src/HOL/Transcendental.thy
author noschinl
Sat, 25 May 2013 13:59:08 +0200
changeset 52139 40fe6b80b481
parent 51641 cd05e9fcc63d
child 53015 a1119cf551e8
permissions -rw-r--r--
add lemma
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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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header{*Power Series, Transcendental Functions etc.*}
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theory Transcendental
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imports Fact Series Deriv NthRoot
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begin
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subsection {* Properties of Power Series *}
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lemma lemma_realpow_diff:
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  fixes y :: "'a::monoid_mult"
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  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
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proof -
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  assume "p \<le> n"
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  hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
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  thus ?thesis by (simp add: power_commutes)
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qed
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lemma lemma_realpow_diff_sumr:
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  fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
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     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
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      y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
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         del: setsum_op_ivl_Suc)
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lemma lemma_realpow_diff_sumr2:
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  fixes y :: "'a::{comm_ring,monoid_mult}" shows
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     "x ^ (Suc n) - y ^ (Suc n) =
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      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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apply (induct n, simp)
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apply (simp del: setsum_op_ivl_Suc)
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apply (subst setsum_op_ivl_Suc)
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apply (subst lemma_realpow_diff_sumr)
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apply (simp add: distrib_left del: setsum_op_ivl_Suc)
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apply (subst mult_left_commute [of "x - y"])
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apply (erule subst)
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apply (simp add: algebra_simps)
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done
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lemma lemma_realpow_rev_sumr:
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     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
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      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
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apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
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apply (rule inj_onI, simp)
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apply auto
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apply (rule_tac x="n - x" in image_eqI, simp, simp)
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done
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
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x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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lemma powser_insidea:
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  fixes x z :: "'a::real_normed_field"
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  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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  assumes 2: "norm z < norm x"
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  shows "summable (\<lambda>n. norm (f n * z ^ n))"
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proof -
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  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
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  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
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    by (rule summable_LIMSEQ_zero)
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  hence "convergent (\<lambda>n. f n * x ^ n)"
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    by (rule convergentI)
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  hence "Cauchy (\<lambda>n. f n * x ^ n)"
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    by (rule convergent_Cauchy)
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  hence "Bseq (\<lambda>n. f n * x ^ n)"
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    by (rule Cauchy_Bseq)
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  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
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    by (simp add: Bseq_def, safe)
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  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
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                   K * norm (z ^ n) * inverse (norm (x ^ n))"
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  proof (intro exI allI impI)
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    fix n::nat assume "0 \<le> n"
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    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
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          norm (f n * x ^ n) * norm (z ^ n)"
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      by (simp add: norm_mult abs_mult)
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    also have "\<dots> \<le> K * norm (z ^ n)"
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      by (simp only: mult_right_mono 4 norm_ge_zero)
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    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
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      by (simp add: x_neq_0)
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    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
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      by (simp only: mult_assoc)
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    finally show "norm (norm (f n * z ^ n)) \<le>
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                  K * norm (z ^ n) * inverse (norm (x ^ n))"
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      by (simp add: mult_le_cancel_right x_neq_0)
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  qed
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  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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  proof -
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    from 2 have "norm (norm (z * inverse x)) < 1"
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      using x_neq_0
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      by (simp add: nonzero_norm_divide divide_inverse [symmetric])
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    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
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      by (rule summable_geometric)
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    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
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      by (rule summable_mult)
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    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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      using x_neq_0
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      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
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                    power_inverse norm_power mult_assoc)
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  qed
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  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
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    by (rule summable_comparison_test)
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qed
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lemma powser_inside:
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  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
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     "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
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      ==> summable (%n. f(n) * (z ^ n))"
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by (rule powser_insidea [THEN summable_norm_cancel])
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lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
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  "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
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   (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
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proof (induct n)
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  case (Suc n)
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  have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
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        (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
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    using Suc.hyps unfolding One_nat_def by auto
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  also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
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  finally show ?case .
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qed auto
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lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
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  shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
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  unfolding sums_def
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proof (rule LIMSEQ_I)
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  fix r :: real assume "0 < r"
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  from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
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diff changeset
   134
  obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   135
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   136
  let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   137
  { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   138
    have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< 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
   139
      using sum_split_even_odd by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   140
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   141
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   142
    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
   143
    proof (cases "even m")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   144
      case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   145
    next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   146
      case False hence "even (Suc m)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   147
      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   148
      have eq: "Suc (2 * (m div 2)) = m" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   149
      hence "even (2 * (m div 2))" using `odd m` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   150
      have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   151
      also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   152
      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
   153
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   154
    ultimately have "(norm (?SUM m - x) < r)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   155
  }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   156
  thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   157
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   158
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   159
lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   160
  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
   161
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   162
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   163
  { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   164
      by (cases B) auto } note if_sum = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   165
  have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   166
  {
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   167
    have "?s 0 = 0" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   168
    have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 38642
diff changeset
   169
    have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" 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
   170
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   171
    have "?s sums y" using sums_if'[OF `f sums y`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   172
    from this[unfolded sums_def, THEN LIMSEQ_Suc]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   173
    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   174
      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   175
                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
31148
7ba7c1f8bc22 Cleaned up Parity a little
nipkow
parents: 31017
diff changeset
   176
                even_Suc Suc_m1 if_eq .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   177
  } from sums_add[OF g_sums this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   178
  show ?thesis unfolding if_sum .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   179
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   180
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   181
subsection {* Alternating series test / Leibniz formula *}
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   182
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   183
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
   184
  fixes a :: "nat \<Rightarrow> real"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   185
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   186
  shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   187
             ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   188
  (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   189
proof -
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   190
  have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_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
   191
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   192
  have "\<forall> n. ?f n \<le> ?f (Suc n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   193
  proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   194
  moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   195
  have "\<forall> n. ?g (Suc n) \<le> ?g n"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   196
  proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   197
    unfolding One_nat_def by auto qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   198
  moreover
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   199
  have "\<forall> n. ?f n \<le> ?g n"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   200
  proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   201
    unfolding One_nat_def by auto qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   202
  moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   203
  have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   204
  proof (rule LIMSEQ_I)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   205
    fix r :: real assume "0 < r"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   206
    with `a ----> 0`[THEN LIMSEQ_D]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   207
    obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   208
    hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   209
    thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   210
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   211
  ultimately
51477
2990382dc066 clean up lemma_nest_unique and renamed to nested_sequence_unique
hoelzl
parents: 50347
diff changeset
   212
  show ?thesis by (rule nested_sequence_unique)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   213
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   214
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   215
lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   216
  assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   217
  and a_monotone: "\<And> n. a (Suc n) \<le> a n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   218
  shows summable: "summable (\<lambda> n. (-1)^n * a n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   219
  and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   220
  and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   221
  and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   222
  and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   223
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   224
  let "?S n" = "(-1)^n * a n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   225
  let "?P n" = "\<Sum>i=0..<n. ?S i"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   226
  let "?f n" = "?P (2 * n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   227
  let "?g n" = "?P (2 * n + 1)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   228
  obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   229
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   230
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   231
  let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   232
  have "?Sa ----> l"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   233
  proof (rule LIMSEQ_I)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   234
    fix r :: real assume "0 < r"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   235
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   236
    with `?f ----> l`[THEN LIMSEQ_D]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   237
    obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   238
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   239
    from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   240
    obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   241
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   242
    { fix n :: nat
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   243
      assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   244
      have "norm (?Sa n - l) < r"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   245
      proof (cases "even n")
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   246
        case True from even_nat_div_two_times_two[OF this]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   247
        have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   248
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   249
        from f[OF this]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   250
        show ?thesis 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
   251
      next
35213
b9866ad4e3be fix looping call to simplifier
huffman
parents: 35038
diff changeset
   252
        case False hence "even (n - 1)" by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   253
        from even_nat_div_two_times_two[OF this]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   254
        have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   255
        hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   256
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   257
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   258
        from g[OF this]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   259
        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost 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
   260
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   261
    }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   262
    thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   263
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   264
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   265
  thus "summable ?S" using summable_def by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   266
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   267
  have "l = suminf ?S" using sums_unique[OF sums_l] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   268
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   269
  { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   270
  { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   271
  show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   272
  show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   273
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   274
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   275
theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   276
  assumes a_zero: "a ----> 0" and "monoseq a"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   277
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   278
  and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   279
  and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   280
  and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   281
  and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   282
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   283
  have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   284
  proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   285
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   286
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   287
    { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   288
    note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   289
    from leibniz[OF mono]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   290
    show ?thesis using `0 \<le> a 0` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   291
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   292
    let ?a = "\<lambda> n. - a n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   293
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   294
    with monoseq_le[OF `monoseq a` `a ----> 0`]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   295
    have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   296
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   297
    { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   298
    note monotone = this
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44319
diff changeset
   299
    note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   300
    have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   301
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   302
    from this[THEN sums_minus]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   303
    have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   304
    hence ?summable unfolding summable_def by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   305
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   306
    have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   307
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   308
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   309
    have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   310
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   311
    have ?pos using `0 \<le> ?a 0` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   312
    moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44319
diff changeset
   313
    moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] 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
   314
    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
   315
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   316
  from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   317
       this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   318
  show ?summable and ?pos and ?neg and ?f and ?g .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   319
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   320
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   321
subsection {* Term-by-Term Differentiability of Power Series *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   322
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   323
definition
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   324
  diffs :: "(nat => 'a::ring_1) => nat => 'a" where
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   325
  "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   326
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   327
text{*Lemma about distributing negation over it*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   328
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   329
by (simp add: diffs_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   330
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   331
lemma sums_Suc_imp:
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   332
  assumes f: "f 0 = 0"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   333
  shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   334
unfolding sums_def
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   335
apply (rule LIMSEQ_imp_Suc)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   336
apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   337
apply (simp only: setsum_shift_bounds_Suc_ivl)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   338
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   339
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   340
lemma diffs_equiv:
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   341
  fixes x :: "'a::{real_normed_vector, ring_1}"
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   342
  shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   343
      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   344
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   345
unfolding diffs_def
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   346
apply (drule summable_sums)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   347
apply (rule sums_Suc_imp, simp_all)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   348
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   349
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   350
lemma lemma_termdiff1:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   351
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   352
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   353
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 38642
diff changeset
   354
by(auto simp add: algebra_simps power_add [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   355
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   356
lemma sumr_diff_mult_const2:
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   357
  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   358
by (simp add: setsum_subtractf)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   359
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   360
lemma lemma_termdiff2:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   361
  fixes h :: "'a :: {field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   362
  assumes h: "h \<noteq> 0" shows
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   363
  "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   364
   h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   365
        (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   366
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   367
apply (simp add: right_diff_distrib diff_divide_distrib h)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   368
apply (simp add: mult_assoc [symmetric])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   369
apply (cases "n", simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   370
apply (simp add: lemma_realpow_diff_sumr2 h
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   371
                 right_diff_distrib [symmetric] mult_assoc
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30082
diff changeset
   372
            del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   373
apply (subst lemma_realpow_rev_sumr)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   374
apply (subst sumr_diff_mult_const2)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   375
apply simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   376
apply (simp only: lemma_termdiff1 setsum_right_distrib)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   377
apply (rule setsum_cong [OF refl])
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   378
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   379
apply (clarify)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   380
apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30082
diff changeset
   381
            del: setsum_op_ivl_Suc power_Suc)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   382
apply (subst mult_assoc [symmetric], subst power_add [symmetric])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   383
apply (simp add: mult_ac)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   384
done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   385
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   386
lemma real_setsum_nat_ivl_bounded2:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   387
  fixes K :: "'a::linordered_semidom"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   388
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   389
  assumes K: "0 \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   390
  shows "setsum f {0..<n-k} \<le> of_nat n * K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   391
apply (rule order_trans [OF setsum_mono])
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   392
apply (rule f, simp)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   393
apply (simp add: mult_right_mono K)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   394
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   395
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   396
lemma lemma_termdiff3:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   397
  fixes h z :: "'a::{real_normed_field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   398
  assumes 1: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   399
  assumes 2: "norm z \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   400
  assumes 3: "norm (z + h) \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   401
  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   402
          \<le> 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
   403
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   404
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   405
        norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   406
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   407
    apply (subst lemma_termdiff2 [OF 1])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   408
    apply (subst norm_mult)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   409
    apply (rule mult_commute)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   410
    done
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   411
  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
   412
  proof (rule mult_right_mono [OF _ norm_ge_zero])
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   413
    from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   414
    have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   415
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   416
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   417
      apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   418
      done
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   419
    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   420
              (z + h) ^ q * z ^ (n - 2 - q))
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   421
          \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   422
      apply (intro
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   423
         order_trans [OF norm_setsum]
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   424
         real_setsum_nat_ivl_bounded2
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   425
         mult_nonneg_nonneg
47489
04e7d09ade7a tuned some proofs;
huffman
parents: 47108
diff changeset
   426
         of_nat_0_le_iff
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   427
         zero_le_power K)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   428
      apply (rule le_Kn, simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   429
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   430
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   431
  also have "\<dots> = 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
   432
    by (simp only: mult_assoc)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   433
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   434
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   435
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   436
lemma lemma_termdiff4:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   437
  fixes f :: "'a::{real_normed_field} \<Rightarrow>
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   438
              'b::real_normed_vector"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   439
  assumes k: "0 < (k::real)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   440
  assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   441
  shows "f -- 0 --> 0"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31271
diff changeset
   442
unfolding LIM_eq diff_0_right
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   443
proof (safe)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   444
  let ?h = "of_real (k / 2)::'a"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   445
  have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   446
  hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   447
  hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   448
  hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   449
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   450
  fix r::real assume r: "0 < r"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   451
  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   452
  proof (cases)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   453
    assume "K = 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   454
    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   455
      by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   456
    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   457
  next
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   458
    assume K_neq_zero: "K \<noteq> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   459
    with zero_le_K have K: "0 < K" by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   460
    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   461
    proof (rule exI, safe)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   462
      from k r K show "0 < min k (r * inverse K / 2)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   463
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   464
    next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   465
      fix x::'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   466
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   467
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   468
        by simp_all
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   469
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   470
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   471
        by (rule mult_strict_left_mono)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   472
      also have "\<dots> = r / 2"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   473
        using K_neq_zero by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   474
      also have "r / 2 < r"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   475
        using r by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   476
      finally show "norm (f x) < r" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   477
    qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   478
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   479
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   480
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   481
lemma lemma_termdiff5:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   482
  fixes g :: "'a::{real_normed_field} \<Rightarrow>
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   483
              nat \<Rightarrow> 'b::banach"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   484
  assumes k: "0 < (k::real)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   485
  assumes f: "summable f"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   486
  assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   487
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   488
proof (rule lemma_termdiff4 [OF k])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   489
  fix h::'a assume "h \<noteq> 0" and "norm h < k"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   490
  hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   491
    by (simp add: le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   492
  hence "\<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
   493
    by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   494
  moreover from f have B: "summable (\<lambda>n. f n * norm h)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   495
    by (rule summable_mult2)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   496
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   497
    by (rule summable_comparison_test)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   498
  hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   499
    by (rule summable_norm)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   500
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   501
    by (rule summable_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   502
  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
   503
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   504
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   505
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   506
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   507
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   508
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   509
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   510
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   511
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   512
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   513
  assumes 2: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   514
  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   515
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   516
proof -
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   517
  from dense [OF 2]
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   518
  obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   519
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   520
    by (rule order_le_less_trans)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   521
  hence r_neq_0: "r \<noteq> 0" by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   522
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   523
  proof (rule lemma_termdiff5)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   524
    show "0 < r - norm x" using r1 by simp
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   525
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   526
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   527
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   528
    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
   529
      by (rule powser_insidea)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   530
    hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   531
      using r
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   532
      by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   533
    hence "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
   534
      by (rule diffs_equiv [THEN sums_summable])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   535
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   536
      = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   537
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   538
      apply (simp add: diffs_def)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   539
      apply (case_tac n, simp_all add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   540
      done
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   541
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   542
      (\<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
   543
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   544
    also have
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   545
      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   546
           r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   547
       (\<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
   548
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   549
      apply (case_tac "n", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   550
      apply (case_tac "nat", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   551
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   552
      done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   553
    finally show
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   554
      "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
   555
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   556
    fix h::'a and n::nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   557
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   558
    assume "norm h < r - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   559
    hence "norm x + norm h < r" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   560
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   561
      by (rule order_le_less_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   562
    show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   563
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   564
      apply (simp only: norm_mult mult_assoc)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   565
      apply (rule mult_left_mono [OF _ norm_ge_zero])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   566
      apply (simp (no_asm) add: mult_assoc [symmetric])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   567
      apply (rule lemma_termdiff3)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   568
      apply (rule h)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   569
      apply (rule r1 [THEN order_less_imp_le])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   570
      apply (rule xh [THEN order_less_imp_le])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   571
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   572
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   573
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   574
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   575
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   576
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   577
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   578
  assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   579
  assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   580
  assumes 4: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   581
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   582
unfolding deriv_def
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   583
proof (rule LIM_zero_cancel)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   584
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   585
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   586
  proof (rule LIM_equal2)
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   587
    show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   588
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   589
    fix h :: 'a
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   590
    assume "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   591
    assume "norm (h - 0) < norm K - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   592
    hence "norm x + norm h < norm K" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   593
    hence 5: "norm (x + h) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   594
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   595
    have A: "summable (\<lambda>n. c n * x ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   596
      by (rule powser_inside [OF 1 4])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   597
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   598
      by (rule powser_inside [OF 1 5])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   599
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   600
      by (rule powser_inside [OF 2 4])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   601
    show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   602
             - (\<Sum>n. diffs c n * x ^ n) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   603
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   604
      apply (subst sums_unique [OF diffs_equiv [OF C]])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   605
      apply (subst suminf_diff [OF B A])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   606
      apply (subst suminf_divide [symmetric])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   607
      apply (rule summable_diff [OF B A])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   608
      apply (subst suminf_diff)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   609
      apply (rule summable_divide)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   610
      apply (rule summable_diff [OF B A])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   611
      apply (rule sums_summable [OF diffs_equiv [OF C]])
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   612
      apply (rule arg_cong [where f="suminf"], rule ext)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29171
diff changeset
   613
      apply (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   614
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   615
  next
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   616
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   617
               of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   618
        by (rule termdiffs_aux [OF 3 4])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   619
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   620
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   621
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   622
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   623
subsection {* Derivability of power series *}
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   624
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   625
lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   626
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   627
  and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   628
  and "summable (f' x0)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   629
  and "summable L" and L_def: "\<And> n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar> f x n - f y n \<bar> \<le> L n * \<bar> x - y \<bar>"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   630
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   631
  unfolding deriv_def
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   632
proof (rule LIM_I)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   633
  fix r :: real assume "0 < r" hence "0 < r/3" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   634
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   635
  obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < 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
   636
    using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   637
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   638
  obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < 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
   639
    using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   640
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   641
  let ?N = "Suc (max N_L N_f')"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   642
  have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   643
    L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   644
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   645
  let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   646
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   647
  let ?r = "r / (3 * real ?N)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   648
  have "0 < 3 * real ?N" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   649
  from divide_pos_pos[OF `0 < r` this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   650
  have "0 < ?r" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   651
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   652
  let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   653
  def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   654
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   655
  have "0 < S'" unfolding S'_def
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   656
  proof (rule iffD2[OF Min_gr_iff])
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   657
    show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   658
    proof (rule ballI)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   659
      fix x assume "x \<in> ?s ` {0..<?N}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   660
      then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   661
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   662
      obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   663
      have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   664
      thus "0 < x" unfolding `x = ?s n` .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   665
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   666
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   667
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   668
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   669
  hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   670
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   671
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   672
  { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   673
    hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   674
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   675
    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
   676
    note div_smbl = summable_divide[OF diff_smbl]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   677
    note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   678
    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
   679
    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
   680
    note div_shft_smbl = summable_divide[OF diff_shft_smbl]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   681
    note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   682
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   683
    { fix n
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   684
      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   685
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   686
      hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   687
    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   688
    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   689
    have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   690
    hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   691
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   692
    have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   693
    also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   694
    proof (rule setsum_strict_mono)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   695
      fix n assume "n \<in> { 0 ..< ?N}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   696
      have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   697
      also have "S \<le> S'" using `S \<le> S'` .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   698
      also have "S' \<le> ?s n" 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
   699
      proof (rule Min_le_iff[THEN iffD2])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   700
        have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   701
        thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" 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
   702
      qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   703
      finally have "\<bar> x \<bar> < ?s n" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   704
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   705
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   706
      have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   707
      with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   708
      show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   709
    qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   710
    also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   711
    also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   712
    also have "\<dots> = r/3" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   713
    finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   714
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   715
    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   716
    have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   717
                    \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   718
    also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   719
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   720
    also have "\<dots> < r /3 + r/3 + r/3"
36842
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
   721
      using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
   722
      by (rule add_strict_mono [OF add_less_le_mono])
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   723
    finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   724
      by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   725
  } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   726
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   727
    unfolding real_norm_def diff_0_right by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   728
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   729
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   730
lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   731
  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   732
  and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   733
  shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   734
  (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   735
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   736
  { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   737
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   738
    have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   739
    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
   740
      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
   741
      proof -
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   742
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   743
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   744
        have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   745
        from powser_insidea[OF converges[OF in_Rball] this] 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
   746
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   747
      { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   748
        show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   749
        proof -
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   750
          have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   751
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   752
          also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   753
          proof (rule mult_left_mono)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   754
            have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   755
            also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   756
            proof (rule setsum_mono)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   757
              fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   758
              { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   759
                hence "\<bar>x\<bar> \<le> R'"  by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   760
                hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   761
              } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   762
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   763
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   764
            qed
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   765
            also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   766
            finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   767
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   768
          qed
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 36776
diff changeset
   769
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   770
          finally show ?thesis .
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   771
        qed }
31881
eba74a5790d2 use DERIV_intros
hoelzl
parents: 31880
diff changeset
   772
      { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   773
          by (auto intro!: DERIV_intros simp del: power_Suc) }
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   774
      { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   775
        have "summable (\<lambda> n. f n * x^n)"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   776
        proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   777
          fix n
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   778
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   779
          show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   780
            by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   781
        qed
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 36776
diff changeset
   782
        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   783
        show "summable (?f x)" 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
   784
      show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   785
      show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   786
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   787
  } note for_subinterval = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   788
  let ?R = "(R + \<bar>x0\<bar>) / 2"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   789
  have "\<bar>x0\<bar> < ?R" using assms by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   790
  hence "- ?R < x0"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   791
  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
   792
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   793
    hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   794
    thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   795
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   796
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   797
    have "- ?R < 0" using assms by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   798
    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
   799
    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
   800
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   801
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   802
  from for_subinterval[OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   803
  show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   804
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   805
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   806
subsection {* Exponential Function *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   807
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
   808
definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
   809
  "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   810
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   811
lemma summable_exp_generic:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   812
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   813
  defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   814
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   815
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   816
  have S_Suc: "\<And>n. S (Suc n) = (x * S 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
   817
    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
   818
  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
   819
    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
   820
  obtain N :: nat where N: "norm x < real N * r"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   821
    using reals_Archimedean3 [OF r0] by fast
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   822
  from r1 show ?thesis
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   823
  proof (rule ratio_test [rule_format])
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   824
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   825
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   826
    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
   827
      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
   828
    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
   829
      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
   830
    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
   831
      using norm_ge_zero by (rule mult_right_mono)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   832
    hence "norm (x * 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
   833
      by (rule order_trans [OF norm_mult_ineq])
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   834
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   835
      by (simp add: pos_divide_le_eq mult_ac)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   836
    thus "norm (S (Suc n)) \<le> r * norm (S n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
   837
      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
   838
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   839
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   840
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   841
lemma summable_norm_exp:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   842
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   843
  shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   844
proof (rule summable_norm_comparison_test [OF exI, rule_format])
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   845
  show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   846
    by (rule summable_exp_generic)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   847
next
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   848
  fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
   849
    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
   850
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   851
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   852
lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   853
by (insert summable_exp_generic [where x=x], simp)
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   854
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   855
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   856
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   857
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   858
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   859
lemma exp_fdiffs:
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   860
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
23431
25ca91279a9b change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents: 23413
diff changeset
   861
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   862
         del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   863
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   864
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   865
by (simp add: diffs_def)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   866
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   867
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
   868
unfolding exp_def scaleR_conv_of_real
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
   869
apply (rule DERIV_cong)
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
   870
apply (rule termdiffs [where K="of_real (1 + norm x)"])
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   871
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   872
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   873
apply (simp del: of_real_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   874
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   875
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
   876
declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
   877
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   878
lemma isCont_exp: "isCont exp x"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   879
  by (rule DERIV_exp [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   880
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   881
lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   882
  by (rule isCont_o2 [OF _ isCont_exp])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   883
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   884
lemma tendsto_exp [tendsto_intros]:
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   885
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   886
  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
   887
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
   888
lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (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
   889
  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
   890
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
   891
lemma continuous_on_exp [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (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
   892
  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
   893
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   894
subsubsection {* Properties of the Exponential Function *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   895
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   896
lemma powser_zero:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   897
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   898
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   899
proof -
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   900
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   901
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   902
  thus ?thesis unfolding One_nat_def by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   903
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   904
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   905
lemma exp_zero [simp]: "exp 0 = 1"
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   906
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   907
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   908
lemma setsum_cl_ivl_Suc2:
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   909
  "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
28069
ba4de3022862 moved lemma into SetInterval where it belongs
nipkow
parents: 27483
diff changeset
   910
by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   911
         del: setsum_cl_ivl_Suc)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   912
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   913
lemma exp_series_add:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   914
  fixes x y :: "'a::{real_field}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   915
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   916
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   917
proof (induct n)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   918
  case 0
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   919
  show ?case
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   920
    unfolding S_def by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   921
next
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   922
  case (Suc n)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   923
  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
   924
    unfolding S_def by (simp del: mult_Suc)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   925
  hence 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
   926
    by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   927
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   928
  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   929
    by (simp only: times_S)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   930
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   931
    by (simp only: Suc)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   932
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   933
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47489
diff changeset
   934
    by (rule distrib_right)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   935
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   936
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   937
    by (simp only: setsum_right_distrib mult_ac)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   938
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   939
                + (\<Sum>i=0..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
   940
    by (simp add: times_S Suc_diff_le)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   941
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   942
             (\<Sum>i=0..Suc n. real 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
   943
    by (subst setsum_cl_ivl_Suc2, simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   944
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   945
             (\<Sum>i=0..Suc 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
   946
    by (subst setsum_cl_ivl_Suc, simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   947
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   948
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   949
             (\<Sum>i=0..Suc n. real (Suc n) *\<^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
   950
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   951
              real_of_nat_add [symmetric], simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   952
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23115
diff changeset
   953
    by (simp only: scaleR_right.setsum)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   954
  finally show
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   955
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
   956
    by (simp del: setsum_cl_ivl_Suc)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   957
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   958
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   959
lemma exp_add: "exp (x + y) = exp x * exp y"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   960
unfolding exp_def
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   961
by (simp only: Cauchy_product summable_norm_exp exp_series_add)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   962
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   963
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   964
by (rule exp_add [symmetric])
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   965
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   966
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   967
unfolding exp_def
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 43335
diff changeset
   968
apply (subst suminf_of_real)
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   969
apply (rule summable_exp_generic)
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   970
apply (simp add: scaleR_conv_of_real)
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   971
done
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   972
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   973
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   974
proof
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   975
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   976
  also assume "exp x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   977
  finally show "False" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   978
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   979
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   980
lemma exp_minus: "exp (- x) = inverse (exp x)"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   981
by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   982
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   983
lemma exp_diff: "exp (x - y) = exp x / exp y"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   984
  unfolding diff_minus divide_inverse
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   985
  by (simp add: exp_add exp_minus)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   986
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   987
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   988
subsubsection {* Properties of the Exponential Function on Reals *}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   989
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   990
text {* Comparisons of @{term "exp x"} with zero. *}
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   991
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   992
text{*Proof: because every exponential can be seen as a square.*}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   993
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   994
proof -
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   995
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   996
  thus ?thesis by (simp add: exp_add [symmetric])
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   997
qed
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   998
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   999
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1000
by (simp add: order_less_le)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1001
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1002
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1003
by (simp add: not_less)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1004
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1005
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1006
by (simp add: not_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1007
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1008
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1009
by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1010
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1011
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
  1012
apply (induct "n")
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47489
diff changeset
  1013
apply (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1014
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1015
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1016
text {* Strict monotonicity of exponential. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1017
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1018
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1019
apply (drule order_le_imp_less_or_eq, auto)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1020
apply (simp add: exp_def)
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 36776
diff changeset
  1021
apply (rule order_trans)
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1022
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1023
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1024
done
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1025
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1026
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1027
proof -
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1028
  assume x: "0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1029
  hence "1 < 1 + x" by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1030
  also from x have "1 + x \<le> exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1031
    by (simp add: exp_ge_add_one_self_aux)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1032
  finally show ?thesis .
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1033
qed
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1034
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1035
lemma exp_less_mono:
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1036
  fixes x y :: real
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1037
  assumes "x < y" shows "exp x < exp y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1038
proof -
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1039
  from `x < y` have "0 < y - x" by simp
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1040
  hence "1 < exp (y - x)" by (rule exp_gt_one)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1041
  hence "1 < exp y / exp x" by (simp only: exp_diff)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1042
  thus "exp x < exp y" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1043
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1044
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1045
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1046
apply (simp add: linorder_not_le [symmetric])
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1047
apply (auto simp add: order_le_less exp_less_mono)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1048
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1049
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1050
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1051
by (auto intro: exp_less_mono exp_less_cancel)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1052
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1053
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1054
by (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1055
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1056
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1057
by (simp add: order_eq_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1058
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1059
text {* Comparisons of @{term "exp x"} with one. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1060
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1061
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1062
  using exp_less_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1063
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1064
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1065
  using exp_less_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1066
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1067
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1068
  using exp_le_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1069
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1070
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1071
  using exp_le_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1072
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1073
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1074
  using exp_inj_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1075
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1076
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1077
proof (rule IVT)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1078
  assume "1 \<le> y"
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1079
  hence "0 \<le> y - 1" by simp
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1080
  hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1081
  thus "y \<le> exp (y - 1)" by simp
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1082
qed (simp_all add: le_diff_eq)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1083
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1084
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1085
proof (rule linorder_le_cases [of 1 y])
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1086
  assume "1 \<le> y" thus "\<exists>x. exp x = y"
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1087
    by (fast dest: lemma_exp_total)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1088
next
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1089
  assume "0 < y" and "y \<le> 1"
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1090
  hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1091
  then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1092
  hence "exp (- x) = y" by (simp add: exp_minus)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1093
  thus "\<exists>x. exp x = y" ..
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1094
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1095
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1096
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  1097
subsection {* Natural Logarithm *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1098
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1099
definition ln :: "real \<Rightarrow> real" where
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1100
  "ln x = (THE u. exp u = x)"
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1101
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1102
lemma ln_exp [simp]: "ln (exp x) = x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1103
  by (simp add: ln_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1104
22654
c2b6b5a9e136 new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents: 22653
diff changeset
  1105
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1106
  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
  1107
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1108
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1109
  by (metis exp_gt_zero exp_ln)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1110
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1111
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1112
  by (erule subst, rule ln_exp)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1113
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1114
lemma ln_one [simp]: "ln 1 = 0"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1115
  by (rule ln_unique, simp)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1116
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1117
lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1118
  by (rule ln_unique, simp add: exp_add)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1119
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1120
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1121
  by (rule ln_unique, simp add: exp_minus)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1122
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1123
lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1124
  by (rule ln_unique, simp add: exp_diff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1125
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1126
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1127
  by (rule ln_unique, simp add: exp_real_of_nat_mult)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1128
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1129
lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1130
  by (subst exp_less_cancel_iff [symmetric], simp)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1131
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1132
lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1133
  by (simp add: linorder_not_less [symmetric])
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1134
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1135
lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1136
  by (simp add: order_eq_iff)
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1137
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1138
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1139
  apply (rule exp_le_cancel_iff [THEN iffD1])
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1140
  apply (simp add: exp_ge_add_one_self_aux)
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1141
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1142
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1143
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1144
  by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1145
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1146
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1147
  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
  1148
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1149
lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1150
  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
  1151
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1152
lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1153
  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
  1154
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1155
lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1156
  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
  1157
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1158
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1159
  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
  1160
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1161
lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1162
  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
  1163
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1164
lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1165
  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
  1166
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1167
lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1168
  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
  1169
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1170
lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1171
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1172
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  1173
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
44308
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1174
  apply (subgoal_tac "isCont ln (exp (ln x))", simp)
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1175
  apply (rule isCont_inverse_function [where f=exp], simp_all)
d2a6f9af02f4 Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents: 44307
diff changeset
  1176
  done
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  1177
45915
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45309
diff changeset
  1178
lemma tendsto_ln [tendsto_intros]:
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45309
diff changeset
  1179
  "\<lbrakk>(f ---> a) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45309
diff changeset
  1180
  by (rule isCont_tendsto_compose [OF isCont_ln])
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45309
diff changeset
  1181
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
  1182
lemma continuous_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
  1183
  "continuous F f \<Longrightarrow> 0 < f (Lim F (\<lambda>x. x)) \<Longrightarrow> continuous F (\<lambda>x. ln (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
  1184
  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
  1185
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
  1186
lemma isCont_ln' [continuous_intros]:
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
  1187
  "continuous (at x) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x) (\<lambda>x. ln (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
  1188
  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
  1189
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
  1190
lemma continuous_within_ln [continuous_intros]:
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
  1191
  "continuous (at x within s) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (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
  1192
  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
  1193
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
  1194
lemma continuous_on_ln [continuous_on_intros]:
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
  1195
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. 0 < f x) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"
270b21f3ae0a move continuous and continuous_on to the HOL image;