros]:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4605
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4606
  shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4607
  unfolding continuous_on_def by (auto intro: tendsto_tan)
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  4608
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  4609
lemma continuous_within_tan [continuous_intros]:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4610
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4611
  shows "continuous (at x within s) f \<Longrightarrow>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4612
    cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  4613
  unfolding continuous_within by (rule tendsto_tan)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  4614
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  4615
lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) \<midarrow>pi/2\<rightarrow> 0"
70365
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70350
diff changeset
  4616
  by (rule tendsto_cong_limit, (rule tendsto_intros)+, simp_all)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4617
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4618
lemma lemma_tan_total: 
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4619
  assumes "0 < y" shows "\<exists>x. 0 < x \<and> x < pi/2 \<and> y < tan x"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4620
proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4621
  obtain s where "0 < s" 
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4622
    and s: "\<And>x. \<lbrakk>x \<noteq> pi/2; norm (x - pi/2) < s\<rbrakk> \<Longrightarrow> norm (cos x / sin x - 0) < inverse y"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4623
    using LIM_D [OF LIM_cos_div_sin, of "inverse y"] that assms by force
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4624
  obtain e where e: "0 < e" "e < s" "e < pi/2"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4625
    using \<open>0 < s\<close> field_lbound_gt_zero pi_half_gt_zero by blast
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4626
  show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4627
  proof (intro exI conjI)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4628
    have "0 < sin e" "0 < cos e"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4629
      using e by (auto intro: cos_gt_zero sin_gt_zero2 simp: mult.commute)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4630
    then 
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4631
    show "y < tan (pi/2 - e)"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4632
      using s [of "pi/2 - e"] e assms
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4633
      by (simp add: tan_def sin_diff cos_diff) (simp add: field_simps split: if_split_asm)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4634
  qed (use e in auto)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4635
qed
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4636
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4637
lemma tan_total_pos: 
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4638
  assumes "0 \<le> y" shows "\<exists>x. 0 \<le> x \<and> x < pi/2 \<and> tan x = y"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4639
proof (cases "y = 0")
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4640
  case True
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4641
  then show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4642
    using pi_half_gt_zero tan_zero by blast
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4643
next
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4644
  case False
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4645
  with assms have "y > 0"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4646
    by linarith
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4647
  obtain x where x: "0 < x" "x < pi/2" "y < tan x"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4648
    using lemma_tan_total \<open>0 < y\<close> by blast
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4649
  have "\<exists>u\<ge>0. u \<le> x \<and> tan u = y"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4650
  proof (intro IVT allI impI)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4651
    show "isCont tan u" if "0 \<le> u \<and> u \<le> x" for u
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4652
    proof -
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4653
      have "cos u \<noteq> 0"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4654
        using antisym_conv2 cos_gt_zero that x(2) by fastforce
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4655
      with assms show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4656
        by (auto intro!: DERIV_tan [THEN DERIV_isCont])
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4657
    qed
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4658
  qed (use assms x in auto)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4659
  then show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4660
    using x(2) by auto
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4661
qed
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4662
    
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4663
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4664
proof (cases "0::real" y rule: le_cases)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4665
  case le
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4666
  then show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4667
    by (meson less_le_trans minus_pi_half_less_zero tan_total_pos)
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4668
next
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4669
  case ge
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4670
  with tan_total_pos [of "-y"] obtain x where "0 \<le> x" "x < pi / 2" "tan x = - y"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4671
    by force
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4672
  then show ?thesis
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4673
    by (rule_tac x="-x" in exI) auto
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4674
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4675
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4676
proposition tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4677
proof -
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4678
  have "u = v" if u: "- (pi / 2) < u" "u < pi / 2" and v: "- (pi / 2) < v" "v < pi / 2"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4679
    and eq: "tan u = tan v" for u v
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4680
  proof (cases u v rule: linorder_cases)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4681
    case less
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4682
    have "\<And>x. u \<le> x \<and> x \<le> v \<longrightarrow> isCont tan x"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4683
      by (metis cos_gt_zero_pi isCont_tan le_less_trans less_irrefl less_le_trans u(1) v(2))
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4684
    then have "continuous_on {u..v} tan"
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4685
      by (simp add: continuous_at_imp_continuous_on)
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4686
    moreover have "\<And>x. u < x \<and> x < v \<Longrightarrow> tan differentiable (at x)"
69022
e2858770997a removal of more redundancies, and fixes
paulson <lp15@cam.ac.uk>
parents: 69020
diff changeset
  4687
      by (metis DERIV_tan cos_gt_zero_pi real_differentiable_def less_numeral_extra(3) order.strict_trans u(1) v(2))
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4688
    ultimately obtain z where "u < z" "z < v" "DERIV tan z :> 0"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4689
      by (metis less Rolle eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4690
    moreover have "cos z \<noteq> 0"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4691
      by (metis (no_types) \<open>u < z\<close> \<open>z < v\<close> cos_gt_zero_pi less_le_trans linorder_not_less not_less_iff_gr_or_eq u(1) v(2))
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4692
    ultimately show ?thesis
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4693
      using DERIV_unique [OF _ DERIV_tan] by fastforce
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4694
  next
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4695
    case greater
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4696
    have "\<And>x. v \<le> x \<and> x \<le> u \<Longrightarrow> isCont tan x"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4697
      by (metis cos_gt_zero_pi isCont_tan le_less_trans less_irrefl less_le_trans u(2) v(1))
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4698
    then have "continuous_on {v..u} tan"
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4699
      by (simp add: continuous_at_imp_continuous_on)
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4700
    moreover have "\<And>x. v < x \<and> x < u \<Longrightarrow> tan differentiable (at x)"
69022
e2858770997a removal of more redundancies, and fixes
paulson <lp15@cam.ac.uk>
parents: 69020
diff changeset
  4701
      by (metis DERIV_tan cos_gt_zero_pi real_differentiable_def less_numeral_extra(3) order.strict_trans u(2) v(1))
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4702
    ultimately obtain z where "v < z" "z < u" "DERIV tan z :> 0"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4703
      by (metis greater Rolle eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4704
    moreover have "cos z \<noteq> 0"
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  4705
      by (metis \<open>v < z\<close> \<open>z < u\<close> cos_gt_zero_pi less_eq_real_def less_le_trans order_less_irrefl u(2) v(1))
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4706
    ultimately show ?thesis
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4707
      using DERIV_unique [OF _ DERIV_tan] by fastforce
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4708
  qed auto
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4709
  then have "\<exists>!x. - (pi / 2) < x \<and> x < pi / 2 \<and> tan x = y" 
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4710
    if x: "- (pi / 2) < x" "x < pi / 2" "tan x = y" for x
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4711
    using that by auto
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4712
  then show ?thesis
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4713
    using lemma_tan_total1 [where y = y]
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4714
    by auto
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4715
qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4716
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4717
lemma tan_monotone:
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4718
  assumes "- (pi/2) < y" and "y < x" and "x < pi/2"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4719
  shows "tan y < tan x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4720
proof -
68635
8094b853a92f fixes and more de-applying
paulson <lp15@cam.ac.uk>
parents: 68634
diff changeset
  4721
  have "DERIV tan x' :> inverse ((cos x')\<^sup>2)" if "y \<le> x'" "x' \<le> x" for x'
8094b853a92f fixes and more de-applying
paulson <lp15@cam.ac.uk>
parents: 68634
diff changeset
  4722
  proof -
8094b853a92f fixes and more de-applying
paulson <lp15@cam.ac.uk>
parents: 68634
diff changeset
  4723
    have "-(pi/2) < x'" and "x' < pi/2"
8094b853a92f fixes and more de-applying
paulson <lp15@cam.ac.uk>
parents: 68634
diff changeset
  4724
      using that assms by auto
8094b853a92f fixes and more de-applying
paulson <lp15@cam.ac.uk>
parents: 68634
diff changeset
  4725
    with cos_gt_zero_pi have "cos x' \<noteq> 0" by force
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4726
    then show "DERIV tan x' :> inverse ((cos x')\<^sup>2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4727
      by (rule DERIV_tan)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4728
  qed
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4729
  from MVT2[OF \<open>y < x\<close> this]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4730
  obtain z where "y < z" and "z < x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4731
    and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4732
  then have "- (pi/2) < z" and "z < pi/2"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4733
    using assms by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4734
  then have "0 < cos z"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4735
    using cos_gt_zero_pi by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4736
  then have inv_pos: "0 < inverse ((cos z)\<^sup>2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4737
    by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4738
  have "0 < x - y" using \<open>y < x\<close> by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4739
  with inv_pos have "0 < tan x - tan y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4740
    unfolding tan_diff by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4741
  then show ?thesis 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
  4742
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4743
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4744
lemma tan_monotone':
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4745
  assumes "- (pi/2) < y"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4746
    and "y < pi/2"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4747
    and "- (pi/2) < x"
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4748
    and "x < pi/2"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4749
  shows "y < x \<longleftrightarrow> tan y < tan x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4750
proof
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4751
  assume "y < x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4752
  then show "tan y < tan x"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4753
    using tan_monotone and \<open>- (pi/2) < y\<close> and \<open>x < pi/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
  4754
next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4755
  assume "tan y < tan x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4756
  show "y < x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4757
  proof (rule ccontr)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4758
    assume "\<not> ?thesis"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4759
    then have "x \<le> y" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4760
    then have "tan x \<le> tan y"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4761
    proof (cases "x = y")
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4762
      case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4763
      then show ?thesis 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
  4764
    next
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4765
      case False
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4766
      then have "x < y" using \<open>x \<le> y\<close> by auto
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4767
      from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi/2\<close>] show ?thesis
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4768
        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
  4769
    qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4770
    then show False
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4771
      using \<open>tan y < tan x\<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
  4772
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4773
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4774
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  4775
lemma tan_inverse: "1 / (tan y) = tan (pi/2 - y)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4776
  unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] 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
  4777
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  4778
lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4779
  by (simp add: tan_def)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4780
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4781
lemma tan_periodic_nat[simp]: "tan (x + real n * pi) = tan x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4782
  for n :: nat
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4783
proof (induct n arbitrary: x)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4784
  case 0
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4785
  then show ?case by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4786
next
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4787
  case (Suc n)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4788
  have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
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
  4789
    unfolding Suc_eq_plus1 of_nat_add  distrib_right by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4790
  show ?case
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4791
    unfolding split_pi_off using Suc by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4792
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4793
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4794
lemma tan_periodic_int[simp]: "tan (x + of_int i * pi) = tan x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4795
proof (cases "0 \<le> i")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4796
  case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4797
  then have i_nat: "of_int i = of_int (nat i)" 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
  4798
  show ?thesis unfolding i_nat
62679
092cb9c96c99 add le_log_of_power and le_log2_of_power by Tobias Nipkow
hoelzl
parents: 62393
diff changeset
  4799
    by (metis of_int_of_nat_eq tan_periodic_nat)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4800
next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4801
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4802
  then have i_nat: "of_int i = - of_int (nat (- i))" 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
  4803
  have "tan x = tan (x + of_int i * pi - of_int i * pi)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4804
    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
  4805
  also have "\<dots> = tan (x + of_int i * pi)"
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
  4806
    unfolding i_nat mult_minus_left diff_minus_eq_add
62679
092cb9c96c99 add le_log_of_power and le_log2_of_power by Tobias Nipkow
hoelzl
parents: 62393
diff changeset
  4807
    by (metis of_int_of_nat_eq tan_periodic_nat)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4808
  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
  4809
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  4810
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46240
diff changeset
  4811
lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
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
  4812
  using tan_periodic_int[of _ "numeral n" ] by simp
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  4813
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4814
lemma tan_minus_45: "tan (-(pi/4)) = -1"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4815
  unfolding tan_def by (simp add: sin_45 cos_45)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4816
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4817
lemma tan_diff:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4818
  "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x - y) \<noteq> 0 \<Longrightarrow> tan (x - y) = (tan x - tan y)/(1 + tan x * tan y)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4819
  for x :: "'a::{real_normed_field,banach}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4820
  using tan_add [of x "-y"] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4821
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4822
lemma tan_pos_pi2_le: "0 \<le> x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4823
  using less_eq_real_def tan_gt_zero by auto
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4824
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4825
lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos x = 1 / sqrt (1 + tan x ^ 2)"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4826
  using cos_gt_zero_pi [of x]
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  4827
  by (simp add: field_split_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4828
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4829
lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin x = tan x / sqrt (1 + tan x ^ 2)"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4830
  using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  4831
  by (force simp: field_split_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4832
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4833
lemma tan_mono_le: "-(pi/2) < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4834
  using less_eq_real_def tan_monotone by auto
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4835
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4836
lemma tan_mono_lt_eq:
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4837
  "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x < tan y \<longleftrightarrow> x < y"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4838
  using tan_monotone' by blast
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4839
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4840
lemma tan_mono_le_eq:
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4841
  "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y \<longleftrightarrow> x \<le> y"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4842
  by (meson tan_mono_le not_le tan_monotone)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4843
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61942
diff changeset
  4844
lemma tan_bound_pi2: "\<bar>x\<bar> < pi/4 \<Longrightarrow> \<bar>tan x\<bar> < 1"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4845
  using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"]
62390
842917225d56 more canonical names
nipkow
parents: 62379
diff changeset
  4846
  by (auto simp: abs_if split: if_split_asm)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4847
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4848
lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  4849
  by (simp add: tan_def sin_diff cos_diff)
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 59647
diff changeset
  4850
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4851
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4852
subsection \<open>Cotangent\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4853
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4854
definition cot :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4855
  where "cot = (\<lambda>x. cos x / sin x)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4856
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4857
lemma cot_of_real: "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4858
  by (simp add: cot_def sin_of_real cos_of_real)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4859
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4860
lemma cot_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4861
  for z :: "'a::{real_normed_field,banach}"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4862
  by (simp add: cot_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4863
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4864
lemma cot_zero [simp]: "cot 0 = 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4865
  by (simp add: cot_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4866
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4867
lemma cot_pi [simp]: "cot pi = 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4868
  by (simp add: cot_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4869
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4870
lemma cot_npi [simp]: "cot (real n * pi) = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4871
  for n :: nat
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4872
  by (simp add: cot_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4873
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4874
lemma cot_minus [simp]: "cot (- x) = - cot x"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4875
  by (simp add: cot_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4876
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4877
lemma cot_periodic [simp]: "cot (x + 2 * pi) = cot x"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4878
  by (simp add: cot_def)
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
  4879
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4880
lemma cot_altdef: "cot x = inverse (tan x)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4881
  by (simp add: cot_def tan_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4882
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4883
lemma tan_altdef: "tan x = inverse (cot x)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4884
  by (simp add: cot_def tan_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4885
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4886
lemma tan_cot': "tan (pi/2 - x) = cot x"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4887
  by (simp add: tan_cot cot_altdef)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4888
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4889
lemma cot_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cot x"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4890
  by (simp add: cot_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4891
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4892
lemma cot_less_zero:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4893
  assumes lb: "- pi/2 < x" and "x < 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4894
  shows "cot x < 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4895
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4896
  have "0 < cot (- x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4897
    using assms by (simp only: cot_gt_zero)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4898
  then show ?thesis by simp
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4899
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4900
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4901
lemma DERIV_cot [simp]: "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4902
  for x :: "'a::{real_normed_field,banach}"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4903
  unfolding cot_def using cos_squared_eq[of x]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4904
  by (auto intro!: derivative_eq_intros) (simp add: divide_inverse power2_eq_square)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4905
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4906
lemma isCont_cot: "sin x \<noteq> 0 \<Longrightarrow> isCont cot x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4907
  for x :: "'a::{real_normed_field,banach}"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4908
  by (rule DERIV_cot [THEN DERIV_isCont])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4909
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4910
lemma isCont_cot' [simp,continuous_intros]:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4911
  "isCont f a \<Longrightarrow> sin (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4912
  for a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4913
  by (rule isCont_o2 [OF _ isCont_cot])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4914
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4915
lemma tendsto_cot [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> sin a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4916
  for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4917
  by (rule isCont_tendsto_compose [OF isCont_cot])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4918
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4919
lemma continuous_cot:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4920
  "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4921
  for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4922
  unfolding continuous_def by (rule tendsto_cot)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4923
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4924
lemma continuous_on_cot [continuous_intros]:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4925
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4926
  shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. sin (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. cot (f x))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4927
  unfolding continuous_on_def by (auto intro: tendsto_cot)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4928
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4929
lemma continuous_within_cot [continuous_intros]:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4930
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4931
  shows "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4932
  unfolding continuous_within by (rule tendsto_cot)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4933
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  4934
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  4935
subsection \<open>Inverse Trigonometric Functions\<close>
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  4936
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4937
definition arcsin :: "real \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4938
  where "arcsin y = (THE x. -(pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4939
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4940
definition arccos :: "real \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4941
  where "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4942
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4943
definition arctan :: "real \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4944
  where "arctan y = (THE x. -(pi/2) < x \<and> x < pi/2 \<and> tan x = y)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4945
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4946
lemma arcsin: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2 \<and> sin (arcsin y) = y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4947
  unfolding arcsin_def by (rule theI' [OF sin_total])
23011
3eae3140b4b2 use THE instead of SOME
huffman
parents: 23007
diff changeset
  4948
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4949
lemma arcsin_pi: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi \<and> sin (arcsin y) = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4950
  by (drule (1) arcsin) (force intro: order_trans)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4951
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4952
lemma sin_arcsin [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin (arcsin y) = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4953
  by (blast dest: arcsin)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4954
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4955
lemma arcsin_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4956
  by (blast dest: arcsin)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4957
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4958
lemma arcsin_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4959
  by (blast dest: arcsin)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4960
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4961
lemma arcsin_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4962
  by (blast dest: arcsin)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4963
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4964
lemma arcsin_lt_bounded:
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4965
  assumes "- 1 < y" "y < 1"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4966
  shows  "- (pi/2) < arcsin y \<and> arcsin y < pi/2"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4967
proof -
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4968
  have "arcsin y \<noteq> pi/2"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4969
    by (metis arcsin assms not_less not_less_iff_gr_or_eq sin_pi_half)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4970
  moreover have "arcsin y \<noteq> - pi/2"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4971
    by (metis arcsin assms minus_divide_left not_less not_less_iff_gr_or_eq sin_minus sin_pi_half)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4972
  ultimately show ?thesis
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4973
    using arcsin_bounded [of y] assms by auto
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4974
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4975
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4976
lemma arcsin_sin: "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> arcsin (sin x) = x"
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4977
  unfolding arcsin_def
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  4978
  using the1_equality [OF sin_total]  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  4979
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4980
lemma arcsin_0 [simp]: "arcsin 0 = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4981
  using arcsin_sin [of 0] by simp
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4982
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4983
lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4984
  using arcsin_sin [of "pi/2"] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4985
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4986
lemma arcsin_minus_1 [simp]: "arcsin (- 1) = - (pi/2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4987
  using arcsin_sin [of "- pi/2"] by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4988
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4989
lemma arcsin_minus: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin (- x) = - arcsin x"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4990
  by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4991
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4992
lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x = arcsin y \<longleftrightarrow> x = y"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  4993
  by (metis abs_le_iff arcsin minus_le_iff)
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4994
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4995
lemma cos_arcsin_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos (arcsin x) \<noteq> 0"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4996
  using arcsin_lt_bounded cos_gt_zero_pi by force
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  4997
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  4998
lemma arccos: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  4999
  unfolding arccos_def by (rule theI' [OF cos_total])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5000
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5001
lemma cos_arccos [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> cos (arccos y) = y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5002
  by (blast dest: arccos)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  5003
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5004
lemma arccos_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5005
  by (blast dest: arccos)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5006
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5007
lemma arccos_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5008
  by (blast dest: arccos)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5009
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5010
lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5011
  by (blast dest: arccos)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5012
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5013
lemma arccos_lt_bounded: 
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5014
  assumes "- 1 < y" "y < 1"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5015
  shows  "0 < arccos y \<and> arccos y < pi"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5016
proof -
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5017
  have "arccos y \<noteq> 0"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5018
    by (metis (no_types) arccos assms(1) assms(2) cos_zero less_eq_real_def less_irrefl)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5019
  moreover have "arccos y \<noteq> -pi"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5020
    by (metis arccos assms(1) assms(2) cos_minus cos_pi not_less not_less_iff_gr_or_eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5021
  ultimately show ?thesis
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5022
    using arccos_bounded [of y] assms
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5023
    by (metis arccos cos_pi not_less not_less_iff_gr_or_eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5024
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5025
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5026
lemma arccos_cos: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> arccos (cos x) = x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5027
  by (auto simp: arccos_def intro!: the1_equality cos_total)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5028
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5029
lemma arccos_cos2: "x \<le> 0 \<Longrightarrow> - pi \<le> x \<Longrightarrow> arccos (cos x) = -x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5030
  by (auto simp: arccos_def intro!: the1_equality cos_total)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5031
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5032
lemma cos_arcsin:
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5033
  assumes "- 1 \<le> x" "x \<le> 1"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5034
  shows "cos (arcsin x) = sqrt (1 - x\<^sup>2)"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5035
proof (rule power2_eq_imp_eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5036
  show "(cos (arcsin x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5037
    by (simp add: square_le_1 assms cos_squared_eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5038
  show "0 \<le> cos (arcsin x)"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5039
    using arcsin assms cos_ge_zero by blast
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5040
  show "0 \<le> sqrt (1 - x\<^sup>2)"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5041
    by (simp add: square_le_1 assms)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5042
qed
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5043
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5044
lemma sin_arccos:
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5045
  assumes "- 1 \<le> x" "x \<le> 1"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5046
  shows "sin (arccos x) = sqrt (1 - x\<^sup>2)"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5047
proof (rule power2_eq_imp_eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5048
  show "(sin (arccos x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5049
    by (simp add: square_le_1 assms sin_squared_eq)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5050
  show "0 \<le> sin (arccos x)"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5051
    by (simp add: arccos_bounded assms sin_ge_zero)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5052
  show "0 \<le> sqrt (1 - x\<^sup>2)"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5053
    by (simp add: square_le_1 assms)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5054
qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5055
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5056
lemma arccos_0 [simp]: "arccos 0 = pi/2"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5057
  by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5058
      pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5059
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5060
lemma arccos_1 [simp]: "arccos 1 = 0"
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5061
  using arccos_cos by force
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5062
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5063
lemma arccos_minus_1 [simp]: "arccos (- 1) = pi"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5064
  by (metis arccos_cos cos_pi order_refl pi_ge_zero)
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5065
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5066
lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos (- x) = pi - arccos x"
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
  5067
  by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5068
      minus_diff_eq uminus_add_conv_diff)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5069
65057
799bbbb3a395 Some new lemmas thanks to Lukas Bulwahn. Also, NEWS.
paulson <lp15@cam.ac.uk>
parents: 65036
diff changeset
  5070
corollary arccos_minus_abs:
799bbbb3a395 Some new lemmas thanks to Lukas Bulwahn. Also, NEWS.
paulson <lp15@cam.ac.uk>
parents: 65036
diff changeset
  5071
  assumes "\<bar>x\<bar> \<le> 1"
799bbbb3a395 Some new lemmas thanks to Lukas Bulwahn. Also, NEWS.
paulson <lp15@cam.ac.uk>
parents: 65036
diff changeset
  5072
  shows "arccos (- x) = pi - arccos x"
799bbbb3a395 Some new lemmas thanks to Lukas Bulwahn. Also, NEWS.
paulson <lp15@cam.ac.uk>
parents: 65036
diff changeset
  5073
using assms by (simp add: arccos_minus)
799bbbb3a395 Some new lemmas thanks to Lukas Bulwahn. Also, NEWS.
paulson <lp15@cam.ac.uk>
parents: 65036
diff changeset
  5074
799bbbb3a395 Some new lemmas thanks to Lukas Bulwahn. Also, NEWS.
paulson <lp15@cam.ac.uk>
parents: 65036
diff changeset
  5075
lemma sin_arccos_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> sin (arccos x) \<noteq> 0"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5076
  using arccos_lt_bounded sin_gt_zero by force
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5077
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5078
lemma arctan: "- (pi/2) < arctan y \<and> arctan y < pi/2 \<and> tan (arctan y) = y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5079
  unfolding arctan_def by (rule theI' [OF tan_total])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5080
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5081
lemma tan_arctan: "tan (arctan y) = y"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5082
  by (simp add: arctan)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5083
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5084
lemma arctan_bounded: "- (pi/2) < arctan y \<and> arctan y < pi/2"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5085
  by (auto simp only: arctan)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5086
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5087
lemma arctan_lbound: "- (pi/2) < arctan y"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5088
  by (simp add: arctan)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5089
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5090
lemma arctan_ubound: "arctan y < pi/2"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5091
  by (auto simp only: arctan)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5092
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5093
lemma arctan_unique:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5094
  assumes "-(pi/2) < x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5095
    and "x < pi/2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5096
    and "tan x = y"
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5097
  shows "arctan y = x"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5098
  using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5099
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5100
lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5101
  by (rule arctan_unique) simp_all
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5102
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5103
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5104
  by (rule arctan_unique) simp_all
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5105
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5106
lemma arctan_minus: "arctan (- x) = - arctan x"
65057
799bbbb3a395 Some new lemmas thanks to Lukas Bulwahn. Also, NEWS.
paulson <lp15@cam.ac.uk>
parents: 65036
diff changeset
  5107
  using arctan [of "x"] by (auto simp: arctan_unique)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5108
44725
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5109
lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5110
  by (intro less_imp_neq [symmetric] cos_gt_zero_pi arctan_lbound arctan_ubound)
44725
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5111
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  5112
lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
44725
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5113
proof (rule power2_eq_imp_eq)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  5114
  have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  5115
  show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
44725
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5116
  show "0 \<le> cos (arctan x)"
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5117
    by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  5118
  have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47489
diff changeset
  5119
    unfolding tan_def by (simp add: distrib_left power_divide)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5120
  then show "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5121
    using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq)
44725
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5122
qed
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5123
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  5124
lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
44725
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5125
  using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5126
  using tan_arctan [of x] unfolding tan_def cos_arctan
d3bf0e33c98a add lemmas cos_arctan and sin_arctan
huffman
parents: 44710
diff changeset
  5127
  by (simp add: eq_divide_eq)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5128
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5129
lemma tan_sec: "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5130
  for x :: "'a::{real_normed_field,banach,field}"
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5131
  by (simp add: add_divide_eq_iff inverse_eq_divide power2_eq_square tan_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  5132
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5133
lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5134
  by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5135
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5136
lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5137
  by (simp only: not_less [symmetric] arctan_less_iff)
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5138
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5139
lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5140
  by (simp only: eq_iff [where 'a=real] arctan_le_iff)
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5141
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5142
lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5143
  using arctan_less_iff [of 0 x] by simp
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5144
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5145
lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5146
  using arctan_less_iff [of x 0] by simp
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5147
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5148
lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5149
  using arctan_le_iff [of 0 x] by simp
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5150
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5151
lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5152
  using arctan_le_iff [of x 0] by simp
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5153
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5154
lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5155
  using arctan_eq_iff [of x 0] by simp
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5156
51482
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5157
lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5158
proof -
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5159
  have "continuous_on (sin ` {- pi/2 .. pi/2}) arcsin"
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  5160
    by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5161
  also have "sin ` {- pi/2 .. pi/2} = {-1 .. 1}"
51482
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5162
  proof safe
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5163
    fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5164
    assume "x \<in> {-1..1}"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5165
    then show "x \<in> sin ` {- pi/2..pi/2}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5166
      using arcsin_lbound arcsin_ubound
56479
91958d4b30f7 revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents: 56409
diff changeset
  5167
      by (intro image_eqI[where x="arcsin x"]) auto
51482
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5168
  qed simp
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5169
  finally show ?thesis .
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5170
qed
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5171
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  5172
lemma continuous_on_arcsin [continuous_intros]:
51482
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5173
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arcsin (f x))"
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5174
  using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5175
  by (auto simp: comp_def subset_eq)
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5176
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5177
lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5178
  using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5179
  by (auto simp: continuous_on_eq_continuous_at subset_eq)
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5180
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5181
lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5182
proof -
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5183
  have "continuous_on (cos ` {0 .. pi}) arccos"
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  5184
    by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos)
51482
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5185
  also have "cos ` {0 .. pi} = {-1 .. 1}"
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5186
  proof safe
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5187
    fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5188
    assume "x \<in> {-1..1}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5189
    then show "x \<in> cos ` {0..pi}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5190
      using arccos_lbound arccos_ubound
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5191
      by (intro image_eqI[where x="arccos x"]) auto
51482
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5192
  qed simp
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5193
  finally show ?thesis .
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5194
qed
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5195
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  5196
lemma continuous_on_arccos [continuous_intros]:
51482
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5197
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arccos (f x))"
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5198
  using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5199
  by (auto simp: comp_def subset_eq)
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5200
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5201
lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5202
  using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
80efd8c49f52 arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents: 51481
diff changeset
  5203
  by (auto simp: continuous_on_eq_continuous_at subset_eq)
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  5204
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  5205
lemma isCont_arctan: "isCont arctan x"
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5206
proof -
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5207
  obtain u where u: "- (pi / 2) < u" "u < arctan x"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5208
    by (meson arctan arctan_less_iff linordered_field_no_lb)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5209
  obtain v where v: "arctan x < v" "v < pi / 2"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5210
    by (meson arctan_less_iff arctan_ubound linordered_field_no_ub)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5211
  have "isCont arctan (tan (arctan x))"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5212
  proof (rule isCont_inverse_function2 [of u "arctan x" v])
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5213
    show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> arctan (tan z) = z"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5214
      using arctan_unique u(1) v(2) by auto
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5215
    then show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> isCont tan z"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5216
      by (metis arctan cos_gt_zero_pi isCont_tan less_irrefl)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5217
  qed (use u v in auto)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5218
  then show ?thesis
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5219
    by (simp add: arctan)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5220
qed
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  5221
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  5222
lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  5223
  by (rule isCont_tendsto_compose [OF isCont_arctan])
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  5224
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  5225
lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  5226
  unfolding continuous_def by (rule tendsto_arctan)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  5227
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5228
lemma continuous_on_arctan [continuous_intros]:
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5229
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
  5230
  unfolding continuous_on_def by (auto intro: tendsto_arctan)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5231
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5232
lemma DERIV_arcsin:
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5233
  assumes "- 1 < x" "x < 1"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5234
  shows "DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5235
proof (rule DERIV_inverse_function)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5236
  show "(sin has_real_derivative sqrt (1 - x\<^sup>2)) (at (arcsin x))"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5237
    by (rule derivative_eq_intros | use assms cos_arcsin in force)+
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5238
  show "sqrt (1 - x\<^sup>2) \<noteq> 0"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5239
    using abs_square_eq_1 assms by force
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5240
qed (use assms isCont_arcsin in auto)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5241
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5242
lemma DERIV_arccos:
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5243
  assumes "- 1 < x" "x < 1"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5244
  shows "DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5245
proof (rule DERIV_inverse_function)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5246
  show "(cos has_real_derivative - sqrt (1 - x\<^sup>2)) (at (arccos x))"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5247
    by (rule derivative_eq_intros | use assms sin_arccos in force)+
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5248
  show "- sqrt (1 - x\<^sup>2) \<noteq> 0"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5249
    using abs_square_eq_1 assms by force
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5250
qed (use assms isCont_arccos in auto)
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  5251
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  5252
lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  5253
proof (rule DERIV_inverse_function)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  5254
  have "inverse ((cos (arctan x))\<^sup>2) = 1 + x\<^sup>2"
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5255
    by (metis arctan cos_arctan_not_zero power_inverse tan_sec)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  5256
  then show "(tan has_real_derivative 1 + x\<^sup>2) (at (arctan x))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  5257
    by (auto intro!: derivative_eq_intros)
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5258
  show "\<And>y. \<lbrakk>x - 1 < y; y < x + 1\<rbrakk> \<Longrightarrow> tan (arctan y) = y"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5259
    using tan_arctan by blast
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5260
  show "1 + x\<^sup>2 \<noteq> 0"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5261
    by (metis power_one sum_power2_eq_zero_iff zero_neq_one)
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5262
qed (use isCont_arctan in auto)
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  5263
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31790
diff changeset
  5264
declare
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  5265
  DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61284
diff changeset
  5266
  DERIV_arcsin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  5267
  DERIV_arccos[THEN DERIV_chain2, derivative_intros]
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61284
diff changeset
  5268
  DERIV_arccos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  5269
  DERIV_arctan[THEN DERIV_chain2, derivative_intros]
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61284
diff changeset
  5270
  DERIV_arctan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31790
diff changeset
  5271
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  5272
lemmas has_derivative_arctan[derivative_intros] = DERIV_arctan[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  5273
  and has_derivative_arccos[derivative_intros] = DERIV_arccos[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  5274
  and has_derivative_arcsin[derivative_intros] = DERIV_arcsin[THEN DERIV_compose_FDERIV]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67574
diff changeset
  5275
61881
b4bfa62e799d Transcendental: use [simp]-canonical form - (pi/2)
hoelzl
parents: 61810
diff changeset
  5276
lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- (pi/2)))"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5277
  by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5278
     (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5279
           intro!: tan_monotone exI[of _ "pi/2"])
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5280
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5281
lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5282
  by (rule filterlim_at_top_at_left[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5283
     (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5284
           intro!: tan_monotone exI[of _ "pi/2"])
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5285
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  5286
lemma tendsto_arctan_at_top: "(arctan \<longlongrightarrow> (pi/2)) at_top"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5287
proof (rule tendstoI)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5288
  fix e :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5289
  assume "0 < e"
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62949
diff changeset
  5290
  define y where "y = pi/2 - min (pi/2) e"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5291
  then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5292
    using \<open>0 < e\<close> by auto
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5293
  show "eventually (\<lambda>x. dist (arctan x) (pi/2) < e) at_top"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5294
  proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5295
    fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5296
    assume "tan y < x"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5297
    then have "arctan (tan y) < arctan x"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5298
      by (simp add: arctan_less_iff)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5299
    with y have "y < arctan x"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5300
      by (subst (asm) arctan_tan) simp_all
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5301
    with arctan_ubound[of x, arith] y \<open>0 < e\<close>
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5302
    show "dist (arctan x) (pi/2) < e"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5303
      by (simp add: dist_real_def)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5304
  qed
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5305
qed
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5306
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  5307
lemma tendsto_arctan_at_bot: "(arctan \<longlongrightarrow> - (pi/2)) at_bot"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5308
  unfolding filterlim_at_bot_mirror arctan_minus
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5309
  by (intro tendsto_minus tendsto_arctan_at_top)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5310
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50326
diff changeset
  5311
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5312
subsection \<open>Prove Totality of the Trigonometric Functions\<close>
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5313
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5314
lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5315
  by (simp add: abs_le_iff)
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5316
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5317
lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5318
  by (simp add: sin_arccos abs_le_iff)
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5319
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5320
lemma sin_mono_less_eq:
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5321
  "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x < sin y \<longleftrightarrow> x < y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5322
  by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5323
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5324
lemma sin_mono_le_eq:
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5325
  "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x \<le> sin y \<longleftrightarrow> x \<le> y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5326
  by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5327
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
  5328
lemma sin_inj_pi:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5329
  "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x = sin y \<Longrightarrow> x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5330
  by (metis arcsin_sin)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5331
70722
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5332
lemma arcsin_le_iff:
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5333
  assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2"
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5334
  shows   "arcsin x \<le> y \<longleftrightarrow> x \<le> sin y"
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5335
proof -
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5336
  have "arcsin x \<le> y \<longleftrightarrow> sin (arcsin x) \<le> sin y"
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5337
    using arcsin_bounded[of x] assms by (subst sin_mono_le_eq) auto
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5338
  also from assms have "sin (arcsin x) = x" by simp
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5339
  finally show ?thesis .
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5340
qed
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5341
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5342
lemma le_arcsin_iff:
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5343
  assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2"
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5344
  shows   "arcsin x \<ge> y \<longleftrightarrow> x \<ge> sin y"
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5345
proof -
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5346
  have "arcsin x \<ge> y \<longleftrightarrow> sin (arcsin x) \<ge> sin y"
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5347
    using arcsin_bounded[of x] assms by (subst sin_mono_le_eq) auto
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5348
  also from assms have "sin (arcsin x) = x" by simp
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5349
  finally show ?thesis .
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5350
qed
ae2528273eeb A couple of new theorems, stolen from AFP entries
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
  5351
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5352
lemma cos_mono_less_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x < cos y \<longleftrightarrow> y < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5353
  by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5354
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5355
lemma cos_mono_le_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x \<le> cos y \<longleftrightarrow> y \<le> x"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5356
  by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear)
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5357
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5358
lemma cos_inj_pi: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x = cos y \<Longrightarrow> x = y"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5359
  by (metis arccos_cos)
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5360
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5361
lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2"
59751
916c0f6c83e3 New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents: 59746
diff changeset
  5362
  by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5363
      cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl)
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5364
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5365
lemma sincos_total_pi_half:
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5366
  assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5367
  shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5368
proof -
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5369
  have x1: "x \<le> 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5370
    using assms by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5371
  with assms have *: "0 \<le> arccos x" "cos (arccos x) = x"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5372
    by (auto simp: arccos)
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  5373
  from assms have "y = sqrt (1 - x\<^sup>2)"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5374
    by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5375
  with x1 * assms arccos_le_pi2 [of x] show ?thesis
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5376
    by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
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
  5377
qed
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5378
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5379
lemma sincos_total_pi:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5380
  assumes "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5381
  shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5382
proof (cases rule: le_cases [of 0 x])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5383
  case le
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5384
  from sincos_total_pi_half [OF le] show ?thesis
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5385
    by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms)
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5386
next
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
  5387
  case ge
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5388
  then have "0 \<le> -x"
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5389
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5390
  then obtain t where t: "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5391
    using sincos_total_pi_half assms
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5392
    by auto (metis \<open>0 \<le> - x\<close> power2_minus)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5393
  show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5394
    by (rule exI [where x = "pi -t"]) (use t in 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
  5395
qed
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
  5396
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5397
lemma sincos_total_2pi_le:
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5398
  assumes "x\<^sup>2 + y\<^sup>2 = 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5399
  shows "\<exists>t. 0 \<le> t \<and> t \<le> 2 * pi \<and> x = cos t \<and> y = sin t"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5400
proof (cases rule: le_cases [of 0 y])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5401
  case le
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5402
  from sincos_total_pi [OF le] show ?thesis
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5403
    by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5404
next
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
  5405
  case ge
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5406
  then have "0 \<le> -y"
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5407
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5408
  then obtain t where t: "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5409
    using sincos_total_pi assms
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5410
    by auto (metis \<open>0 \<le> - y\<close> power2_minus)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5411
  show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5412
    by (rule exI [where x = "2 * pi - t"]) (use t in 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
  5413
qed
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5414
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5415
lemma sincos_total_2pi:
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5416
  assumes "x\<^sup>2 + y\<^sup>2 = 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5417
  obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5418
proof -
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5419
  from sincos_total_2pi_le [OF assms]
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5420
  obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t"
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5421
    by blast
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5422
  show ?thesis
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5423
    by (cases "t = 2 * pi") (use t that in \<open>force+\<close>)
59746
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5424
qed
ddae5727c5a9 new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
  5425
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61942
diff changeset
  5426
lemma arcsin_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5427
  by (rule trans [OF sin_mono_less_eq [symmetric]]) (use arcsin_ubound arcsin_lbound in auto)
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5428
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61942
diff changeset
  5429
lemma arcsin_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5430
  using arcsin_less_mono not_le by blast
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5431
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5432
lemma arcsin_less_arcsin: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5433
  using arcsin_less_mono by auto
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5434
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5435
lemma arcsin_le_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5436
  using arcsin_le_mono by auto
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5437
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5438
lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x < arccos y \<longleftrightarrow> y < x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5439
  by (rule trans [OF cos_mono_less_eq [symmetric]]) (use arccos_ubound arccos_lbound in auto)
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5440
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61942
diff changeset
  5441
lemma arccos_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5442
  using arccos_less_mono [of y x] by (simp add: not_le [symmetric])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5443
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5444
lemma arccos_less_arccos: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5445
  using arccos_less_mono by auto
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5446
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5447
lemma arccos_le_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5448
  using arccos_le_mono by auto
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5449
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5450
lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 \<and> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x = arccos y \<longleftrightarrow> x = y"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5451
  using cos_arccos_abs by fastforce
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5452
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5453
68499
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5454
lemma arccos_cos_eq_abs:
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5455
  assumes "\<bar>\<theta>\<bar> \<le> pi"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5456
  shows "arccos (cos \<theta>) = \<bar>\<theta>\<bar>"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  5457
  unfolding arccos_def
68499
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5458
proof (intro the_equality conjI; clarify?)
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5459
  show "cos \<bar>\<theta>\<bar> = cos \<theta>"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5460
    by (simp add: abs_real_def)
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5461
  show "x = \<bar>\<theta>\<bar>" if "cos x = cos \<theta>" "0 \<le> x" "x \<le> pi" for x
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5462
    by (simp add: \<open>cos \<bar>\<theta>\<bar> = cos \<theta>\<close> assms cos_inj_pi that)
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5463
qed (use assms in auto)
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5464
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5465
lemma arccos_cos_eq_abs_2pi:
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5466
  obtains k where "arccos (cos \<theta>) = \<bar>\<theta> - of_int k * (2 * pi)\<bar>"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5467
proof -
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5468
  define k where "k \<equiv>  \<lfloor>(\<theta> + pi) / (2 * pi)\<rfloor>"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5469
  have lepi: "\<bar>\<theta> - of_int k * (2 * pi)\<bar> \<le> pi"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5470
    using floor_divide_lower [of "2*pi" "\<theta> + pi"] floor_divide_upper [of "2*pi" "\<theta> + pi"]
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5471
    by (auto simp: k_def abs_if algebra_simps)
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5472
  have "arccos (cos \<theta>) = arccos (cos (\<theta> - of_int k * (2 * pi)))"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5473
    using cos_int_2pin sin_int_2pin by (simp add: cos_diff mult.commute)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  5474
  also have "\<dots> = \<bar>\<theta> - of_int k * (2 * pi)\<bar>"
68499
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5475
    using arccos_cos_eq_abs lepi by blast
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  5476
  finally show ?thesis
68499
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5477
    using that by metis
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5478
qed
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5479
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5480
lemma cos_limit_1:
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5481
  assumes "(\<lambda>j. cos (\<theta> j)) \<longlonglongrightarrow> 1"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5482
  shows "\<exists>k. (\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> 0"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5483
proof -
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5484
  have "\<forall>\<^sub>F j in sequentially. cos (\<theta> j) \<in> {- 1..1}"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5485
    by auto
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5486
  then have "(\<lambda>j. arccos (cos (\<theta> j))) \<longlonglongrightarrow> arccos 1"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5487
    using continuous_on_tendsto_compose [OF continuous_on_arccos' assms] by auto
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5488
  moreover have "\<And>j. \<exists>k. arccos (cos (\<theta> j)) = \<bar>\<theta> j - of_int k * (2 * pi)\<bar>"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5489
    using arccos_cos_eq_abs_2pi by metis
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5490
  then have "\<exists>k. \<forall>j. arccos (cos (\<theta> j)) = \<bar>\<theta> j - of_int (k j) * (2 * pi)\<bar>"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5491
    by metis
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5492
  ultimately have "\<exists>k. (\<lambda>j. \<bar>\<theta> j - of_int (k j) * (2 * pi)\<bar>) \<longlonglongrightarrow> 0"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5493
    by auto
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5494
  then show ?thesis
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5495
    by (simp add: tendsto_rabs_zero_iff)
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5496
qed
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5497
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5498
lemma cos_diff_limit_1:
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5499
  assumes "(\<lambda>j. cos (\<theta> j - \<Theta>)) \<longlonglongrightarrow> 1"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5500
  obtains k where "(\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5501
proof -
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5502
  obtain k where "(\<lambda>j. (\<theta> j - \<Theta>) - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> 0"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5503
    using cos_limit_1 [OF assms] by auto
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5504
  then have "(\<lambda>j. \<Theta> + ((\<theta> j - \<Theta>) - of_int (k j) * (2 * pi))) \<longlonglongrightarrow> \<Theta> + 0"
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5505
    by (rule tendsto_add [OF tendsto_const])
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5506
  with that show ?thesis
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  5507
    by auto
68499
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5508
qed
d4312962161a Rationalisation of complex transcendentals, esp the Arg function
paulson <lp15@cam.ac.uk>
parents: 68100
diff changeset
  5509
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5510
subsection \<open>Machin's formula\<close>
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5511
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5512
lemma arctan_one: "arctan 1 = pi / 4"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5513
  by (rule arctan_unique) (simp_all add: tan_45 m2pi_less_pi)
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5514
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5515
lemma tan_total_pi4:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5516
  assumes "\<bar>x\<bar> < 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5517
  shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5518
proof
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5519
  show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5520
    unfolding arctan_one [symmetric] arctan_minus [symmetric]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5521
    unfolding arctan_less_iff
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  5522
    using assms by (auto simp: arctan)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5523
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5524
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5525
lemma arctan_add:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5526
  assumes "\<bar>x\<bar> \<le> 1" "\<bar>y\<bar> < 1"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5527
  shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5528
proof (rule arctan_unique [symmetric])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5529
  have "- (pi / 4) \<le> arctan x" "- (pi / 4) < arctan y"
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5530
    unfolding arctan_one [symmetric] arctan_minus [symmetric]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5531
    unfolding arctan_le_iff arctan_less_iff
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5532
    using assms by auto
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5533
  from add_le_less_mono [OF this] show 1: "- (pi/2) < arctan x + arctan y"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5534
    by simp
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5535
  have "arctan x \<le> pi / 4" "arctan y < pi / 4"
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5536
    unfolding arctan_one [symmetric]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5537
    unfolding arctan_le_iff arctan_less_iff
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5538
    using assms by auto
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5539
  from add_le_less_mono [OF this] show 2: "arctan x + arctan y < pi/2"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5540
    by simp
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5541
  show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
59869
3b5b53eb78ba arcsin and arccos lemmas
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  5542
    using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5543
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5544
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5545
lemma arctan_double: "\<bar>x\<bar> < 1 \<Longrightarrow> 2 * arctan x = arctan ((2 * x) / (1 - x\<^sup>2))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5546
  by (metis arctan_add linear mult_2 not_less power2_eq_square)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5547
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5548
theorem machin: "pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5549
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5550
  have "\<bar>1 / 5\<bar> < (1 :: real)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5551
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5552
  from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (1 / 5) = arctan (5 / 12)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5553
    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
  5554
  moreover
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5555
  have "\<bar>5 / 12\<bar> < (1 :: real)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5556
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5557
  from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (5 / 12) = arctan (120 / 119)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5558
    by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  5559
  moreover
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5560
  have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5561
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5562
  from arctan_add[OF this] have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5563
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5564
  ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5565
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5566
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5567
    unfolding arctan_one by algebra
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5568
qed
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  5569
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5570
lemma machin_Euler: "5 * arctan (1 / 7) + 2 * arctan (3 / 79) = pi / 4"
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5571
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5572
  have 17: "\<bar>1 / 7\<bar> < (1 :: real)" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5573
  with arctan_double have "2 * arctan (1 / 7) = arctan (7 / 24)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  5574
    by simp (simp add: field_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5575
  moreover
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5576
  have "\<bar>7 / 24\<bar> < (1 :: real)" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5577
  with arctan_double have "2 * arctan (7 / 24) = arctan (336 / 527)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5578
    by simp (simp add: field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5579
  moreover
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5580
  have "\<bar>336 / 527\<bar> < (1 :: real)" by auto
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5581
  from arctan_add[OF less_imp_le[OF 17] this]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5582
  have "arctan(1/7) + arctan (336 / 527) = arctan (2879 / 3353)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5583
    by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5584
  ultimately have I: "5 * arctan (1 / 7) = arctan (2879 / 3353)" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5585
  have 379: "\<bar>3 / 79\<bar> < (1 :: real)" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5586
  with arctan_double have II: "2 * arctan (3 / 79) = arctan (237 / 3116)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5587
    by simp (simp add: field_simps)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5588
  have *: "\<bar>2879 / 3353\<bar> < (1 :: real)" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5589
  have "\<bar>237 / 3116\<bar> < (1 :: real)" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5590
  from arctan_add[OF less_imp_le[OF *] this] have "arctan (2879/3353) + arctan (237/3116) = pi/4"
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5591
    by (simp add: arctan_one)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5592
  with I II show ?thesis by auto
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5593
qed
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5594
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5595
(*But could also prove MACHIN_GAUSS:
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5596
  12 * arctan(1/18) + 8 * arctan(1/57) - 5 * arctan(1/239) = pi/4*)
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5597
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5598
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5599
subsection \<open>Introducing the inverse tangent 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
  5600
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5601
lemma monoseq_arctan_series:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5602
  fixes x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5603
  assumes "\<bar>x\<bar> \<le> 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5604
  shows "monoseq (\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5605
    (is "monoseq ?a")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5606
proof (cases "x = 0")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5607
  case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5608
  then show ?thesis by (auto simp: monoseq_def)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5609
next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5610
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5611
  have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5612
    using assms 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
  5613
  show "monoseq ?a"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5614
  proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5615
    have mono: "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5616
        1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5617
      if "0 \<le> x" and "x \<le> 1" for n and x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5618
    proof (rule mult_mono)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5619
      show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5620
        by (rule frac_le) simp_all
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5621
      show "0 \<le> 1 / real (Suc (n * 2))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5622
        by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5623
      show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5624
        by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5625
      show "0 \<le> x ^ Suc (Suc n * 2)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5626
        by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5627
    qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5628
    show ?thesis
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5629
    proof (cases "0 \<le> x")
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5630
      case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5631
      from mono[OF this \<open>x \<le> 1\<close>, THEN allI]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5632
      show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5633
        unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5634
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5635
      case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5636
      then have "0 \<le> - x" and "- x \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5637
        using \<open>-1 \<le> x\<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
  5638
      from mono[OF this]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5639
      have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5640
          1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5641
        using \<open>0 \<le> -x\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5642
      then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5643
        unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5644
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5645
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5646
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5647
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5648
lemma zeroseq_arctan_series:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5649
  fixes x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5650
  assumes "\<bar>x\<bar> \<le> 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5651
  shows "(\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1)) \<longlonglongrightarrow> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5652
    (is "?a \<longlonglongrightarrow> 0")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5653
proof (cases "x = 0")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5654
  case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5655
  then show ?thesis 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
  5656
next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5657
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5658
  have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5659
    using assms by auto
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
  5660
  show "?a \<longlonglongrightarrow> 0"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5661
  proof (cases "\<bar>x\<bar> < 1")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5662
    case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5663
    then have "norm x < 1" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5664
    from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]]
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
  5665
    have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0"
31790
05c92381363c corrected and unified thm names
nipkow
parents: 31338
diff changeset
  5666
      unfolding inverse_eq_divide Suc_eq_plus1 by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5667
    then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5668
      using pos2 by (rule LIMSEQ_linear)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5669
  next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5670
    case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5671
    then have "x = -1 \<or> x = 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5672
      using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5673
    then have n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5674
      unfolding One_nat_def by auto
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44319
diff changeset
  5675
    from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5676
    show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5677
      unfolding n_eq Suc_eq_plus1 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
  5678
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5679
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5680
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5681
lemma summable_arctan_series:
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  5682
  fixes n :: nat
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5683
  assumes "\<bar>x\<bar> \<le> 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5684
  shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5685
    (is "summable (?c x)")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5686
  by (rule summable_Leibniz(1),
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5687
      rule zeroseq_arctan_series[OF assms],
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5688
      rule monoseq_arctan_series[OF assms])
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5689
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5690
lemma DERIV_arctan_series:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5691
  assumes "\<bar>x\<bar> < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5692
  shows "DERIV (\<lambda>x'. \<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x' ^ (k * 2 + 1))) x :>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5693
      (\<Sum>k. (-1)^k * x^(k * 2))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5694
    (is "DERIV ?arctan _ :> ?Int")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5695
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5696
  let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5697
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5698
  have n_even: "even n \<Longrightarrow> 2 * (n div 2) = n" for n :: nat
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5699
    by presburger
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5700
  then have if_eq: "?f n * real (Suc n) * x'^n =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5701
      (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5702
    for n x'
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5703
    by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5704
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5705
  have summable_Integral: "summable (\<lambda> n. (- 1) ^ n * x^(2 * n))" if "\<bar>x\<bar> < 1" for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5706
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5707
    from that have "x\<^sup>2 < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5708
      by (simp add: abs_square_less_1)
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
  5709
    have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5710
      by (rule summable_Leibniz(1))
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5711
        (auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5712
    then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5713
      by (simp only: power_mult)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5714
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5715
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67268
diff changeset
  5716
  have sums_even: "(sums) f = (sums) (\<lambda> n. if even n then f (n div 2) else 0)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5717
    for f :: "nat \<Rightarrow> real"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5718
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5719
    have "f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x" for x :: real
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5720
    proof
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5721
      assume "f sums x"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5722
      from sums_if[OF sums_zero this] show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5723
        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
  5724
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5725
      assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
63170
eae6549dbea2 tuned proofs, to allow unfold_abs_def;
wenzelm
parents: 63145
diff changeset
  5726
      from LIMSEQ_linear[OF this[simplified sums_def] pos2, simplified sum_split_even_odd[simplified mult.commute]]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5727
      show "f sums x"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5728
        unfolding sums_def 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
  5729
    qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5730
    then show ?thesis ..
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5731
  qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5732
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5733
  have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5734
    unfolding if_eq mult.commute[of _ 2]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5735
      suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5736
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5737
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5738
  have arctan_eq: "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x" for x
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5739
  proof -
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
  5740
    have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
  5741
      (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5742
      using n_even by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5743
    have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5744
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5745
    then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5746
      unfolding if_eq' idx_eq suminf_def
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5747
        sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5748
      by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5749
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5750
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5751
  have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum>n. ?f n * real (Suc n) * x^n)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5752
  proof (rule DERIV_power_series')
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5753
    show "x \<in> {- 1 <..< 1}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5754
      using \<open>\<bar> x \<bar> < 1\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5755
    show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5756
      if x'_bounds: "x' \<in> {- 1 <..< 1}" for x' :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5757
    proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5758
      from that have "\<bar>x'\<bar> < 1" by auto
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
  5759
      then show ?thesis
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
  5760
        using that sums_summable sums_if [OF sums_0 [of "\<lambda>x. 0"] summable_sums [OF summable_Integral]]   
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
  5761
        by (auto simp add: if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5762
    qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5763
  qed auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5764
  then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5765
    by (simp only: Int_eq arctan_eq)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5766
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5767
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5768
lemma arctan_series:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5769
  assumes "\<bar>x\<bar> \<le> 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5770
  shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5771
    (is "_ = suminf (\<lambda> n. ?c x n)")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5772
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5773
  let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5774
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5775
  have DERIV_arctan_suminf: "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5776
    if "0 < r" and "r < 1" and "\<bar>x\<bar> < r" for r x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5777
  proof (rule DERIV_arctan_series)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5778
    from that show "\<bar>x\<bar> < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5779
      using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5780
  qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5781
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5782
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5783
    fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5784
    assume "\<bar>x\<bar> \<le> 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5785
    note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5786
  } note arctan_series_borders = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5787
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5788
  have when_less_one: "arctan x = (\<Sum>k. ?c x k)" if "\<bar>x\<bar> < 1" for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5789
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5790
    obtain r where "\<bar>x\<bar> < r" and "r < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5791
      using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5792
    then have "0 < r" and "- r < x" and "x < r" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5793
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5794
    have suminf_eq_arctan_bounded: "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5795
      if "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" for x a b
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5796
    proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5797
      from that have "\<bar>x\<bar> < r" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5798
      show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5799
      proof (rule DERIV_isconst2[of "a" "b"])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5800
        show "a < b" and "a \<le> x" and "x \<le> b"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5801
          using \<open>a < b\<close> \<open>a \<le> x\<close> \<open>x \<le> b\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5802
        have "\<forall>x. - r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5803
        proof (rule allI, rule impI)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5804
          fix x
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5805
          assume "-r < x \<and> x < r"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5806
          then have "\<bar>x\<bar> < r" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5807
          with \<open>r < 1\<close> have "\<bar>x\<bar> < 1" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5808
          have "\<bar>- (x\<^sup>2)\<bar> < 1" using abs_square_less_1 \<open>\<bar>x\<bar> < 1\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5809
          then have "(\<lambda>n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5810
            unfolding real_norm_def[symmetric] by (rule geometric_sums)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5811
          then have "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5812
            unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5813
          then have suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5814
            using sums_unique unfolding inverse_eq_divide by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5815
          have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5816
            unfolding suminf_c'_eq_geom
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5817
            by (rule DERIV_arctan_suminf[OF \<open>0 < r\<close> \<open>r < 1\<close> \<open>\<bar>x\<bar> < r\<close>])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5818
          from DERIV_diff [OF this DERIV_arctan] show "DERIV (\<lambda>x. suminf (?c x) - arctan x) x :> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5819
            by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5820
        qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5821
        then have DERIV_in_rball: "\<forall>y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5822
          using \<open>-r < a\<close> \<open>b < r\<close> by auto
68638
87d1bff264df de-applying and meta-quantifying
paulson <lp15@cam.ac.uk>
parents: 68635
diff changeset
  5823
        then show "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5824
          using \<open>\<bar>x\<bar> < r\<close> by auto
69020
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  5825
        show "continuous_on {a..b} (\<lambda>x. suminf (?c x) - arctan x)"
4f94e262976d elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents: 68774
diff changeset
  5826
          using DERIV_in_rball DERIV_atLeastAtMost_imp_continuous_on 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
  5827
      qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5828
    qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5829
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5830
    have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5831
      unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5832
      by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5833
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5834
    have "suminf (?c x) - arctan x = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5835
    proof (cases "x = 0")
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5836
      case True
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5837
      then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5838
        using suminf_arctan_zero by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5839
    next
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5840
      case False
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5841
      then have "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5842
        by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5843
      have "suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>) = suminf (?c 0) - arctan 0"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  5844
        by (rule suminf_eq_arctan_bounded[where x1=0 and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5845
          (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5846
      moreover
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5847
      have "suminf (?c x) - arctan x = suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  5848
        by (rule suminf_eq_arctan_bounded[where x1=x and a1="- \<bar>x\<bar>" and b1="\<bar>x\<bar>"])
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5849
           (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>- \<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5850
      ultimately show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5851
        using suminf_arctan_zero 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
  5852
    qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5853
    then show ?thesis by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5854
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5855
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5856
  show "arctan x = suminf (\<lambda>n. ?c x n)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5857
  proof (cases "\<bar>x\<bar> < 1")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5858
    case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5859
    then show ?thesis by (rule when_less_one)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5860
  next
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5861
    case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5862
    then have "\<bar>x\<bar> = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5863
    let ?a = "\<lambda>x n. \<bar>1 / real (n * 2 + 1) * x^(n * 2 + 1)\<bar>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5864
    let ?diff = "\<lambda>x n. \<bar>arctan x - (\<Sum>i<n. ?c x i)\<bar>"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5865
    have "?diff 1 n \<le> ?a 1 n" for n :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5866
    proof -
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5867
      have "0 < (1 :: real)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5868
      moreover
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5869
      have "?diff x n \<le> ?a x n" if "0 < x" and "x < 1" for x :: real
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5870
      proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5871
        from that have "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5872
          by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5873
        from \<open>0 < x\<close> have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5874
          by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5875
        note bounds = mp[OF arctan_series_borders(2)[OF \<open>\<bar>x\<bar> \<le> 1\<close>] this, unfolded when_less_one[OF \<open>\<bar>x\<bar> < 1\<close>, symmetric], THEN spec]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5876
        have "0 < 1 / real (n*2+1) * x^(n*2+1)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5877
          by (rule mult_pos_pos) (simp_all only: zero_less_power[OF \<open>0 < x\<close>], auto)
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5878
        then have a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5879
          by (rule abs_of_pos)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5880
        show ?thesis
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5881
        proof (cases "even n")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5882
          case True
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5883
          then have sgn_pos: "(-1)^n = (1::real)" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5884
          from \<open>even n\<close> obtain m where "n = 2 * m" ..
58709
efdc6c533bd3 prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents: 58656
diff changeset
  5885
          then have "2 * m = n" ..
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5886
          from bounds[of m, unfolded this atLeastAtMost_iff]
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  5887
          have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n + 1. (?c x i)) - (\<Sum>i<n. (?c x i))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5888
            by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5889
          also have "\<dots> = ?c x n" by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5890
          also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5891
          finally show ?thesis .
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5892
        next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5893
          case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5894
          then have sgn_neg: "(-1)^n = (-1::real)" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5895
          from \<open>odd n\<close> obtain m where "n = 2 * m + 1" ..
58709
efdc6c533bd3 prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents: 58656
diff changeset
  5896
          then have m_def: "2 * m + 1 = n" ..
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5897
          then have m_plus: "2 * (m + 1) = n + 1" by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5898
          from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5899
          have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n. (?c x i)) - (\<Sum>i<n+1. (?c x i))" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5900
          also have "\<dots> = - ?c x n" by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5901
          also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5902
          finally show ?thesis .
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  5903
        qed
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5904
      qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5905
      hence "\<forall>x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5906
      moreover have "isCont (\<lambda> x. ?a x n - ?diff x n) x" for x
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
  5907
        unfolding diff_conv_add_uminus divide_inverse
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  5908
        by (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68603
diff changeset
  5909
          continuous_at_within_inverse isCont_mult isCont_power continuous_const isCont_sum
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
  5910
          simp del: add_uminus_conv_diff)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5911
      ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5912
        by (rule LIM_less_bound)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5913
      then show ?thesis by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5914
    qed
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
  5915
    have "?a 1 \<longlonglongrightarrow> 0"
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44319
diff changeset
  5916
      unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
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
  5917
      by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc)
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
  5918
    have "?diff 1 \<longlonglongrightarrow> 0"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5919
    proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5920
      fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5921
      assume "0 < r"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5922
      obtain N :: nat where N_I: "N \<le> n \<Longrightarrow> ?a 1 n < r" for n
61969
e01015e49041 more symbols;
wenzelm
parents: 61944
diff changeset
  5923
        using LIMSEQ_D[OF \<open>?a 1 \<longlonglongrightarrow> 0\<close> \<open>0 < r\<close>] by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5924
      have "norm (?diff 1 n - 0) < r" if "N \<le> n" for n
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5925
        using \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF that] by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5926
      then show "\<exists>N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < 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
  5927
    qed
44710
9caf6883f1f4 remove redundant lemmas about LIMSEQ
huffman
parents: 44568
diff changeset
  5928
    from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5929
    have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5930
    then have "arctan 1 = (\<Sum>i. ?c 1 i)" by (rule sums_unique)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  5931
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5932
    show ?thesis
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5933
    proof (cases "x = 1")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5934
      case True
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5935
      then show ?thesis by (simp add: \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close>)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5936
    next
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5937
      case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5938
      then have "x = -1" using \<open>\<bar>x\<bar> = 1\<close> by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  5939
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5940
      have "- (pi/2) < 0" using pi_gt_zero 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
  5941
      have "- (2 * pi) < 0" using pi_gt_zero by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  5942
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5943
      have c_minus_minus: "?c (- 1) i = - ?c 1 i" for i by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5944
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5945
      have "arctan (- 1) = arctan (tan (-(pi / 4)))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5946
        unfolding tan_45 tan_minus ..
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5947
      also have "\<dots> = - (pi / 4)"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5948
        by (rule arctan_tan) (auto simp: order_less_trans[OF \<open>- (pi/2) < 0\<close> pi_gt_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5949
      also have "\<dots> = - (arctan (tan (pi / 4)))"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5950
        unfolding neg_equal_iff_equal
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5951
        by (rule arctan_tan[symmetric]) (auto simp: order_less_trans[OF \<open>- (2 * pi) < 0\<close> pi_gt_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5952
      also have "\<dots> = - (arctan 1)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5953
        unfolding tan_45 ..
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5954
      also have "\<dots> = - (\<Sum> i. ?c 1 i)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5955
        using \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close> by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5956
      also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5957
        using suminf_minus[OF sums_summable[OF \<open>(?c 1) sums (arctan 1)\<close>]]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5958
        unfolding c_minus_minus by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5959
      finally show ?thesis using \<open>x = -1\<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
  5960
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5961
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5962
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5963
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5964
lemma arctan_half: "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5965
  for x :: real
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5966
proof -
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5967
  obtain y where low: "- (pi/2) < y" and high: "y < pi/2" and y_eq: "tan y = x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5968
    using tan_total by blast
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  5969
  then have low2: "- (pi/2) < y / 2" and high2: "y / 2 < pi/2"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5970
    by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5971
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5972
  have "0 < cos y" by (rule cos_gt_zero_pi[OF low high])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  5973
  then have "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5974
    by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5975
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5976
  have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5977
    unfolding tan_def power_divide ..
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5978
  also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5979
    using \<open>cos y \<noteq> 0\<close> by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5980
  also have "\<dots> = 1 / (cos y)\<^sup>2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5981
    unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
53076
47c9aff07725 more symbols;
wenzelm
parents: 53015
diff changeset
  5982
  finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  5983
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5984
  have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5985
    unfolding tan_def using \<open>cos y \<noteq> 0\<close> by (simp add: field_simps)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5986
  also have "\<dots> = tan y / (1 + 1 / cos y)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5987
    using \<open>cos y \<noteq> 0\<close> unfolding add_divide_distrib by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5988
  also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5989
    unfolding cos_sqrt ..
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5990
  also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5991
    unfolding real_sqrt_divide by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5992
  finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  5993
    unfolding \<open>1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2\<close> .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5994
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5995
  have "arctan x = y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5996
    using arctan_tan low high y_eq by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5997
  also have "\<dots> = 2 * (arctan (tan (y/2)))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5998
    using arctan_tan[OF low2 high2] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  5999
  also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6000
    unfolding tan_half by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6001
  finally show ?thesis
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  6002
    unfolding eq \<open>tan y = x\<close> .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  6003
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  6004
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6005
lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6006
  by (simp only: arctan_less_iff)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6007
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6008
lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6009
  by (simp only: arctan_le_iff)
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  6010
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  6011
lemma arctan_inverse:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6012
  assumes "x \<noteq> 0"
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  6013
  shows "arctan (1 / x) = sgn x * pi/2 - arctan x"
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  6014
proof (rule arctan_unique)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6015
  have \<section>: "x > 0 \<Longrightarrow> arctan x < pi"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6016
    using arctan_bounded [of x] by linarith 
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  6017
  show "- (pi/2) < sgn x * pi/2 - arctan x"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6018
    using assms by (auto simp: sgn_real_def arctan algebra_simps \<section>)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  6019
  show "sgn x * pi/2 - arctan x < pi/2"
44746
9e4f7d3b5376 add lemmas about arctan;
huffman
parents: 44745
diff changeset
  6020
    using arctan_bounded [of "- x"] assms
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6021
    by (auto simp: algebra_simps sgn_real_def arctan_minus)
68603
73eeb3f31406 De-applying
paulson <lp15@cam.ac.uk>
parents: 68601
diff changeset
  6022
  show "tan (sgn x * pi/2 - arctan x) = 1 / x"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6023
    unfolding tan_inverse [of "arctan x", unfolded tan_arctan] sgn_real_def
56479
91958d4b30f7 revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents: 56409
diff changeset
  6024
    by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  6025
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  6026
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6027
theorem pi_series: "pi / 4 = (\<Sum>k. (-1)^k * 1 / real (k * 2 + 1))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6028
  (is "_ = ?SUM")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  6029
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6030
  have "pi / 4 = arctan 1"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6031
    using arctan_one by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6032
  also have "\<dots> = ?SUM"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6033
    using arctan_series[of 1] 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
  6034
  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
  6035
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  6036
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  6037
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  6038
subsection \<open>Existence of Polar Coordinates\<close>
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  6039
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52139
diff changeset
  6040
lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6041
  by (rule power2_le_imp_le [OF _ zero_le_one])
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6042
    (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  6043
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6044
lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a \<and> y = r * sin a"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6045
proof -
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6046
  have polar_ex1: "\<exists>r a. x = r * cos a \<and> y = r * sin a" if "0 < y" for y
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6047
  proof -
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6048
    have "x = sqrt (x\<^sup>2 + y\<^sup>2) * cos (arccos (x / sqrt (x\<^sup>2 + y\<^sup>2)))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6049
      by (simp add: cos_arccos_abs [OF cos_x_y_le_one])
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6050
    moreover have "y = sqrt (x\<^sup>2 + y\<^sup>2) * sin (arccos (x / sqrt (x\<^sup>2 + y\<^sup>2)))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6051
      using that
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6052
      by (simp add: sin_arccos_abs [OF cos_x_y_le_one] power_divide right_diff_distrib flip: real_sqrt_mult)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6053
    ultimately show ?thesis
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6054
      by blast
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6055
  qed
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6056
  show ?thesis
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6057
  proof (cases "0::real" y rule: linorder_cases)
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  6058
    case less
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6059
    then show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6060
      by (rule polar_ex1)
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6061
  next
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6062
    case equal
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6063
    then show ?thesis
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6064
      by (force simp: intro!: cos_zero sin_zero)
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6065
  next
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6066
    case greater
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6067
    with polar_ex1 [where y="-y"] show ?thesis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6068
      by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6069
  qed
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
  6070
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  6071
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6072
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6073
subsection \<open>Basics about polynomial functions: products, extremal behaviour and root counts\<close>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6074
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6075
lemma pairs_le_eq_Sigma: "{(i, j). i + j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m - r))"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6076
  for m :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6077
  by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6078
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6079
lemma sum_up_index_split: "(\<Sum>k\<le>m + n. f k) = (\<Sum>k\<le>m. f k) + (\<Sum>k = Suc m..m + n. f k)"
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  6080
  by (metis atLeast0AtMost Suc_eq_plus1 le0 sum.ub_add_nat)
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6081
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6082
lemma Sigma_interval_disjoint: "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6083
  for w :: "'a::order"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6084
  by auto
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6085
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6086
lemma product_atMost_eq_Un: "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6087
  for m :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6088
  by auto
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6089
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6090
lemma polynomial_product: (*with thanks to Chaitanya Mangla*)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6091
  fixes x :: "'a::idom"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6092
  assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6093
    and n: "\<And>j. j > n \<Longrightarrow> b j = 0"
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
  6094
  shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6095
         (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6096
proof -
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6097
  have "\<And>i j. \<lbrakk>m + n - i < j; a i \<noteq> 0\<rbrakk> \<Longrightarrow> b j = 0"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6098
    by (meson le_add_diff leI le_less_trans m n)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6099
  then have \<section>: "(\<Sum>(i,j)\<in>(SIGMA i:{..m+n}. {m+n - i<..m+n}). a i * x ^ i * (b j * x ^ j)) = 0"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6100
    by (clarsimp simp add: sum_Un Sigma_interval_disjoint intro!: sum.neutral)
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6101
  have "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = (\<Sum>i\<le>m. \<Sum>j\<le>n. (a i * x ^ i) * (b j * x ^ j))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6102
    by (rule sum_product)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6103
  also have "\<dots> = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6104
    using assms by (auto simp: sum_up_index_split)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6105
  also have "\<dots> = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6106
    by (simp add: add_ac sum.Sigma product_atMost_eq_Un sum_Un Sigma_interval_disjoint \<section>)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6107
  also have "\<dots> = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6108
    by (auto simp: pairs_le_eq_Sigma sum.Sigma)
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6109
  also have "... = (\<Sum>k\<le>m + n. \<Sum>i\<le>k. a i * x ^ i * (b (k - i) * x ^ (k - i)))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6110
    by (rule sum.triangle_reindex_eq)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6111
  also have "\<dots> = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6112
    by (auto simp: algebra_simps sum_distrib_left simp flip: power_add intro!: sum.cong)
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6113
  finally show ?thesis .
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6114
qed
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6115
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
  6116
lemma polynomial_product_nat:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6117
  fixes x :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6118
  assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6119
    and n: "\<And>j. j > n \<Longrightarrow> b j = 0"
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
  6120
  shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6121
         (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6122
  using polynomial_product [of m a n b x] assms
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6123
  by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric]
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6124
      of_nat_eq_iff Int.int_sum [symmetric])
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6125
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6126
lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6127
  fixes x :: "'a::idom"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6128
  assumes "1 \<le> n"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6129
  shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6130
    (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6131
proof -
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6132
  have h: "bij_betw (\<lambda>(i,j). (j,i)) ((SIGMA i : atMost n. lessThan i)) (SIGMA j : lessThan n. {Suc j..n})"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6133
    by (auto simp: bij_betw_def inj_on_def)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6134
  have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = (\<Sum>i\<le>n. a i * (x^i - y^i))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6135
    by (simp add: right_diff_distrib sum_subtractf)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6136
  also have "\<dots> = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6137
    by (simp add: power_diff_sumr2 mult.assoc)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6138
  also have "\<dots> = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6139
    by (simp add: sum_distrib_left)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6140
  also have "\<dots> = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6141
    by (simp add: sum.Sigma)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6142
  also have "\<dots> = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))"
69654
bc758f4f09e5 uniform naming
nipkow
parents: 69593
diff changeset
  6143
    by (auto simp: sum.reindex_bij_betw [OF h, symmetric] intro: sum.cong_simp)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6144
  also have "\<dots> = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6145
    by (simp add: sum.Sigma)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6146
  also have "\<dots> = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6147
    by (simp add: sum_distrib_left mult_ac)
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6148
  finally show ?thesis .
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6149
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6150
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6151
lemma polyfun_diff_alt: (*COMPLEX_SUB_POLYFUN_ALT in HOL Light*)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6152
  fixes x :: "'a::idom"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6153
  assumes "1 \<le> n"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6154
  shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6155
    (x - y) * ((\<Sum>j<n. \<Sum>k<n-j. a(j + k + 1) * y^k * x^j))"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6156
proof -
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6157
  have "(\<Sum>i=Suc j..n. a i * y^(i - j - 1)) = (\<Sum>k<n-j. a(j+k+1) * y^k)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6158
    if "j < n" for j :: nat
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6159
  proof -
71585
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6160
    have "\<And>k. k < n - j \<Longrightarrow> k \<in> (\<lambda>i. i - Suc j) ` {Suc j..n}"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6161
      by (rule_tac x="k + Suc j" in image_eqI, auto)
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6162
    then have h: "bij_betw (\<lambda>i. i - (j + 1)) {Suc j..n} (lessThan (n-j))"
4b1021677f15 tidying up some horrible proofs
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  6163
      by (auto simp: bij_betw_def inj_on_def)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6164
    then show ?thesis
69654
bc758f4f09e5 uniform naming
nipkow
parents: 69593
diff changeset
  6165
      by (auto simp: sum.reindex_bij_betw [OF h, symmetric] intro: sum.cong_simp)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6166
  qed
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6167
  then show ?thesis
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6168
    by (simp add: polyfun_diff [OF assms] sum_distrib_right)
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6169
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6170
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6171
lemma polyfun_linear_factor:  (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6172
  fixes a :: "'a::idom"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6173
  shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i) + (\<Sum>i\<le>n. c(i) * a^i)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6174
proof (cases "n = 0")
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6175
  case True then show ?thesis
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6176
    by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6177
next
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6178
  case False
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6179
  have "(\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) = (z - a) * (\<Sum>i<n. b i * z^i) + (\<Sum>i\<le>n. c i * a^i)) \<longleftrightarrow>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6180
        (\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) = (z - a) * (\<Sum>i<n. b i * z^i))"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6181
    by (simp add: algebra_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6182
  also have "\<dots> \<longleftrightarrow>
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6183
    (\<exists>b. \<forall>z. (z - a) * (\<Sum>j<n. (\<Sum>i = Suc j..n. c i * a^(i - Suc j)) * z^j) =
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6184
      (z - a) * (\<Sum>i<n. b i * z^i))"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6185
    using False by (simp add: polyfun_diff)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6186
  also have "\<dots> = True" by auto
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6187
  finally show ?thesis
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6188
    by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6189
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6190
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6191
lemma polyfun_linear_factor_root:  (*COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT in HOL Light*)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6192
  fixes a :: "'a::idom"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6193
  assumes "(\<Sum>i\<le>n. c(i) * a^i) = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6194
  obtains b where "\<And>z. (\<Sum>i\<le>n. c i * z^i) = (z - a) * (\<Sum>i<n. b i * z^i)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6195
  using polyfun_linear_factor [of c n a] assms by auto
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6196
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6197
(*The material of this section, up until this point, could go into a new theory of polynomials
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6198
  based on Main alone. The remaining material involves limits, continuity, series, etc.*)
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60141
diff changeset
  6199
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6200
lemma isCont_polynom: "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w^i) a"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6201
  for c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6202
  by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6203
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6204
lemma zero_polynom_imp_zero_coeffs:
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6205
  fixes c :: "nat \<Rightarrow> 'a::{ab_semigroup_mult,real_normed_div_algebra}"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6206
  assumes "\<And>w. (\<Sum>i\<le>n. c i * w^i) = 0"  "k \<le> n"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6207
  shows "c k = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6208
  using assms
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6209
proof (induction n arbitrary: c k)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6210
  case 0
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6211
  then show ?case
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6212
    by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6213
next
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6214
  case (Suc n c k)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6215
  have [simp]: "c 0 = 0" using Suc.prems(1) [of 0]
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6216
    by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6217
  have "(\<Sum>i\<le>Suc n. c i * w^i) = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" for w
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6218
  proof -
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6219
    have "(\<Sum>i\<le>Suc n. c i * w^i) = (\<Sum>i\<le>n. c (Suc i) * w ^ Suc i)"
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  6220
      unfolding Set_Interval.sum.atMost_Suc_shift
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6221
      by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6222
    also have "\<dots> = w * (\<Sum>i\<le>n. c (Suc i) * w^i)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6223
      by (simp add: sum_distrib_left ac_simps)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6224
    finally show ?thesis .
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6225
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6226
  then have w: "\<And>w. w \<noteq> 0 \<Longrightarrow> (\<Sum>i\<le>n. c (Suc i) * w^i) = 0"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6227
    using Suc  by auto
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  6228
  then have "(\<lambda>h. \<Sum>i\<le>n. c (Suc i) * h^i) \<midarrow>0\<rightarrow> 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6229
    by (simp cong: LIM_cong)  \<comment> \<open>the case \<open>w = 0\<close> by continuity\<close>
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6230
  then have "(\<Sum>i\<le>n. c (Suc i) * 0^i) = 0"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6231
    using isCont_polynom [of 0 "\<lambda>i. c (Suc i)" n] LIM_unique
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6232
    by (force simp: Limits.isCont_iff)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6233
  then have "\<And>w. (\<Sum>i\<le>n. c (Suc i) * w^i) = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6234
    using w by metis
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6235
  then have "\<And>i. i \<le> n \<Longrightarrow> c (Suc i) = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6236
    using Suc.IH [of "\<lambda>i. c (Suc i)"] by blast
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  6237
  then show ?case using \<open>k \<le> Suc n\<close>
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6238
    by (cases k) auto
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6239
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6240
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6241
lemma polyfun_rootbound: (*COMPLEX_POLYFUN_ROOTBOUND in HOL Light*)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6242
  fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6243
  assumes "c k \<noteq> 0" "k\<le>n"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6244
  shows "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<and> card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6245
  using assms
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6246
proof (induction n arbitrary: c k)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6247
  case 0
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6248
  then show ?case
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6249
    by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6250
next
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6251
  case (Suc m c k)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6252
  let ?succase = ?case
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6253
  show ?case
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6254
  proof (cases "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = {}")
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6255
    case True
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6256
    then show ?succase
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6257
      by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6258
  next
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6259
    case False
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6260
    then obtain z0 where z0: "(\<Sum>i\<le>Suc m. c(i) * z0^i) = 0"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6261
      by blast
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6262
    then obtain b where b: "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = (w - z0) * (\<Sum>i\<le>m. b i * w^i)"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6263
      using polyfun_linear_factor_root [OF z0, unfolded lessThan_Suc_atMost]
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6264
      by blast
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6265
    then have eq: "{z. (\<Sum>i\<le>Suc m. c i * z^i) = 0} = insert z0 {z. (\<Sum>i\<le>m. b i * z^i) = 0}"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6266
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6267
    have "\<not> (\<forall>k\<le>m. b k = 0)"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6268
    proof
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6269
      assume [simp]: "\<forall>k\<le>m. b k = 0"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6270
      then have "\<And>w. (\<Sum>i\<le>m. b i * w^i) = 0"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6271
        by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6272
      then have "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = 0"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6273
        using b by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6274
      then have "\<And>k. k \<le> Suc m \<Longrightarrow> c k = 0"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6275
        using zero_polynom_imp_zero_coeffs by blast
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6276
      then show False using Suc.prems by blast
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6277
    qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6278
    then obtain k' where bk': "b k' \<noteq> 0" "k' \<le> m"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6279
      by blast
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6280
    show ?succase
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6281
      using Suc.IH [of b k'] bk'
70097
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69654
diff changeset
  6282
      by (simp add: eq card_insert_if del: sum.atMost_Suc)
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6283
    qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6284
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6285
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6286
lemma
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6287
  fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6288
  assumes "c k \<noteq> 0" "k\<le>n"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6289
  shows polyfun_roots_finite: "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6290
    and polyfun_roots_card: "card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6291
  using polyfun_rootbound assms by auto
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6292
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6293
lemma polyfun_finite_roots: (*COMPLEX_POLYFUN_FINITE_ROOTS in HOL Light*)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6294
  fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6295
  shows "finite {x. (\<Sum>i\<le>n. c i * x^i) = 0} \<longleftrightarrow> (\<exists>i\<le>n. c i \<noteq> 0)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6296
    (is "?lhs = ?rhs")
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6297
proof
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6298
  assume ?lhs
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6299
  moreover have "\<not> finite {x. (\<Sum>i\<le>n. c i * x^i) = 0}" if "\<forall>i\<le>n. c i = 0"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6300
  proof -
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6301
    from that have "\<And>x. (\<Sum>i\<le>n. c i * x^i) = 0"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6302
      by simp
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6303
    then show ?thesis
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6304
      using ex_new_if_finite [OF infinite_UNIV_char_0 [where 'a='a]]
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6305
      by auto
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6306
  qed
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6307
  ultimately show ?rhs by metis
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6308
next
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6309
  assume ?rhs
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6310
  with polyfun_rootbound show ?lhs by blast
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6311
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6312
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6313
lemma polyfun_eq_0: "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = 0) \<longleftrightarrow> (\<forall>i\<le>n. c i = 0)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6314
  for c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6315
  (*COMPLEX_POLYFUN_EQ_0 in HOL Light*)
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6316
  using zero_polynom_imp_zero_coeffs by auto
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6317
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6318
lemma polyfun_eq_coeffs: "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>i\<le>n. c i = d i)"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6319
  for c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6320
proof -
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6321
  have "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>x. (\<Sum>i\<le>n. (c i - d i) * x^i) = 0)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
  6322
    by (simp add: left_diff_distrib Groups_Big.sum_subtractf)
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6323
  also have "\<dots> \<longleftrightarrow> (\<forall>i\<le>n. c i - d i = 0)"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6324
    by (rule polyfun_eq_0)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6325
  finally show ?thesis
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6326
    by simp
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6327
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6328
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6329
lemma polyfun_eq_const: (*COMPLEX_POLYFUN_EQ_CONST in HOL Light*)
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6330
  fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6331
  shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = k) \<longleftrightarrow> c 0 = k \<and> (\<forall>i \<in> {1..n}. c i = 0)"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6332
    (is "?lhs = ?rhs")
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6333
proof -
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6334
  have *: "\<forall>x. (\<Sum>i\<le>n. (if i=0 then k else 0) * x^i) = k"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6335
    by (induct n) auto
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6336
  show ?thesis
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6337
  proof
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6338
    assume ?lhs
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6339
    with * have "(\<forall>i\<le>n. c i = (if i=0 then k else 0))"
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6340
      by (simp add: polyfun_eq_coeffs [symmetric])
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  6341
    then show ?rhs by simp
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6342
  next
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  6343
    assume ?rhs
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  6344
    then show ?lhs by (induct n) auto
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6345
  qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6346
qed
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6347
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6348
lemma root_polyfun:
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  6349
  fixes z :: "'a::idom"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6350
  assumes "1 \<le> n"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  6351
  shows "z^n = a \<longleftrightarrow> (\<Sum>i\<le>n. (if i = 0 then -a else if i=n then 1 else 0) * z^i) = 0"
70097
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69654
diff changeset
  6352
  using assms by (cases n) (simp_all add: sum.atLeast_Suc_atMost atLeast0AtMost [symmetric])
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6353
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6354
lemma
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6355
  assumes "SORT_CONSTRAINT('a::{idom,real_normed_div_algebra})"
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6356
    and "1 \<le> n"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  6357
  shows finite_roots_unity: "finite {z::'a. z^n = 1}"
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63467
diff changeset
  6358
    and card_roots_unity: "card {z::'a. z^n = 1} \<le> n"
63558
0aa33085c8b1 misc tuning and modernization;
wenzelm
parents: 63540
diff changeset
  6359
  using polyfun_rootbound [of "\<lambda>i. if i = 0 then -1 else if i=n then 1 else 0" n n] assms(2)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6360
  by (auto simp: root_polyfun [OF assms(2)])
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59869
diff changeset
  6361
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  6362
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6363
subsection \<open>Hyperbolic functions\<close>
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6364
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6365
definition sinh :: "'a :: {banach, real_normed_algebra_1} \<Rightarrow> 'a" where
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6366
  "sinh x = (exp x - exp (-x)) /\<^sub>R 2"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6367
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6368
definition cosh :: "'a :: {banach, real_normed_algebra_1} \<Rightarrow> 'a" where
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6369
  "cosh x = (exp x + exp (-x)) /\<^sub>R 2"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6370
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6371
definition tanh :: "'a :: {banach, real_normed_field} \<Rightarrow> 'a" where
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6372
  "tanh x = sinh x / cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6373
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6374
definition arsinh :: "'a :: {banach, real_normed_algebra_1, ln} \<Rightarrow> 'a" where
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6375
  "arsinh x = ln (x + (x^2 + 1) powr of_real (1/2))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6376
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6377
definition arcosh :: "'a :: {banach, real_normed_algebra_1, ln} \<Rightarrow> 'a" where
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6378
  "arcosh x = ln (x + (x^2 - 1) powr of_real (1/2))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6379
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6380
definition artanh :: "'a :: {banach, real_normed_field, ln} \<Rightarrow> 'a" where
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6381
  "artanh x = ln ((1 + x) / (1 - x)) / 2"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6382
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6383
lemma arsinh_0 [simp]: "arsinh 0 = 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6384
  by (simp add: arsinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6385
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6386
lemma arcosh_1 [simp]: "arcosh 1 = 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6387
  by (simp add: arcosh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6388
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6389
lemma artanh_0 [simp]: "artanh 0 = 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6390
  by (simp add: artanh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6391
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6392
lemma tanh_altdef:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6393
  "tanh x = (exp x - exp (-x)) / (exp x + exp (-x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6394
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6395
  have "tanh x = (2 *\<^sub>R sinh x) / (2 *\<^sub>R cosh x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6396
    by (simp add: tanh_def scaleR_conv_of_real)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6397
  also have "2 *\<^sub>R sinh x = exp x - exp (-x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6398
    by (simp add: sinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6399
  also have "2 *\<^sub>R cosh x = exp x + exp (-x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6400
    by (simp add: cosh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6401
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6402
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6403
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6404
lemma tanh_real_altdef: "tanh (x::real) = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6405
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6406
  have [simp]: "exp (2 * x) = exp x * exp x" "exp (x * 2) = exp x * exp x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6407
    by (subst exp_add [symmetric]; simp)+
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6408
  have "tanh x = (2 * exp (-x) * sinh x) / (2 * exp (-x) * cosh x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6409
    by (simp add: tanh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6410
  also have "2 * exp (-x) * sinh x = 1 - exp (-2*x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6411
    by (simp add: exp_minus field_simps sinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6412
  also have "2 * exp (-x) * cosh x = 1 + exp (-2*x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6413
    by (simp add: exp_minus field_simps cosh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6414
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6415
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6416
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6417
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6418
lemma sinh_converges: "(\<lambda>n. if even n then 0 else x ^ n /\<^sub>R fact n) sums sinh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6419
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6420
  have "(\<lambda>n. (x ^ n /\<^sub>R fact n - (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) sums sinh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6421
    unfolding sinh_def by (intro sums_scaleR_right sums_diff exp_converges)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6422
  also have "(\<lambda>n. (x ^ n /\<^sub>R fact n - (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) =
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6423
               (\<lambda>n. if even n then 0 else x ^ n /\<^sub>R fact n)" by auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6424
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6425
qed
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6426
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6427
lemma cosh_converges: "(\<lambda>n. if even n then x ^ n /\<^sub>R fact n else 0) sums cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6428
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6429
  have "(\<lambda>n. (x ^ n /\<^sub>R fact n + (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) sums cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6430
    unfolding cosh_def by (intro sums_scaleR_right sums_add exp_converges)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6431
  also have "(\<lambda>n. (x ^ n /\<^sub>R fact n + (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) =
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6432
               (\<lambda>n. if even n then x ^ n /\<^sub>R fact n else 0)" by auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6433
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6434
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6435
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6436
lemma sinh_0 [simp]: "sinh 0 = 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6437
  by (simp add: sinh_def)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6438
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6439
lemma cosh_0 [simp]: "cosh 0 = 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6440
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6441
  have "cosh 0 = (1/2) *\<^sub>R (1 + 1)" by (simp add: cosh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6442
  also have "\<dots> = 1" by (rule scaleR_half_double)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6443
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6444
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6445
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6446
lemma tanh_0 [simp]: "tanh 0 = 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6447
  by (simp add: tanh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6448
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6449
lemma sinh_minus [simp]: "sinh (- x) = -sinh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6450
  by (simp add: sinh_def algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6451
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6452
lemma cosh_minus [simp]: "cosh (- x) = cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6453
  by (simp add: cosh_def algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6454
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6455
lemma tanh_minus [simp]: "tanh (-x) = -tanh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6456
  by (simp add: tanh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6457
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6458
lemma sinh_ln_real: "x > 0 \<Longrightarrow> sinh (ln x :: real) = (x - inverse x) / 2"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6459
  by (simp add: sinh_def exp_minus)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6460
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6461
lemma cosh_ln_real: "x > 0 \<Longrightarrow> cosh (ln x :: real) = (x + inverse x) / 2"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6462
  by (simp add: cosh_def exp_minus)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6463
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6464
lemma tanh_ln_real:
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6465
  "tanh (ln x :: real) = (x ^ 2 - 1) / (x ^ 2 + 1)" if "x > 0"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6466
proof -
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6467
  from that have "(x * 2 - inverse x * 2) * (x\<^sup>2 + 1) =
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6468
    (x\<^sup>2 - 1) * (2 * x + 2 * inverse x)"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6469
    by (simp add: field_simps power2_eq_square)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6470
  moreover have "x\<^sup>2 + 1 > 0"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6471
    using that by (simp add: ac_simps add_pos_nonneg)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6472
  moreover have "2 * x + 2 * inverse x > 0"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6473
    using that by (simp add: add_pos_pos)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6474
  ultimately have "(x * 2 - inverse x * 2) /
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6475
    (2 * x + 2 * inverse x) =
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6476
    (x\<^sup>2 - 1) / (x\<^sup>2 + 1)"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6477
    by (simp add: frac_eq_eq)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6478
  with that show ?thesis
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6479
    by (simp add: tanh_def sinh_ln_real cosh_ln_real)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6480
qed
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6481
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6482
lemma has_field_derivative_scaleR_right [derivative_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6483
  "(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_field_derivative (c *\<^sub>R D)) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6484
  unfolding has_field_derivative_def
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6485
  using has_derivative_scaleR_right[of f "\<lambda>x. D * x" F c]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6486
  by (simp add: mult_scaleR_left [symmetric] del: mult_scaleR_left)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6487
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6488
lemma has_field_derivative_sinh [THEN DERIV_chain2, derivative_intros]:
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6489
  "(sinh has_field_derivative cosh x) (at (x :: 'a :: {banach, real_normed_field}))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6490
  unfolding sinh_def cosh_def by (auto intro!: derivative_eq_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6491
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6492
lemma has_field_derivative_cosh [THEN DERIV_chain2, derivative_intros]:
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6493
  "(cosh has_field_derivative sinh x) (at (x :: 'a :: {banach, real_normed_field}))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6494
  unfolding sinh_def cosh_def by (auto intro!: derivative_eq_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6495
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6496
lemma has_field_derivative_tanh [THEN DERIV_chain2, derivative_intros]:
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6497
  "cosh x \<noteq> 0 \<Longrightarrow> (tanh has_field_derivative 1 - tanh x ^ 2)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6498
                     (at (x :: 'a :: {banach, real_normed_field}))"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6499
  unfolding tanh_def by (auto intro!: derivative_eq_intros simp: power2_eq_square field_split_simps)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6500
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6501
lemma has_derivative_sinh [derivative_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6502
  fixes g :: "'a \<Rightarrow> ('a :: {banach, real_normed_field})"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6503
  assumes "(g has_derivative (\<lambda>x. Db * x)) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6504
  shows   "((\<lambda>x. sinh (g x)) has_derivative (\<lambda>y. (cosh (g x) * Db) * y)) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6505
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6506
  have "((\<lambda>x. - g x) has_derivative (\<lambda>y. -(Db * y))) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6507
    using assms by (intro derivative_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6508
  also have "(\<lambda>y. -(Db * y)) = (\<lambda>x. (-Db) * x)" by (simp add: fun_eq_iff)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6509
  finally have "((\<lambda>x. sinh (g x)) has_derivative
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6510
    (\<lambda>y. (exp (g x) * Db * y - exp (-g x) * (-Db) * y) /\<^sub>R 2)) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6511
    unfolding sinh_def by (intro derivative_intros assms)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6512
  also have "(\<lambda>y. (exp (g x) * Db * y - exp (-g x) * (-Db) * y) /\<^sub>R 2) = (\<lambda>y. (cosh (g x) * Db) * y)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6513
    by (simp add: fun_eq_iff cosh_def algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6514
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6515
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6516
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6517
lemma has_derivative_cosh [derivative_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6518
  fixes g :: "'a \<Rightarrow> ('a :: {banach, real_normed_field})"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6519
  assumes "(g has_derivative (\<lambda>y. Db * y)) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6520
  shows   "((\<lambda>x. cosh (g x)) has_derivative (\<lambda>y. (sinh (g x) * Db) * y)) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6521
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6522
  have "((\<lambda>x. - g x) has_derivative (\<lambda>y. -(Db * y))) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6523
    using assms by (intro derivative_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6524
  also have "(\<lambda>y. -(Db * y)) = (\<lambda>y. (-Db) * y)" by (simp add: fun_eq_iff)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6525
  finally have "((\<lambda>x. cosh (g x)) has_derivative
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6526
    (\<lambda>y. (exp (g x) * Db * y + exp (-g x) * (-Db) * y) /\<^sub>R 2)) (at x within s)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6527
    unfolding cosh_def by (intro derivative_intros assms)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6528
  also have "(\<lambda>y. (exp (g x) * Db * y + exp (-g x) * (-Db) * y) /\<^sub>R 2) = (\<lambda>y. (sinh (g x) * Db) * y)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6529
    by (simp add: fun_eq_iff sinh_def algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6530
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6531
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6532
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6533
lemma sinh_plus_cosh: "sinh x + cosh x = exp x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6534
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6535
  have "sinh x + cosh x = (1 / 2) *\<^sub>R (exp x + exp x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6536
    by (simp add: sinh_def cosh_def algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6537
  also have "\<dots> = exp x" by (rule scaleR_half_double)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6538
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6539
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6540
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6541
lemma cosh_plus_sinh: "cosh x + sinh x = exp x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6542
  by (subst add.commute) (rule sinh_plus_cosh)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6543
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6544
lemma cosh_minus_sinh: "cosh x - sinh x = exp (-x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6545
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6546
  have "cosh x - sinh x = (1 / 2) *\<^sub>R (exp (-x) + exp (-x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6547
    by (simp add: sinh_def cosh_def algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6548
  also have "\<dots> = exp (-x)" by (rule scaleR_half_double)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6549
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6550
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6551
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6552
lemma sinh_minus_cosh: "sinh x - cosh x = -exp (-x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6553
  using cosh_minus_sinh[of x] by (simp add: algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6554
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6555
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6556
context
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6557
  fixes x :: "'a :: {real_normed_field, banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6558
begin
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6559
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6560
lemma sinh_zero_iff: "sinh x = 0 \<longleftrightarrow> exp x \<in> {1, -1}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6561
  by (auto simp: sinh_def field_simps exp_minus power2_eq_square square_eq_1_iff)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6562
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6563
lemma cosh_zero_iff: "cosh x = 0 \<longleftrightarrow> exp x ^ 2 = -1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6564
  by (auto simp: cosh_def exp_minus field_simps power2_eq_square eq_neg_iff_add_eq_0)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6565
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6566
lemma cosh_square_eq: "cosh x ^ 2 = sinh x ^ 2 + 1"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6567
  by (simp add: cosh_def sinh_def algebra_simps power2_eq_square exp_add [symmetric]
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6568
                scaleR_conv_of_real)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6569
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6570
lemma sinh_square_eq: "sinh x ^ 2 = cosh x ^ 2 - 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6571
  by (simp add: cosh_square_eq)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6572
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6573
lemma hyperbolic_pythagoras: "cosh x ^ 2 - sinh x ^ 2 = 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6574
  by (simp add: cosh_square_eq)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6575
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6576
lemma sinh_add: "sinh (x + y) = sinh x * cosh y + cosh x * sinh y"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6577
  by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6578
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6579
lemma sinh_diff: "sinh (x - y) = sinh x * cosh y - cosh x * sinh y"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6580
  by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric])
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6581
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6582
lemma cosh_add: "cosh (x + y) = cosh x * cosh y + sinh x * sinh y"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6583
  by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric])
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6584
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6585
lemma cosh_diff: "cosh (x - y) = cosh x * cosh y - sinh x * sinh y"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6586
  by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6587
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6588
lemma tanh_add:
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6589
  "tanh (x + y) = (tanh x + tanh y) / (1 + tanh x * tanh y)"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6590
  if "cosh x \<noteq> 0" "cosh y \<noteq> 0"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6591
proof -
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6592
  have "(sinh x * cosh y + cosh x * sinh y) * (1 + sinh x * sinh y / (cosh x * cosh y)) =
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6593
    (cosh x * cosh y + sinh x * sinh y) * ((sinh x * cosh y + sinh y * cosh x) / (cosh y * cosh x))"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6594
    using that by (simp add: field_split_simps)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6595
  also have "(sinh x * cosh y + sinh y * cosh x) / (cosh y * cosh x) = sinh x / cosh x + sinh y / cosh y"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6596
    using that by (simp add: field_split_simps)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6597
  finally have "(sinh x * cosh y + cosh x * sinh y) * (1 + sinh x * sinh y / (cosh x * cosh y)) =
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6598
    (sinh x / cosh x + sinh y / cosh y) * (cosh x * cosh y + sinh x * sinh y)"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6599
    by simp
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6600
  then show ?thesis
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6601
    using that by (auto simp add: tanh_def sinh_add cosh_add eq_divide_eq)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6602
     (simp_all add: field_split_simps)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6603
qed
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6604
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6605
lemma sinh_double: "sinh (2 * x) = 2 * sinh x * cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6606
  using sinh_add[of x] by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6607
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6608
lemma cosh_double: "cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6609
  using cosh_add[of x] by (simp add: power2_eq_square)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6610
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6611
end
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6612
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6613
lemma sinh_field_def: "sinh z = (exp z - exp (-z)) / (2 :: 'a :: {banach, real_normed_field})"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6614
  by (simp add: sinh_def scaleR_conv_of_real)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6615
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6616
lemma cosh_field_def: "cosh z = (exp z + exp (-z)) / (2 :: 'a :: {banach, real_normed_field})"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6617
  by (simp add: cosh_def scaleR_conv_of_real)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6618
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6619
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6620
subsubsection \<open>More specific properties of the real functions\<close>
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6621
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6622
lemma sinh_real_zero_iff [simp]: "sinh (x::real) = 0 \<longleftrightarrow> x = 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6623
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6624
  have "(-1 :: real) < 0" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6625
  also have "0 < exp x" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6626
  finally have "exp x \<noteq> -1" by (intro notI) simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6627
  thus ?thesis by (subst sinh_zero_iff) simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6628
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6629
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6630
lemma plus_inverse_ge_2:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6631
  fixes x :: real
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6632
  assumes "x > 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6633
  shows   "x + inverse x \<ge> 2"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6634
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6635
  have "0 \<le> (x - 1) ^ 2" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6636
  also have "\<dots> = x^2 - 2*x + 1" by (simp add: power2_eq_square algebra_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6637
  finally show ?thesis using assms by (simp add: field_simps power2_eq_square)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6638
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6639
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6640
lemma sinh_real_nonneg_iff [simp]: "sinh (x :: real) \<ge> 0 \<longleftrightarrow> x \<ge> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6641
  by (simp add: sinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6642
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6643
lemma sinh_real_pos_iff [simp]: "sinh (x :: real) > 0 \<longleftrightarrow> x > 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6644
  by (simp add: sinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6645
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6646
lemma sinh_real_nonpos_iff [simp]: "sinh (x :: real) \<le> 0 \<longleftrightarrow> x \<le> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6647
  by (simp add: sinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6648
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6649
lemma sinh_real_neg_iff [simp]: "sinh (x :: real) < 0 \<longleftrightarrow> x < 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6650
  by (simp add: sinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6651
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6652
lemma cosh_real_ge_1: "cosh (x :: real) \<ge> 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6653
  using plus_inverse_ge_2[of "exp x"] by (simp add: cosh_def exp_minus)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6654
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6655
lemma cosh_real_pos [simp]: "cosh (x :: real) > 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6656
  using cosh_real_ge_1[of x] by simp
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6657
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6658
lemma cosh_real_nonneg[simp]: "cosh (x :: real) \<ge> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6659
  using cosh_real_ge_1[of x] by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6660
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6661
lemma cosh_real_nonzero [simp]: "cosh (x :: real) \<noteq> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6662
  using cosh_real_ge_1[of x] by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6663
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6664
lemma tanh_real_nonneg_iff [simp]: "tanh (x :: real) \<ge> 0 \<longleftrightarrow> x \<ge> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6665
  by (simp add: tanh_def field_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6666
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6667
lemma tanh_real_pos_iff [simp]: "tanh (x :: real) > 0 \<longleftrightarrow> x > 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6668
  by (simp add: tanh_def field_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6669
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6670
lemma tanh_real_nonpos_iff [simp]: "tanh (x :: real) \<le> 0 \<longleftrightarrow> x \<le> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6671
  by (simp add: tanh_def field_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6672
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6673
lemma tanh_real_neg_iff [simp]: "tanh (x :: real) < 0 \<longleftrightarrow> x < 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6674
  by (simp add: tanh_def field_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6675
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6676
lemma tanh_real_zero_iff [simp]: "tanh (x :: real) = 0 \<longleftrightarrow> x = 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6677
  by (simp add: tanh_def field_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6678
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6679
lemma arsinh_real_def: "arsinh (x::real) = ln (x + sqrt (x^2 + 1))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6680
  by (simp add: arsinh_def powr_half_sqrt)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6681
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6682
lemma arcosh_real_def: "x \<ge> 1 \<Longrightarrow> arcosh (x::real) = ln (x + sqrt (x^2 - 1))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6683
  by (simp add: arcosh_def powr_half_sqrt)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6684
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6685
lemma arsinh_real_aux: "0 < x + sqrt (x ^ 2 + 1 :: real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6686
proof (cases "x < 0")
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6687
  case True
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6688
  have "(-x) ^ 2 = x ^ 2" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6689
  also have "x ^ 2 < x ^ 2 + 1" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6690
  finally have "sqrt ((-x) ^ 2) < sqrt (x ^ 2 + 1)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6691
    by (rule real_sqrt_less_mono)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6692
  thus ?thesis using True by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6693
qed (auto simp: add_nonneg_pos)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6694
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6695
lemma arsinh_minus_real [simp]: "arsinh (-x::real) = -arsinh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6696
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6697
  have "arsinh (-x) = ln (sqrt (x\<^sup>2 + 1) - x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6698
    by (simp add: arsinh_real_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6699
  also have "sqrt (x^2 + 1) - x = inverse (sqrt (x^2 + 1) + x)"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6700
    using arsinh_real_aux[of x] by (simp add: field_split_simps algebra_simps power2_eq_square)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6701
  also have "ln \<dots> = -arsinh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6702
    using arsinh_real_aux[of x] by (simp add: arsinh_real_def ln_inverse)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6703
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6704
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6705
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6706
lemma artanh_minus_real [simp]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6707
  assumes "abs x < 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6708
  shows   "artanh (-x::real) = -artanh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6709
  using assms by (simp add: artanh_def ln_div field_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6710
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6711
lemma sinh_less_cosh_real: "sinh (x :: real) < cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6712
  by (simp add: sinh_def cosh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6713
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6714
lemma sinh_le_cosh_real: "sinh (x :: real) \<le> cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6715
  by (simp add: sinh_def cosh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6716
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6717
lemma tanh_real_lt_1: "tanh (x :: real) < 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6718
  by (simp add: tanh_def sinh_less_cosh_real)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6719
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6720
lemma tanh_real_gt_neg1: "tanh (x :: real) > -1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6721
proof -
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6722
  have "- cosh x < sinh x" by (simp add: sinh_def cosh_def field_split_simps)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6723
  thus ?thesis by (simp add: tanh_def field_simps)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6724
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6725
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6726
lemma tanh_real_bounds: "tanh (x :: real) \<in> {-1<..<1}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6727
  using tanh_real_lt_1 tanh_real_gt_neg1 by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6728
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6729
context
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6730
  fixes x :: real
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6731
begin
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6732
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6733
lemma arsinh_sinh_real: "arsinh (sinh x) = x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6734
  by (simp add: arsinh_real_def powr_def sinh_square_eq sinh_plus_cosh)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6735
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6736
lemma arcosh_cosh_real: "x \<ge> 0 \<Longrightarrow> arcosh (cosh x) = x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6737
  by (simp add: arcosh_real_def powr_def cosh_square_eq cosh_real_ge_1 cosh_plus_sinh)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6738
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6739
lemma artanh_tanh_real: "artanh (tanh x) = x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6740
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6741
  have "artanh (tanh x) = ln (cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x))) / 2"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6742
    by (simp add: artanh_def tanh_def field_split_simps)
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6743
  also have "cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x)) =
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6744
               (cosh x + sinh x) / (cosh x - sinh x)" by simp
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6745
  also have "\<dots> = (exp x)^2"
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6746
    by (simp add: cosh_plus_sinh cosh_minus_sinh exp_minus field_simps power2_eq_square)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6747
  also have "ln ((exp x)^2) / 2 = x" by (simp add: ln_realpow)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6748
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6749
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6750
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6751
end
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6752
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6753
lemma sinh_real_strict_mono: "strict_mono (sinh :: real \<Rightarrow> real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6754
  by (rule pos_deriv_imp_strict_mono derivative_intros)+ auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6755
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6756
lemma cosh_real_strict_mono:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6757
  assumes "0 \<le> x" and "x < (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6758
  shows   "cosh x < cosh y"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6759
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6760
  from assms have "\<exists>z>x. z < y \<and> cosh y - cosh x = (y - x) * sinh z"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6761
    by (intro MVT2) (auto dest: connectedD_interval intro!: derivative_eq_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6762
  then obtain z where z: "z > x" "z < y" "cosh y - cosh x = (y - x) * sinh z" by blast
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6763
  note \<open>cosh y - cosh x = (y - x) * sinh z\<close>
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6764
  also from \<open>z > x\<close> and assms have "(y - x) * sinh z > 0" by (intro mult_pos_pos) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6765
  finally show "cosh x < cosh y" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6766
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6767
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6768
lemma tanh_real_strict_mono: "strict_mono (tanh :: real \<Rightarrow> real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6769
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6770
  have *: "tanh x ^ 2 < 1" for x :: real
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6771
    using tanh_real_bounds[of x] by (simp add: abs_square_less_1 abs_if)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6772
  show ?thesis
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6773
    by (rule pos_deriv_imp_strict_mono) (insert *, auto intro!: derivative_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6774
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6775
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6776
lemma sinh_real_abs [simp]: "sinh (abs x :: real) = abs (sinh x)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6777
  by (simp add: abs_if)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6778
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6779
lemma cosh_real_abs [simp]: "cosh (abs x :: real) = cosh x"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6780
  by (simp add: abs_if)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6781
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6782
lemma tanh_real_abs [simp]: "tanh (abs x :: real) = abs (tanh x)"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6783
  by (auto simp: abs_if)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6784
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6785
lemma sinh_real_eq_iff [simp]: "sinh x = sinh y \<longleftrightarrow> x = (y :: real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6786
  using sinh_real_strict_mono by (simp add: strict_mono_eq)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6787
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6788
lemma tanh_real_eq_iff [simp]: "tanh x = tanh y \<longleftrightarrow> x = (y :: real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6789
  using tanh_real_strict_mono by (simp add: strict_mono_eq)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6790
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6791
lemma cosh_real_eq_iff [simp]: "cosh x = cosh y \<longleftrightarrow> abs x = abs (y :: real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6792
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6793
  have "cosh x = cosh y \<longleftrightarrow> x = y" if "x \<ge> 0" "y \<ge> 0" for x y :: real
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6794
    using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x] that
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6795
    by (cases x y rule: linorder_cases) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6796
  from this[of "abs x" "abs y"] show ?thesis by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6797
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6798
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6799
lemma sinh_real_le_iff [simp]: "sinh x \<le> sinh y \<longleftrightarrow> x \<le> (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6800
  using sinh_real_strict_mono by (simp add: strict_mono_less_eq)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6801
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6802
lemma cosh_real_nonneg_le_iff: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> cosh x \<le> cosh y \<longleftrightarrow> x \<le> (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6803
  using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6804
  by (cases x y rule: linorder_cases) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6805
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6806
lemma cosh_real_nonpos_le_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> cosh x \<le> cosh y \<longleftrightarrow> x \<ge> (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6807
  using cosh_real_nonneg_le_iff[of "-x" "-y"] by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6808
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6809
lemma tanh_real_le_iff [simp]: "tanh x \<le> tanh y \<longleftrightarrow> x \<le> (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6810
  using tanh_real_strict_mono by (simp add: strict_mono_less_eq)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6811
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6812
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6813
lemma sinh_real_less_iff [simp]: "sinh x < sinh y \<longleftrightarrow> x < (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6814
  using sinh_real_strict_mono by (simp add: strict_mono_less)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6815
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6816
lemma cosh_real_nonneg_less_iff: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> cosh x < cosh y \<longleftrightarrow> x < (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6817
  using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6818
  by (cases x y rule: linorder_cases) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6819
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6820
lemma cosh_real_nonpos_less_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> cosh x < cosh y \<longleftrightarrow> x > (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6821
  using cosh_real_nonneg_less_iff[of "-x" "-y"] by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6822
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6823
lemma tanh_real_less_iff [simp]: "tanh x < tanh y \<longleftrightarrow> x < (y::real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6824
  using tanh_real_strict_mono by (simp add: strict_mono_less)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6825
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6826
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6827
subsubsection \<open>Limits\<close>
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6828
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6829
lemma sinh_real_at_top: "filterlim (sinh :: real \<Rightarrow> real) at_top at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6830
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6831
  have *: "((\<lambda>x. - exp (- x)) \<longlongrightarrow> (-0::real)) at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6832
    by (intro tendsto_minus filterlim_compose[OF exp_at_bot] filterlim_uminus_at_bot_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6833
  have "filterlim (\<lambda>x. (1 / 2) * (-exp (-x) + exp x) :: real) at_top at_top"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6834
    by (rule filterlim_tendsto_pos_mult_at_top[OF _ _
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6835
               filterlim_tendsto_add_at_top[OF *]] tendsto_const)+ (auto simp: exp_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6836
  also have "(\<lambda>x. (1 / 2) * (-exp (-x) + exp x) :: real) = sinh"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6837
    by (simp add: fun_eq_iff sinh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6838
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6839
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6840
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6841
lemma sinh_real_at_bot: "filterlim (sinh :: real \<Rightarrow> real) at_bot at_bot"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6842
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6843
  have "filterlim (\<lambda>x. -sinh x :: real) at_bot at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6844
    by (simp add: filterlim_uminus_at_top [symmetric] sinh_real_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6845
  also have "(\<lambda>x. -sinh x :: real) = (\<lambda>x. sinh (-x))" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6846
  finally show ?thesis by (subst filterlim_at_bot_mirror)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6847
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6848
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6849
lemma cosh_real_at_top: "filterlim (cosh :: real \<Rightarrow> real) at_top at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6850
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6851
  have *: "((\<lambda>x. exp (- x)) \<longlongrightarrow> (0::real)) at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6852
    by (intro filterlim_compose[OF exp_at_bot] filterlim_uminus_at_bot_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6853
  have "filterlim (\<lambda>x. (1 / 2) * (exp (-x) + exp x) :: real) at_top at_top"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6854
    by (rule filterlim_tendsto_pos_mult_at_top[OF _ _
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6855
               filterlim_tendsto_add_at_top[OF *]] tendsto_const)+ (auto simp: exp_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6856
  also have "(\<lambda>x. (1 / 2) * (exp (-x) + exp x) :: real) = cosh"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6857
    by (simp add: fun_eq_iff cosh_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6858
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6859
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6860
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6861
lemma cosh_real_at_bot: "filterlim (cosh :: real \<Rightarrow> real) at_top at_bot"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6862
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6863
  have "filterlim (\<lambda>x. cosh (-x) :: real) at_top at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6864
    by (simp add: cosh_real_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6865
  thus ?thesis by (subst filterlim_at_bot_mirror)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6866
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6867
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6868
lemma tanh_real_at_top: "(tanh \<longlongrightarrow> (1::real)) at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6869
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6870
  have "((\<lambda>x::real. (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))) \<longlongrightarrow> (1 - 0) / (1 + 0)) at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6871
    by (intro tendsto_intros filterlim_compose[OF exp_at_bot]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6872
              filterlim_tendsto_neg_mult_at_bot[OF tendsto_const] filterlim_ident) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6873
  also have "(\<lambda>x::real. (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))) = tanh"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6874
    by (rule ext) (simp add: tanh_real_altdef)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6875
  finally show ?thesis by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6876
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6877
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6878
lemma tanh_real_at_bot: "(tanh \<longlongrightarrow> (-1::real)) at_bot"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6879
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6880
  have "((\<lambda>x::real. -tanh x) \<longlongrightarrow> -1) at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6881
    by (intro tendsto_minus tanh_real_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6882
  also have "(\<lambda>x. -tanh x :: real) = (\<lambda>x. tanh (-x))" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6883
  finally show ?thesis by (subst filterlim_at_bot_mirror)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6884
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6885
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6886
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6887
subsubsection \<open>Properties of the inverse hyperbolic functions\<close>
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6888
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6889
lemma isCont_sinh: "isCont sinh (x :: 'a :: {real_normed_field, banach})"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6890
  unfolding sinh_def [abs_def] by (auto intro!: continuous_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6891
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6892
lemma isCont_cosh: "isCont cosh (x :: 'a :: {real_normed_field, banach})"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6893
  unfolding cosh_def [abs_def] by (auto intro!: continuous_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6894
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6895
lemma isCont_tanh: "cosh x \<noteq> 0 \<Longrightarrow> isCont tanh (x :: 'a :: {real_normed_field, banach})"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6896
  unfolding tanh_def [abs_def]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6897
  by (auto intro!: continuous_intros isCont_divide isCont_sinh isCont_cosh)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6898
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6899
lemma continuous_on_sinh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6900
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6901
  assumes "continuous_on A f"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6902
  shows   "continuous_on A (\<lambda>x. sinh (f x))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6903
  unfolding sinh_def using assms by (intro continuous_intros)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6904
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6905
lemma continuous_on_cosh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6906
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6907
  assumes "continuous_on A f"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6908
  shows   "continuous_on A (\<lambda>x. cosh (f x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6909
  unfolding cosh_def using assms by (intro continuous_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6910
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6911
lemma continuous_sinh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6912
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6913
  assumes "continuous F f"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6914
  shows   "continuous F (\<lambda>x. sinh (f x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6915
  unfolding sinh_def using assms by (intro continuous_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6916
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6917
lemma continuous_cosh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6918
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6919
  assumes "continuous F f"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6920
  shows   "continuous F (\<lambda>x. cosh (f x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6921
  unfolding cosh_def using assms by (intro continuous_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6922
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6923
lemma continuous_on_tanh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6924
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6925
  assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> cosh (f x) \<noteq> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6926
  shows   "continuous_on A (\<lambda>x. tanh (f x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6927
  unfolding tanh_def using assms by (intro continuous_intros) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6928
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6929
lemma continuous_at_within_tanh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6930
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6931
  assumes "continuous (at x within A) f" "cosh (f x) \<noteq> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6932
  shows   "continuous (at x within A) (\<lambda>x. tanh (f x))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6933
  unfolding tanh_def using assms by (intro continuous_intros continuous_divide) auto
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6934
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6935
lemma continuous_tanh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6936
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6937
  assumes "continuous F f" "cosh (f (Lim F (\<lambda>x. x))) \<noteq> 0"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6938
  shows   "continuous F (\<lambda>x. tanh (f x))"
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6939
  unfolding tanh_def using assms by (intro continuous_intros continuous_divide) auto
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6940
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6941
lemma tendsto_sinh [tendsto_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6942
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6943
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sinh (f x)) \<longlongrightarrow> sinh a) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6944
  by (rule isCont_tendsto_compose [OF isCont_sinh])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6945
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6946
lemma tendsto_cosh [tendsto_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6947
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6948
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cosh (f x)) \<longlongrightarrow> cosh a) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6949
  by (rule isCont_tendsto_compose [OF isCont_cosh])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6950
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6951
lemma tendsto_tanh [tendsto_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6952
  fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6953
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cosh a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tanh (f x)) \<longlongrightarrow> tanh a) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6954
  by (rule isCont_tendsto_compose [OF isCont_tanh])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6955
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6956
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6957
lemma arsinh_real_has_field_derivative [derivative_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6958
  fixes x :: real
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6959
  shows "(arsinh has_field_derivative (1 / (sqrt (x ^ 2 + 1)))) (at x within A)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6960
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6961
  have pos: "1 + x ^ 2 > 0" by (intro add_pos_nonneg) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6962
  from pos arsinh_real_aux[of x] show ?thesis unfolding arsinh_def [abs_def]
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6963
    by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt field_split_simps)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6964
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6965
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6966
lemma arcosh_real_has_field_derivative [derivative_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6967
  fixes x :: real
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6968
  assumes "x > 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6969
  shows   "(arcosh has_field_derivative (1 / (sqrt (x ^ 2 - 1)))) (at x within A)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6970
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6971
  from assms have "x + sqrt (x\<^sup>2 - 1) > 0" by (simp add: add_pos_pos)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6972
  thus ?thesis using assms unfolding arcosh_def [abs_def]
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6973
    by (auto intro!: derivative_eq_intros
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6974
             simp: powr_minus powr_half_sqrt field_split_simps power2_eq_1_iff)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6975
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6976
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6977
lemma artanh_real_has_field_derivative [derivative_intros]:
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6978
  "(artanh has_field_derivative (1 / (1 - x ^ 2))) (at x within A)" if
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6979
    "\<bar>x\<bar> < 1" for x :: real
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6980
proof -
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6981
  from that have "- 1 < x" "x < 1" by linarith+
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  6982
  hence "(artanh has_field_derivative (4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4))
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6983
           (at x within A)" unfolding artanh_def [abs_def]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6984
    by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6985
  also have "(4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4) = 1 / ((1 + x) * (1 - x))"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6986
    using \<open>-1 < x\<close> \<open>x < 1\<close> by (simp add: frac_eq_eq)
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6987
  also have "(1 + x) * (1 - x) = 1 - x ^ 2"
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70723
diff changeset
  6988
    by (simp add: algebra_simps power2_eq_square)
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6989
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6990
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6991
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6992
lemma continuous_on_arsinh [continuous_intros]: "continuous_on A (arsinh :: real \<Rightarrow> real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6993
  by (rule DERIV_continuous_on derivative_intros)+
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6994
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6995
lemma continuous_on_arcosh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6996
  assumes "A \<subseteq> {1..}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6997
  shows   "continuous_on A (arcosh :: real \<Rightarrow> real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6998
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  6999
  have pos: "x + sqrt (x ^ 2 - 1) > 0" if "x \<ge> 1" for x
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7000
    using that by (intro add_pos_nonneg) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7001
  show ?thesis
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7002
  unfolding arcosh_def [abs_def]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7003
  by (intro continuous_on_subset [OF _ assms] continuous_on_ln continuous_on_add
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7004
               continuous_on_id continuous_on_powr')
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7005
     (auto dest: pos simp: powr_half_sqrt intro!: continuous_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7006
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7007
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7008
lemma continuous_on_artanh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7009
  assumes "A \<subseteq> {-1<..<1}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7010
  shows   "continuous_on A (artanh :: real \<Rightarrow> real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7011
  unfolding artanh_def [abs_def]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7012
  by (intro continuous_on_subset [OF _ assms]) (auto intro!: continuous_intros)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7013
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7014
lemma continuous_on_arsinh' [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7015
  fixes f :: "real \<Rightarrow> real"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7016
  assumes "continuous_on A f"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7017
  shows   "continuous_on A (\<lambda>x. arsinh (f x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7018
  by (rule continuous_on_compose2[OF continuous_on_arsinh assms]) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7019
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7020
lemma continuous_on_arcosh' [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7021
  fixes f :: "real \<Rightarrow> real"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7022
  assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7023
  shows   "continuous_on A (\<lambda>x. arcosh (f x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7024
  by (rule continuous_on_compose2[OF continuous_on_arcosh assms(1) order.refl])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7025
     (use assms(2) in auto)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7026
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7027
lemma continuous_on_artanh' [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7028
  fixes f :: "real \<Rightarrow> real"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7029
  assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {-1<..<1}"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7030
  shows   "continuous_on A (\<lambda>x. artanh (f x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7031
  by (rule continuous_on_compose2[OF continuous_on_artanh assms(1) order.refl])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7032
     (use assms(2) in auto)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7033
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7034
lemma isCont_arsinh [continuous_intros]: "isCont arsinh (x :: real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7035
  using continuous_on_arsinh[of UNIV] by (auto simp: continuous_on_eq_continuous_at)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7036
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7037
lemma isCont_arcosh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7038
  assumes "x > 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7039
  shows   "isCont arcosh (x :: real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7040
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7041
  have "continuous_on {1::real<..} arcosh"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7042
    by (rule continuous_on_arcosh) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7043
  with assms show ?thesis by (auto simp: continuous_on_eq_continuous_at)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7044
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7045
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7046
lemma isCont_artanh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7047
  assumes "x > -1" "x < 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7048
  shows   "isCont artanh (x :: real)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7049
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7050
  have "continuous_on {-1<..<(1::real)} artanh"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7051
    by (rule continuous_on_artanh) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7052
  with assms show ?thesis by (auto simp: continuous_on_eq_continuous_at)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7053
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7054
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7055
lemma tendsto_arsinh [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. arsinh (f x)) \<longlongrightarrow> arsinh a) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7056
  for f :: "_ \<Rightarrow> real"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7057
  by (rule isCont_tendsto_compose [OF isCont_arsinh])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7058
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7059
lemma tendsto_arcosh_strong [tendsto_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7060
  fixes f :: "_ \<Rightarrow> real"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7061
  assumes "(f \<longlongrightarrow> a) F" "a \<ge> 1" "eventually (\<lambda>x. f x \<ge> 1) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7062
  shows   "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7063
  by (rule continuous_on_tendsto_compose[OF continuous_on_arcosh[OF order.refl]])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7064
     (use assms in auto)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7065
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7066
lemma tendsto_arcosh:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7067
  fixes f :: "_ \<Rightarrow> real"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7068
  assumes "(f \<longlongrightarrow> a) F" "a > 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7069
  shows "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7070
  by (rule isCont_tendsto_compose [OF isCont_arcosh]) (use assms in auto)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7071
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7072
lemma tendsto_arcosh_at_left_1: "(arcosh \<longlongrightarrow> 0) (at_right (1::real))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7073
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7074
  have "(arcosh \<longlongrightarrow> arcosh 1) (at_right (1::real))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7075
    by (rule tendsto_arcosh_strong) (auto simp: eventually_at intro!: exI[of _ 1])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7076
  thus ?thesis by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7077
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7078
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  7079
lemma tendsto_artanh [tendsto_intros]:
67574
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7080
  fixes f :: "'a \<Rightarrow> real"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7081
  assumes "(f \<longlongrightarrow> a) F" "a > -1" "a < 1"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7082
  shows   "((\<lambda>x. artanh (f x)) \<longlongrightarrow> artanh a) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7083
  by (rule isCont_tendsto_compose [OF isCont_artanh]) (use assms in auto)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7084
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7085
lemma continuous_arsinh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7086
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arsinh (f x :: real))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7087
  unfolding continuous_def by (rule tendsto_arsinh)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7088
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7089
(* TODO: This rule does not work for one-sided continuity at 1 *)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7090
lemma continuous_arcosh_strong [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7091
  assumes "continuous F f" "eventually (\<lambda>x. f x \<ge> 1) F"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7092
  shows   "continuous F (\<lambda>x. arcosh (f x :: real))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7093
proof (cases "F = bot")
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7094
  case False
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7095
  show ?thesis
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7096
    unfolding continuous_def
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7097
  proof (intro tendsto_arcosh_strong)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7098
    show "1 \<le> f (Lim F (\<lambda>x. x))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7099
      using assms False unfolding continuous_def by (rule tendsto_lowerbound)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7100
  qed (insert assms, auto simp: continuous_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7101
qed auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7102
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7103
lemma continuous_arcosh:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7104
  "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) > 1 \<Longrightarrow> continuous F (\<lambda>x. arcosh (f x :: real))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7105
  unfolding continuous_def by (rule tendsto_arcosh) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7106
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7107
lemma continuous_artanh [continuous_intros]:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7108
  "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<in> {-1<..<1} \<Longrightarrow> continuous F (\<lambda>x. artanh (f x :: real))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7109
  unfolding continuous_def by (rule tendsto_artanh) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7110
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7111
lemma arsinh_real_at_top:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7112
  "filterlim (arsinh :: real \<Rightarrow> real) at_top at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7113
proof (subst filterlim_cong[OF refl refl])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7114
  show "filterlim (\<lambda>x. ln (x + sqrt (1 + x\<^sup>2))) at_top at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7115
    by (intro filterlim_compose[OF ln_at_top filterlim_at_top_add_at_top] filterlim_ident
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7116
              filterlim_compose[OF sqrt_at_top] filterlim_tendsto_add_at_top[OF tendsto_const]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7117
              filterlim_pow_at_top) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7118
qed (auto intro!: eventually_mono[OF eventually_ge_at_top[of 1]] simp: arsinh_real_def add_ac)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7119
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7120
lemma arsinh_real_at_bot:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7121
  "filterlim (arsinh :: real \<Rightarrow> real) at_bot at_bot"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7122
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7123
  have "filterlim (\<lambda>x::real. -arsinh x) at_bot at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7124
    by (subst filterlim_uminus_at_top [symmetric]) (rule arsinh_real_at_top)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7125
  also have "(\<lambda>x::real. -arsinh x) = (\<lambda>x. arsinh (-x))" by simp
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7126
  finally show ?thesis
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7127
    by (subst filterlim_at_bot_mirror)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7128
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7129
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7130
lemma arcosh_real_at_top:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7131
  "filterlim (arcosh :: real \<Rightarrow> real) at_top at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7132
proof (subst filterlim_cong[OF refl refl])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7133
  show "filterlim (\<lambda>x. ln (x + sqrt (-1 + x\<^sup>2))) at_top at_top"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7134
    by (intro filterlim_compose[OF ln_at_top filterlim_at_top_add_at_top] filterlim_ident
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7135
              filterlim_compose[OF sqrt_at_top] filterlim_tendsto_add_at_top[OF tendsto_const]
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7136
              filterlim_pow_at_top) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7137
qed (auto intro!: eventually_mono[OF eventually_ge_at_top[of 1]] simp: arcosh_real_def)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7138
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7139
lemma artanh_real_at_left_1:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7140
  "filterlim (artanh :: real \<Rightarrow> real) at_top (at_left 1)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7141
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7142
  have *: "filterlim (\<lambda>x::real. (1 + x) / (1 - x)) at_top (at_left 1)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7143
    by (rule LIM_at_top_divide)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7144
       (auto intro!: tendsto_eq_intros eventually_mono[OF eventually_at_left_real[of 0]])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7145
  have "filterlim (\<lambda>x::real. (1/2) * ln ((1 + x) / (1 - x))) at_top (at_left 1)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7146
    by (intro filterlim_tendsto_pos_mult_at_top[OF tendsto_const] *
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7147
                 filterlim_compose[OF ln_at_top]) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7148
  also have "(\<lambda>x::real. (1/2) * ln ((1 + x) / (1 - x))) = artanh"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7149
    by (simp add: artanh_def [abs_def])
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7150
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7151
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7152
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7153
lemma artanh_real_at_right_1:
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7154
  "filterlim (artanh :: real \<Rightarrow> real) at_bot (at_right (-1))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7155
proof -
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7156
  have "?thesis \<longleftrightarrow> filterlim (\<lambda>x::real. -artanh x) at_top (at_right (-1))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7157
    by (simp add: filterlim_uminus_at_bot)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7158
  also have "\<dots> \<longleftrightarrow> filterlim (\<lambda>x::real. artanh (-x)) at_top (at_right (-1))"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7159
    by (intro filterlim_cong refl eventually_mono[OF eventually_at_right_real[of "-1" "1"]]) auto
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7160
  also have "\<dots> \<longleftrightarrow> filterlim (artanh :: real \<Rightarrow> real) at_top (at_left 1)"
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7161
    by (simp add: filterlim_at_left_to_right)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7162
  also have \<dots> by (rule artanh_real_at_left_1)
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7163
  finally show ?thesis .
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7164
qed
4a3d657adc62 Added hyperbolic functions
eberlm <eberlm@in.tum.de>
parents: 67573
diff changeset
  7165
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7166
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7167
subsection \<open>Simprocs for root and power literals\<close>
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7168
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7169
lemma numeral_powr_numeral_real [simp]:
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7170
  "numeral m powr numeral n = (numeral m ^ numeral n :: real)"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7171
  by (simp add: powr_numeral)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7172
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7173
context
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7174
begin
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  7175
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  7176
private lemma sqrt_numeral_simproc_aux:
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7177
  assumes "m * m \<equiv> n"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7178
  shows   "sqrt (numeral n :: real) \<equiv> numeral m"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7179
proof -
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7180
  have "numeral n \<equiv> numeral m * (numeral m :: real)" by (simp add: assms [symmetric])
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7181
  moreover have "sqrt \<dots> \<equiv> numeral m" by (subst real_sqrt_abs2) simp
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7182
  ultimately show "sqrt (numeral n :: real) \<equiv> numeral m" by simp
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7183
qed
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7184
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  7185
private lemma root_numeral_simproc_aux:
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7186
  assumes "Num.pow m n \<equiv> x"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7187
  shows   "root (numeral n) (numeral x :: real) \<equiv> numeral m"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7188
  by (subst assms [symmetric], subst numeral_pow, subst real_root_pos2) simp_all
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7189
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7190
private lemma powr_numeral_simproc_aux:
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7191
  assumes "Num.pow y n = x"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7192
  shows   "numeral x powr (m / numeral n :: real) \<equiv> numeral y powr m"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7193
  by (subst assms [symmetric], subst numeral_pow, subst powr_numeral [symmetric])
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7194
     (simp, subst powr_powr, simp_all)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7195
68601
7828f3b85156 de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  7196
private lemma numeral_powr_inverse_eq:
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7197
  "numeral x powr (inverse (numeral n)) = numeral x powr (1 / numeral n :: real)"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7198
  by simp
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7199
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7200
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7201
ML \<open>
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7202
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7203
signature ROOT_NUMERAL_SIMPROC = sig
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7204
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7205
val sqrt : int option -> int -> int option
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7206
val sqrt' : int option -> int -> int option
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7207
val nth_root : int option -> int -> int -> int option
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7208
val nth_root' : int option -> int -> int -> int option
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7209
val sqrt_simproc : Proof.context -> cterm -> thm option
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7210
val root_simproc : int * int -> Proof.context -> cterm -> thm option
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7211
val powr_simproc : int * int -> Proof.context -> cterm -> thm option
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7212
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
  7213
end
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7214
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7215
structure Root_Numeral_Simproc : ROOT_NUMERAL_SIMPROC = struct
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7216
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7217
fun iterate NONE p f x =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7218
      let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7219
        fun go x = if p x then x else go (f x)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7220
      in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7221
        SOME (go x)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7222
      end
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7223
  | iterate (SOME threshold) p f x =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7224
      let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7225
        fun go (threshold, x) = 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7226
          if p x then SOME x else if threshold = 0 then NONE else go (threshold - 1, f x)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7227
      in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7228
        go (threshold, x)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7229
      end  
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7230
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7231
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7232
fun nth_root _ 1 x = SOME x
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7233
  | nth_root _ _ 0 = SOME 0
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7234
  | nth_root _ _ 1 = SOME 1
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7235
  | nth_root threshold n x =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7236
  let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7237
    fun newton_step y = ((n - 1) * y + x div Integer.pow (n - 1) y) div n
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7238
    fun is_root y = Integer.pow n y <= x andalso x < Integer.pow n (y + 1)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7239
  in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7240
    if x < n then
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7241
      SOME 1
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7242
    else if x < Integer.pow n 2 then 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7243
      SOME 1 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7244
    else 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7245
      let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7246
        val y = Real.floor (Math.pow (Real.fromInt x, Real.fromInt 1 / Real.fromInt n))
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7247
      in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7248
        if is_root y then
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7249
          SOME y
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7250
        else
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7251
          iterate threshold is_root newton_step ((x + n - 1) div n)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7252
      end
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7253
  end
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7254
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7255
fun nth_root' _ 1 x = SOME x
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7256
  | nth_root' _ _ 0 = SOME 0
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7257
  | nth_root' _ _ 1 = SOME 1
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7258
  | nth_root' threshold n x = if x < n then NONE else if x < Integer.pow n 2 then NONE else
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7259
      case nth_root threshold n x of
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7260
        NONE => NONE
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7261
      | SOME y => if Integer.pow n y = x then SOME y else NONE
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7262
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7263
fun sqrt _ 0 = SOME 0
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7264
  | sqrt _ 1 = SOME 1
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7265
  | sqrt threshold n =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7266
    let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7267
      fun aux (a, b) = if n >= b * b then aux (b, b * b) else (a, b)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7268
      val (lower_root, lower_n) = aux (1, 2)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7269
      fun newton_step x = (x + n div x) div 2
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7270
      fun is_sqrt r = r*r <= n andalso n < (r+1)*(r+1)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7271
      val y = Real.floor (Math.sqrt (Real.fromInt n))
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7272
    in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7273
      if is_sqrt y then 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7274
        SOME y
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7275
      else
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7276
        Option.mapPartial (iterate threshold is_sqrt newton_step o (fn x => x * lower_root)) 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7277
          (sqrt threshold (n div lower_n))
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7278
    end
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7280
fun sqrt' threshold x =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7281
  case sqrt threshold x of
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7282
    NONE => NONE
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7283
  | SOME y => if y * y = x then SOME y else NONE
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7284
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7285
fun sqrt_simproc ctxt ct =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7286
  let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7287
    val n = ct |> Thm.term_of |> dest_comb |> snd |> dest_comb |> snd |> HOLogic.dest_numeral
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7288
  in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7289
    case sqrt' (SOME 10000) n of
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7290
      NONE => NONE
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7291
    | SOME m => 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7292
        SOME (Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt o HOLogic.mk_numeral) [m, n])
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7293
                  @{thm sqrt_numeral_simproc_aux})
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7294
  end
68642
d812b6ee711b Made simproc for sqrt/root of numeral more robust
Manuel Eberl <eberlm@in.tum.de>
parents: 68638
diff changeset
  7295
    handle TERM _ => NONE
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7296
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7297
fun root_simproc (threshold1, threshold2) ctxt ct =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7298
  let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7299
    val [n, x] = 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7300
      ct |> Thm.term_of |> strip_comb |> snd |> map (dest_comb #> snd #> HOLogic.dest_numeral)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7301
  in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7302
    if n > threshold1 orelse x > threshold2 then NONE else
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7303
      case nth_root' (SOME 100) n x of
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7304
        NONE => NONE
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7305
      | SOME m => 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7306
          SOME (Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt o HOLogic.mk_numeral) [m, n, x])
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7307
            @{thm root_numeral_simproc_aux})
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7308
  end
68642
d812b6ee711b Made simproc for sqrt/root of numeral more robust
Manuel Eberl <eberlm@in.tum.de>
parents: 68638
diff changeset
  7309
    handle TERM _ => NONE
d812b6ee711b Made simproc for sqrt/root of numeral more robust
Manuel Eberl <eberlm@in.tum.de>
parents: 68638
diff changeset
  7310
         | Match => NONE
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7311
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7312
fun powr_simproc (threshold1, threshold2) ctxt ct =
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7313
  let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7314
    val eq_thm = Conv.try_conv (Conv.rewr_conv @{thm numeral_powr_inverse_eq}) ct
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7315
    val ct = Thm.dest_equals_rhs (Thm.cprop_of eq_thm)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7316
    val (_, [x, t]) = strip_comb (Thm.term_of ct)
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7317
    val (_, [m, n]) = strip_comb t
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7318
    val [x, n] = map (dest_comb #> snd #> HOLogic.dest_numeral) [x, n]
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7319
  in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7320
    if n > threshold1 orelse x > threshold2 then NONE else
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7321
      case nth_root' (SOME 100) n x of
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7322
        NONE => NONE
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7323
      | SOME y => 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7324
          let
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7325
            val [y, n, x] = map HOLogic.mk_numeral [y, n, x]
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7326
            val thm = Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt) [y, n, x, m])
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7327
              @{thm powr_numeral_simproc_aux}
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7328
          in
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7329
            SOME (@{thm transitive} OF [eq_thm, thm])
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7330
          end
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7331
  end
68642
d812b6ee711b Made simproc for sqrt/root of numeral more robust
Manuel Eberl <eberlm@in.tum.de>
parents: 68638
diff changeset
  7332
    handle TERM _ => NONE
d812b6ee711b Made simproc for sqrt/root of numeral more robust
Manuel Eberl <eberlm@in.tum.de>
parents: 68638
diff changeset
  7333
         | Match => NONE
66279
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7334
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7335
end
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7336
\<close>
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7337
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7338
end
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7339
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7340
simproc_setup sqrt_numeral ("sqrt (numeral n)") = 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7341
  \<open>K Root_Numeral_Simproc.sqrt_simproc\<close>
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7342
  
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7343
simproc_setup root_numeral ("root (numeral n) (numeral x)") = 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7344
  \<open>K (Root_Numeral_Simproc.root_simproc (200, Integer.pow 200 2))\<close>
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7345
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7346
simproc_setup powr_divide_numeral 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7347
  ("numeral x powr (m / numeral n :: real)" | "numeral x powr (inverse (numeral n) :: real)") = 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7348
    \<open>K (Root_Numeral_Simproc.powr_simproc (200, Integer.pow 200 2))\<close>
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7349
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7350
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7351
lemma "root 100 1267650600228229401496703205376 = 2"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7352
  by simp
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7353
    
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7354
lemma "sqrt 196 = 14" 
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7355
  by simp
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7356
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7357
lemma "256 powr (7 / 4 :: real) = 16384"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7358
  by simp
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7359
    
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7360
lemma "27 powr (inverse 3) = (3::real)"
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7361
  by simp
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7362
2dba15d3c402 Simprocs for roots of numerals
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  7363
end