src/HOL/Transcendental.thy
author hoelzl
Wed, 19 Mar 2014 15:34:57 +0100
changeset 56213 e5720d3c18f0
parent 56193 c726ecfb22b6
child 56217 dc429a5b13c4
permissions -rw-r--r--
further renaming in Series
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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_sumr2:
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  fixes y :: "'a::{comm_ring,monoid_mult}"
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  shows
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    "x ^ (Suc n) - y ^ (Suc n) =
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      (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
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proof (induct n)
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  case (Suc n)
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  have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x ^ n) - y * (y * y ^ n)"
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    by simp
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  also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x ^ n)"
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    by (simp add: algebra_simps)
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  also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    by (simp only: Suc)
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  also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    by (simp only: mult_left_commute)
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  also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
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    by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
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  finally show ?case .
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qed simp
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corollary power_diff_sumr2: --{* @{text COMPLEX_POLYFUN} in HOL Light *}
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  fixes x :: "'a::{comm_ring,monoid_mult}"
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  shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
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using lemma_realpow_diff_sumr2[of x "n - 1" y]
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by (cases "n = 0") (simp_all add: field_simps)
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lemma lemma_realpow_rev_sumr:
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   "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) =
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    (\<Sum>p<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 (auto simp: image_iff Bex_def intro!: inj_onI)
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  apply arith
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  done
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lemma power_diff_1_eq:
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  fixes x :: "'a::{comm_ring,monoid_mult}"
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  shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
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using lemma_realpow_diff_sumr2 [of x _ 1] 
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  by (cases n) auto
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cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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lemma one_diff_power_eq':
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  fixes x :: "'a::{comm_ring,monoid_mult}"
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  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
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using lemma_realpow_diff_sumr2 [of 1 _ x] 
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  by (cases n) auto
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lemma one_diff_power_eq:
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  fixes x :: "'a::{comm_ring,monoid_mult}"
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  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
55719
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by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
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    74
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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_div_algebra"
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  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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    and 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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    86
    by (rule summable_LIMSEQ_zero)
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    87
  hence "convergent (\<lambda>n. f n * x ^ n)"
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    88
    by (rule convergentI)
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    89
  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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    92
    by (rule Cauchy_Bseq)
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    93
  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
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    94
    by (simp add: Bseq_def, safe)
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    95
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
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    96
                   K * norm (z ^ n) * inverse (norm (x ^ n))"
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    97
  proof (intro exI allI impI)
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    98
    fix n::nat
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    99
    assume "0 \<le> n"
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   100
    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   101
          norm (f n * x ^ n) * norm (z ^ n)"
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      by (simp add: norm_mult abs_mult)
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   103
    also have "\<dots> \<le> K * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   104
      by (simp only: mult_right_mono 4 norm_ge_zero)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   105
    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
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   106
      by (simp add: x_neq_0)
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   107
    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
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   108
      by (simp only: mult_assoc)
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   109
    finally show "norm (norm (f n * z ^ n)) \<le>
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   110
                  K * norm (z ^ n) * inverse (norm (x ^ n))"
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   111
      by (simp add: mult_le_cancel_right x_neq_0)
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   112
  qed
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   113
  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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   114
  proof -
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   115
    from 2 have "norm (norm (z * inverse x)) < 1"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   116
      using x_neq_0
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   117
      by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
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   118
    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
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   119
      by (rule summable_geometric)
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   120
    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
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   121
      by (rule summable_mult)
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    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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   123
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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parents: 23069
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   124
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   125
                    power_inverse norm_power mult_assoc)
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   126
  qed
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   127
  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
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   128
    by (rule summable_comparison_test)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
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   129
qed
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parents: 15013
diff changeset
   130
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   131
lemma powser_inside:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   132
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   133
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   134
    "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   135
      summable (\<lambda>n. f n * (z ^ n))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   136
  by (rule powser_insidea [THEN summable_norm_cancel])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   137
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   138
lemma sum_split_even_odd:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   139
  fixes f :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   140
  shows
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   141
    "(\<Sum>i<2 * n. if even i then f i else g i) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   142
     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   143
proof (induct n)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   144
  case 0
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   145
  then show ?case by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   146
next
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   147
  case (Suc n)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   148
  have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   149
    (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   150
    using Suc.hyps unfolding One_nat_def by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   151
  also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   152
    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
   153
  finally show ?case .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   154
qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   155
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   156
lemma sums_if':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   157
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   158
  assumes "g sums x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   159
  shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   160
  unfolding sums_def
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   161
proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   162
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   163
  assume "0 < r"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   164
  from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   165
  obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   166
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   167
  let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   168
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   169
    fix m
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   170
    assume "m \<ge> 2 * no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   171
    hence "m div 2 \<ge> no" by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   172
    have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   173
      using sum_split_even_odd by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   174
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   175
      using no_eq unfolding sum_eq using `m div 2 \<ge> no` 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
   176
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   177
    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
   178
    proof (cases "even m")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   179
      case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   180
      show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   181
        unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   182
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   183
      case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   184
      hence "even (Suc m)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   185
      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   186
        odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   187
      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
   188
      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
   189
      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
   190
      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
   191
      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
   192
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   193
    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
   194
  }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   195
  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
   196
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   197
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   198
lemma sums_if:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   199
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   200
  assumes "g sums x" and "f sums y"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   201
  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
   202
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   203
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   204
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   205
    fix B T E
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   206
    have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   207
      by (cases B) auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   208
  } note if_sum = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   209
  have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   210
    using sums_if'[OF `g sums x`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   211
  {
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 38642
diff changeset
   212
    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
   213
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   214
    have "?s sums y" using sums_if'[OF `f sums y`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   215
    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
   216
    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   217
      by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum_reindex if_eq sums_def cong del: if_cong)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   218
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   219
  from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   220
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   221
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   222
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
   223
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   224
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
   225
  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
   226
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   227
  shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. -1^i*a i) ----> l) \<and>
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   228
             ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. -1^i*a i) ----> l)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   229
  (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   230
proof (rule nested_sequence_unique)
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   231
  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
   232
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   233
  show "\<forall>n. ?f n \<le> ?f (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   234
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   235
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   236
    show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   237
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   238
  show "\<forall>n. ?g (Suc n) \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   239
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   240
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   241
    show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   242
      unfolding One_nat_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   243
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   244
  show "\<forall>n. ?f n \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   245
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   246
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   247
    show "?f n \<le> ?g n" using fg_diff a_pos
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   248
      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
   249
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   250
  show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   251
  proof (rule LIMSEQ_I)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   252
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   253
    assume "0 < r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   254
    with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   255
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   256
    hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   257
    thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   258
  qed
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   259
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   260
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   261
lemma summable_Leibniz':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   262
  fixes a :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   263
  assumes a_zero: "a ----> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   264
    and a_pos: "\<And> n. 0 \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   265
    and a_monotone: "\<And> n. a (Suc n) \<le> a n"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   266
  shows summable: "summable (\<lambda> n. (-1)^n * a n)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   267
    and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   268
    and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   269
    and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   270
    and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   271
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   272
  let ?S = "\<lambda>n. (-1)^n * a n"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   273
  let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   274
  let ?f = "\<lambda>n. ?P (2 * n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   275
  let ?g = "\<lambda>n. ?P (2 * n + 1)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   276
  obtain l :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   277
    where below_l: "\<forall> n. ?f n \<le> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   278
      and "?f ----> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   279
      and above_l: "\<forall> n. l \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   280
      and "?g ----> l"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   281
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   282
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   283
  let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   284
  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
   285
  proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   286
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   287
    assume "0 < r"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   288
    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
   289
    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
   290
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   291
    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
   292
    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
   293
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   294
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   295
      fix n :: nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   296
      assume "n \<ge> (max (2 * f_no) (2 * g_no))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   297
      hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" 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
   298
      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
   299
      proof (cases "even n")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   300
        case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   301
        from even_nat_div_two_times_two[OF this]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   302
        have n_eq: "2 * (n div 2) = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   303
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   304
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   305
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   306
        from f[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   307
          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
   308
      next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   309
        case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   310
        hence "even (n - 1)" by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   311
        from even_nat_div_two_times_two[OF this]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   312
        have n_eq: "2 * ((n - 1) div 2) = n - 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   313
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   314
        hence range_eq: "n - 1 + 1 = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   315
          using odd_pos[OF False] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   316
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   317
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   318
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   319
        from g[OF this] show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   320
          unfolding n_eq range_eq .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   321
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   322
    }
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   323
    thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   324
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   325
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   326
    unfolding sums_def .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   327
  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
   328
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   329
  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
   330
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   331
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   332
  show "suminf ?S \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   333
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   334
  show "?f n \<le> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   335
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   336
  show "?g ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   337
    using `?g ----> l` `l = suminf ?S` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   338
  show "?f ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   339
    using `?f ----> l` `l = suminf ?S` 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
   340
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   341
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   342
theorem summable_Leibniz:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   343
  fixes a :: "nat \<Rightarrow> real"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   344
  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
   345
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   346
    and "0 < a 0 \<longrightarrow>
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   347
      (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i<2*n. -1^i * a i .. \<Sum>i<2*n+1. -1^i * a i})" (is "?pos")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   348
    and "a 0 < 0 \<longrightarrow>
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   349
      (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i<2*n+1. -1^i * a i .. \<Sum>i<2*n. -1^i * a i})" (is "?neg")
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   350
    and "(\<lambda>n. \<Sum>i<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   351
    and "(\<lambda>n. \<Sum>i<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   352
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   353
  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
   354
  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
   355
    case True
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   356
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   357
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   358
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   359
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   360
      have "a (Suc n) \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   361
        using ord[where n="Suc n" and m=n] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   362
    } note mono = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   363
    note leibniz = summable_Leibniz'[OF `a ----> 0` ge0]
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   364
    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
   365
    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
   366
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   367
    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
   368
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   369
    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
   370
    have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   371
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   372
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   373
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   374
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   375
      have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   376
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   377
    } note monotone = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   378
    note leibniz =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   379
      summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   380
        OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   381
    have "summable (\<lambda> n. (-1)^n * ?a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   382
      using leibniz(1) by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   383
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   384
      unfolding summable_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   385
    from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   386
      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
   387
    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
   388
    moreover
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   389
    have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   390
      unfolding minus_diff_minus by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   391
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   392
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   393
    have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   394
      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
   395
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   396
    have ?pos using `0 \<le> ?a 0` by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   397
    moreover have ?neg
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   398
      using leibniz(2,4)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   399
      unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   400
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   401
    moreover have ?f and ?g
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   402
      using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   403
      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
   404
    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
   405
  qed
54576
e877eec2b698 tidied more proofs
paulson
parents: 54575
diff changeset
   406
  then show ?summable and ?pos and ?neg and ?f and ?g 
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   407
    by safe
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   408
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   409
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   410
subsection {* Term-by-Term Differentiability of Power Series *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   411
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   412
definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   413
  where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   414
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   415
text{*Lemma about distributing negation over it*}
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   416
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   417
  by (simp add: diffs_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   418
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   419
lemma sums_Suc_imp:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   420
  "(f::nat \<Rightarrow> 'a::real_normed_vector) 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   421
  using sums_Suc_iff[of f] by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   422
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   423
lemma diffs_equiv:
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   424
  fixes x :: "'a::{real_normed_vector, ring_1}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   425
  shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   426
      (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   427
  unfolding diffs_def
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   428
  by (simp add: summable_sums sums_Suc_imp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   429
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   430
lemma lemma_termdiff1:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   431
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   432
  "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   433
   (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   434
  by (auto simp add: algebra_simps power_add [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   435
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   436
lemma sumr_diff_mult_const2:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   437
  "setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   438
  by (simp add: setsum_subtractf)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   439
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   440
lemma lemma_termdiff2:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   441
  fixes h :: "'a :: {field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   442
  assumes h: "h \<noteq> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   443
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   444
    "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   445
     h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   446
          (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   447
  apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   448
  apply (simp add: right_diff_distrib diff_divide_distrib h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   449
  apply (simp add: mult_assoc [symmetric])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   450
  apply (cases "n", simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   451
  apply (simp add: lemma_realpow_diff_sumr2 h
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   452
                   right_diff_distrib [symmetric] mult_assoc
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   453
              del: power_Suc setsum_lessThan_Suc of_nat_Suc)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   454
  apply (subst lemma_realpow_rev_sumr)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   455
  apply (subst sumr_diff_mult_const2)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   456
  apply simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   457
  apply (simp only: lemma_termdiff1 setsum_right_distrib)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   458
  apply (rule setsum_cong [OF refl])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   459
  apply (simp add: less_iff_Suc_add)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   460
  apply (clarify)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   461
  apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   462
              del: setsum_lessThan_Suc power_Suc)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   463
  apply (subst mult_assoc [symmetric], subst power_add [symmetric])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   464
  apply (simp add: mult_ac)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   465
  done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   466
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   467
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
   468
  fixes K :: "'a::linordered_semidom"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   469
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   470
    and K: "0 \<le> K"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   471
  shows "setsum f {..<n-k} \<le> of_nat n * K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   472
  apply (rule order_trans [OF setsum_mono])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   473
  apply (rule f, simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   474
  apply (simp add: mult_right_mono K)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   475
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   476
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   477
lemma lemma_termdiff3:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   478
  fixes h z :: "'a::{real_normed_field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   479
  assumes 1: "h \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   480
    and 2: "norm z \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   481
    and 3: "norm (z + h) \<le> K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   482
  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
   483
          \<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
   484
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   485
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   486
        norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   487
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   488
    by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult_commute norm_mult)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   489
  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
   490
  proof (rule mult_right_mono [OF _ norm_ge_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   491
    from norm_ge_zero 2 have K: "0 \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   492
      by (rule order_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   493
    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
   494
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   495
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   496
      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
   497
      done
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   498
    show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   499
          \<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
   500
      apply (intro
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   501
         order_trans [OF norm_setsum]
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   502
         real_setsum_nat_ivl_bounded2
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   503
         mult_nonneg_nonneg
47489
04e7d09ade7a tuned some proofs;
huffman
parents: 47108
diff changeset
   504
         of_nat_0_le_iff
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   505
         zero_le_power K)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   506
      apply (rule le_Kn, simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   507
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   508
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   509
  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
   510
    by (simp only: mult_assoc)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   511
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   512
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   513
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   514
lemma lemma_termdiff4:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   515
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   516
  assumes k: "0 < (k::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   517
    and 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
   518
  shows "f -- 0 --> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   519
proof (rule tendsto_norm_zero_cancel)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   520
  show "(\<lambda>h. norm (f h)) -- 0 --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   521
  proof (rule real_tendsto_sandwich)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   522
    show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   523
      by simp
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   524
    show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   525
      using k by (auto simp add: eventually_at dist_norm le)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   526
    show "(\<lambda>h. 0) -- (0::'a) --> (0::real)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   527
      by (rule tendsto_const)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   528
    have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   529
      by (intro tendsto_intros)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   530
    then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   531
      by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   532
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   533
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   534
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   535
lemma lemma_termdiff5:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   536
  fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   537
  assumes k: "0 < (k::real)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   538
  assumes f: "summable f"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   539
  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
   540
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   541
proof (rule lemma_termdiff4 [OF k])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   542
  fix h::'a
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   543
  assume "h \<noteq> 0" and "norm h < k"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   544
  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
   545
    by (simp add: le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   546
  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
   547
    by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   548
  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
   549
    by (rule summable_mult2)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   550
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   551
    by (rule summable_comparison_test)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   552
  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
   553
    by (rule summable_norm)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   554
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
   555
    by (rule suminf_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   556
  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
   557
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   558
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   559
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   560
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   561
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   562
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   563
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   564
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   565
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   566
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   567
    and 2: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   568
  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
   569
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   570
proof -
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   571
  from dense [OF 2]
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   572
  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
   573
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   574
    by (rule order_le_less_trans)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   575
  hence r_neq_0: "r \<noteq> 0" by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   576
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   577
  proof (rule lemma_termdiff5)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   578
    show "0 < r - norm x" using r1 by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   579
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   580
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   581
    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
   582
      by (rule powser_insidea)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   583
    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
   584
      using r
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   585
      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
   586
    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
   587
      by (rule diffs_equiv [THEN sums_summable])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   588
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   589
      (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   590
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   591
      apply (simp add: diffs_def)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   592
      apply (case_tac n, simp_all add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   593
      done
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   594
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   595
      (\<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
   596
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   597
    also have
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   598
      "(\<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
   599
           r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   600
       (\<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
   601
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   602
      apply (case_tac "n", simp)
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 54576
diff changeset
   603
      apply (rename_tac nat)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   604
      apply (case_tac "nat", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   605
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   606
      done
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   607
    finally
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   608
    show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   609
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   610
    fix h::'a and n::nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   611
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   612
    assume "norm h < r - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   613
    hence "norm x + norm h < r" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   614
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   615
      by (rule order_le_less_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   616
    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
   617
          \<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
   618
      apply (simp only: norm_mult mult_assoc)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   619
      apply (rule mult_left_mono [OF _ norm_ge_zero])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   620
      apply (simp add: mult_assoc [symmetric])
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   621
      apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   622
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   623
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   624
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   625
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   626
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   627
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   628
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   629
      and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   630
      and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   631
      and 4: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   632
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   633
  unfolding deriv_def
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   634
proof (rule LIM_zero_cancel)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   635
  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
   636
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   637
  proof (rule LIM_equal2)
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   638
    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
   639
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   640
    fix h :: 'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   641
    assume "norm (h - 0) < norm K - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   642
    hence "norm x + norm h < norm K" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   643
    hence 5: "norm (x + h) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   644
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   645
    have "summable (\<lambda>n. c n * x ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   646
      and "summable (\<lambda>n. c n * (x + h) ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   647
      and "summable (\<lambda>n. diffs c n * x ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   648
      using 1 2 4 5 by (auto elim: powser_inside)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   649
    then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   650
          (\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h - of_nat n * c n * x ^ (n - Suc 0))"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   651
      by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   652
    then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   653
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   654
      by (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   655
  next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   656
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   657
      by (rule termdiffs_aux [OF 3 4])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   658
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   659
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   660
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   661
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   662
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
   663
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   664
lemma DERIV_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   665
  fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   666
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   667
    and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   668
    and "summable (f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   669
    and "summable L"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   670
    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>"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   671
  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
   672
  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
   673
proof (rule LIM_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   674
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   675
  assume "0 < r" hence "0 < r/3" by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   676
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   677
  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
   678
    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
   679
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   680
  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
   681
    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
   682
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   683
  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
   684
  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
   685
    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
   686
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   687
  let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   688
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   689
  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
   690
  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
   691
  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
   692
  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
   693
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   694
  let ?s = "\<lambda>n. SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   695
  def S' \<equiv> "Min (?s ` {..< ?N })"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   696
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   697
  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
   698
  proof (rule iffD2[OF Min_gr_iff])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   699
    show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   700
    proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   701
      fix x
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   702
      assume "x \<in> ?s ` {..<?N}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   703
      then obtain n where "x = ?s n" and "n \<in> {..<?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   704
        using image_iff[THEN iffD1] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   705
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   706
      obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   707
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   708
      have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   709
      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
   710
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   711
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   712
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   713
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   714
  hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   715
    and "S \<le> S'" using x0_in_I and `0 < S'`
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   716
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   717
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   718
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   719
    fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   720
    assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   721
    hence x_in_I: "x0 + x \<in> { a <..< b }"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   722
      using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   723
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   724
    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
   725
    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
   726
    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
   727
    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
   728
    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
   729
    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
   730
    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
   731
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   732
    { fix n
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   733
      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   734
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   735
        unfolding abs_divide .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   736
      hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   737
        using `x \<noteq> 0` by auto }
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   738
    note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF `summable L`]]
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   739
    then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
   740
      by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF `summable L`]]])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   741
    then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   742
      using L_estimate by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   743
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   744
    have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" ..
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   745
    also have "\<dots> < (\<Sum>n<?N. ?r)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   746
    proof (rule setsum_strict_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   747
      fix n
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   748
      assume "n \<in> {..< ?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   749
      have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   750
      also have "S \<le> S'" using `S \<le> S'` .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   751
      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
   752
      proof (rule Min_le_iff[THEN iffD2])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   753
        have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   754
          using `n \<in> {..< ?N}` by auto
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   755
        thus "\<exists> a \<in> (?s ` {..<?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
   756
      qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   757
      finally have "\<bar>x\<bar> < ?s n" .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   758
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   759
      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
   760
      have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   761
      with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   762
        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
   763
    qed auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   764
    also have "\<dots> = of_nat (card {..<?N}) * ?r"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   765
      by (rule setsum_constant)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   766
    also have "\<dots> = real ?N * ?r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   767
      unfolding real_eq_of_nat 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
   768
    also have "\<dots> = r/3" by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   769
    finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   770
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   771
    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   772
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   773
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   774
      unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   775
      using suminf_divide[OF diff_smbl, symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   776
    also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   777
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   778
      unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   779
      apply (subst (5) add_commute)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   780
      by (rule abs_triangle_ineq)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   781
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   782
      using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   783
    also have "\<dots> < r /3 + r/3 + r/3"
36842
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
   784
      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
   785
      by (rule add_strict_mono [OF add_less_le_mono])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   786
    finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   787
      by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   788
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   789
  thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   790
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   791
    using `0 < S` unfolding real_norm_def diff_0_right 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
   792
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   793
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   794
lemma DERIV_power_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   795
  fixes f :: "nat \<Rightarrow> real"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   796
  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   797
    and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   798
  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
   799
  (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
   800
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   801
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   802
    fix R'
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   803
    assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   804
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   805
      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
   806
    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
   807
    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
   808
      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
   809
      proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   810
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   811
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   812
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   813
          using `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   814
        have "norm R' < norm ((R' + R) / 2)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   815
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   816
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   817
          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
   818
      qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   819
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   820
        fix n x y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   821
        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
   822
        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
   823
        proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   824
          have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   825
            (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   826
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   827
            by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   828
          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
   829
          proof (rule mult_left_mono)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   830
            have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   831
              by (rule setsum_abs)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   832
            also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   833
            proof (rule setsum_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   834
              fix p
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   835
              assume "p \<in> {..<Suc n}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   836
              hence "p \<le> n" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   837
              {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   838
                fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   839
                fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   840
                assume "x \<in> {-R'<..<R'}"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   841
                hence "\<bar>x\<bar> \<le> R'"  by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   842
                hence "\<bar>x^n\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   843
                  unfolding power_abs by (rule power_mono, auto)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   844
              }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   845
              from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   846
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   847
                unfolding abs_mult by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   848
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   849
                unfolding power_add[symmetric] using `p \<le> n` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   850
            qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   851
            also have "\<dots> = real (Suc n) * R' ^ n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   852
              unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   853
            finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   854
              unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   855
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   856
              unfolding abs_mult[symmetric] by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   857
          qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   858
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   859
            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
   860
          finally show ?thesis .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   861
        qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   862
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   863
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   864
        fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   865
        show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   866
          by (auto intro!: DERIV_intros simp del: power_Suc)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   867
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   868
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   869
        fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   870
        assume "x \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   871
        hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   872
          using assms `R' < R` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   873
        have "summable (\<lambda> n. f n * x^n)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   874
        proof (rule summable_comparison_test, intro exI allI impI)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   875
          fix n
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   876
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   877
            by (rule mult_left_mono) auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   878
          show "norm (f n * x ^ n) \<le> norm (f n * real (Suc n) * x ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   879
            unfolding real_norm_def abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   880
            by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   881
        qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`])
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 36776
diff changeset
   882
        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   883
        show "summable (?f x)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   884
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   885
      show "summable (?f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   886
        using converges[OF `x0 \<in> {-R <..< R}`] .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   887
      show "x0 \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   888
        using `x0 \<in> {-R' <..< R'}` .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   889
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   890
  } 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
   891
  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
   892
  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
   893
  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
   894
  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
   895
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   896
    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
   897
    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
   898
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   899
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   900
    have "- ?R < 0" using assms by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   901
    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
   902
    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
   903
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   904
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   905
    using assms by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   906
  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
   907
  show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   908
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   909
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   910
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   911
subsection {* Exponential Function *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   912
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   913
definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   914
  where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   915
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   916
lemma summable_exp_generic:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   917
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   918
  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
   919
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   920
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   921
  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
   922
    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
   923
  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
   924
    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
   925
  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
   926
    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
   927
  from r1 show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   928
  proof (rule summable_ratio_test [rule_format])
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   929
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   930
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   931
    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
   932
      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
   933
    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
   934
      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
   935
    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
   936
      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
   937
    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
   938
      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
   939
    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
   940
      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
   941
    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
   942
      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
   943
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   944
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   945
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   946
lemma summable_norm_exp:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   947
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   948
  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
   949
proof (rule summable_norm_comparison_test [OF exI, rule_format])
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   950
  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
   951
    by (rule summable_exp_generic)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   952
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   953
  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
   954
    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
   955
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   956
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   957
lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   958
  using summable_exp_generic [where x=x] by simp
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   959
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   960
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   961
  unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   962
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   963
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   964
lemma exp_fdiffs:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   965
      "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   966
  by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   967
        del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   968
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   969
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   970
  by (simp add: diffs_def)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   971
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   972
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   973
  unfolding exp_def scaleR_conv_of_real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   974
  apply (rule DERIV_cong)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   975
  apply (rule termdiffs [where K="of_real (1 + norm x)"])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   976
  apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   977
  apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   978
  apply (simp del: of_real_add)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   979
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   980
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
   981
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
   982
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   983
lemma isCont_exp: "isCont exp x"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   984
  by (rule DERIV_exp [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   985
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   986
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
   987
  by (rule isCont_o2 [OF _ isCont_exp])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   988
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   989
lemma tendsto_exp [tendsto_intros]:
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   990
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
   991
  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
   992
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   993
lemma continuous_exp [continuous_intros]:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   994
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
   995
  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
   996
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   997
lemma continuous_on_exp [continuous_on_intros]:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   998
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51477
diff changeset
   999
  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
  1000
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1001
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1002
subsubsection {* Properties of the Exponential Function *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1003
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1004
lemma powser_zero:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1005
  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
  1006
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1007
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1008
  have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1009
    by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
  1010
  thus ?thesis unfolding One_nat_def by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1011
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1012
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1013
lemma exp_zero [simp]: "exp 0 = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1014
  unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1015
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1016
lemma exp_series_add:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1017
  fixes x y :: "'a::{real_field}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1018
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1019
  shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1020
proof (induct n)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1021
  case 0
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1022
  show ?case
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1023
    unfolding S_def by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1024
next
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1025
  case (Suc n)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1026
  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
  1027
    unfolding S_def by (simp del: mult_Suc)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1028
  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
  1029
    by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1030
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1031
  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
  1032
    by (simp only: times_S)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1033
  also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1034
    by (simp only: Suc)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1035
  also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i))
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1036
                + y * (\<Sum>i\<le>n. S x i * S y (n-i))"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47489
diff changeset
  1037
    by (rule distrib_right)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1038
  also have "\<dots> = (\<Sum>i\<le>n. (x * S x i) * S y (n-i))
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1039
                + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1040
    by (simp only: setsum_right_distrib mult_ac)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1041
  also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1042
                + (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1043
    by (simp add: times_S Suc_diff_le)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1044
  also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1045
             (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1046
    by (subst setsum_atMost_Suc_shift) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1047
  also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1048
             (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1049
    by simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1050
  also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1051
             (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1052
             (\<Sum>i\<le>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
  1053
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1054
                   real_of_nat_add [symmetric]) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1055
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23115
diff changeset
  1056
    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
  1057
  finally show
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1058
    "S (x + y) (Suc n) = (\<Sum>i\<le>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
  1059
    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
  1060
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1061
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1062
lemma exp_add: "exp (x + y) = exp x * exp y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1063
  unfolding exp_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1064
  by (simp only: Cauchy_product summable_norm_exp exp_series_add)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1065
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1066
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1067
  by (rule exp_add [symmetric])
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1068
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1069
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1070
  unfolding exp_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1071
  apply (subst suminf_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1072
  apply (rule summable_exp_generic)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1073
  apply (simp add: scaleR_conv_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1074
  done
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1075
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1076
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1077
proof
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1078
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1079
  also assume "exp x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1080
  finally show "False" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1081
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1082
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1083
lemma exp_minus: "exp (- x) = inverse (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1084
  by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1085
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1086
lemma exp_diff: "exp (x - y) = exp x / exp y"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
  1087
  using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1088
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1089
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1090
subsubsection {* Properties of the Exponential Function on Reals *}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1091
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1092
text {* Comparisons of @{term "exp x"} with zero. *}
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1093
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1094
text{*Proof: because every exponential can be seen as a square.*}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1095
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1096
proof -
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1097
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1098
  thus ?thesis by (simp add: exp_add [symmetric])
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1099
qed
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1100
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1101
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1102
  by (simp add: order_less_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1103
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1104
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1105
  by (simp add: not_less)
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1106
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1107
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1108
  by (simp add: not_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1109
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1110
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1111
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1112
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1113
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1114
  by (induct n) (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
  1115
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1116
text {* Strict monotonicity of exponential. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1117
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1118
lemma exp_ge_add_one_self_aux: 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1119
  assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1120
using order_le_imp_less_or_eq [OF assms]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1121
proof 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1122
  assume "0 < x"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1123
  have "1+x \<le> (\<Sum>n<2. inverse (real (fact n)) * x ^ n)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1124
    by (auto simp add: numeral_2_eq_2)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1125
  also have "... \<le> (\<Sum>n. inverse (real (fact n)) * x ^ n)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1126
    apply (rule setsum_le_suminf [OF summable_exp])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1127
    using `0 < x`
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1128
    apply (auto  simp add:  zero_le_mult_iff)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1129
    done
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1130
  finally show "1+x \<le> exp x" 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1131
    by (simp add: exp_def)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1132
next
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1133
  assume "0 = x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1134
  then show "1 + x \<le> exp x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1135
    by auto
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1136
qed
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1137
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1138
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1139
proof -
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1140
  assume x: "0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1141
  hence "1 < 1 + x" by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1142
  also from x have "1 + x \<le> exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1143
    by (simp add: exp_ge_add_one_self_aux)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1144
  finally show ?thesis .
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1145
qed
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1146
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1147
lemma exp_less_mono:
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1148
  fixes x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1149
  assumes "x < y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1150
  shows "exp x < exp y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1151
proof -
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1152
  from `x < y` have "0 < y - x" by simp
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1153
  hence "1 < exp (y - x)" by (rule exp_gt_one)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1154
  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
  1155
  thus "exp x < exp y" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1156
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1157
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1158
lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1159
  unfolding linorder_not_le [symmetric]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1160
  by (auto simp add: order_le_less exp_less_mono)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1161
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1162
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1163
  by (auto intro: exp_less_mono exp_less_cancel)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1164
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1165
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1166
  by (auto simp add: linorder_not_less [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1167
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1168
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1169
  by (simp add: order_eq_iff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1170
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1171
text {* Comparisons of @{term "exp x"} with one. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1172
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1173
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1174
  using exp_less_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1175
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1176
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1177
  using exp_less_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1178
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1179
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
  1180
  using exp_le_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1181
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1182
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
  1183
  using exp_le_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1184
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1185
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1186
  using exp_inj_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1187
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1188
lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<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
  1189
proof (rule IVT)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1190
  assume "1 \<le> y"
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1191
  hence "0 \<le> y - 1" by simp
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1192
  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
  1193
  thus "y \<le> exp (y - 1)" by simp
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1194
qed (simp_all add: le_diff_eq)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1195
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1196
lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1197
proof (rule linorder_le_cases [of 1 y])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1198
  assume "1 \<le> y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1199
  thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1200
next
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1201
  assume "0 < y" and "y \<le> 1"
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1202
  hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1203
  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
  1204
  hence "exp (- x) = y" by (simp add: exp_minus)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1205
  thus "\<exists>x. exp x = y" ..
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1206
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1207
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1208
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  1209
subsection {* Natural Logarithm *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1210
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1211
definition ln :: "real \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1212
  where "ln x = (THE u. exp u = x)"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1213
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1214
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
  1215
  by (simp add: ln_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1216
22654
c2b6b5a9e136 new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents: 22653
diff changeset
  1217
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
  1218
  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
  1219
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1220
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
  1221
  by (metis exp_gt_zero exp_ln)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1222