src/HOL/Transcendental.thy
author immler
Mon, 13 Oct 2014 18:55:05 +0200
changeset 58656 7f14d5d9b933
parent 58410 6d46ad54a2ab
child 58709 efdc6c533bd3
permissions -rw-r--r--
relaxed class constraints for exp
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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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lemma root_test_convergence:
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  fixes f :: "nat \<Rightarrow> 'a::banach"
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  assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup"
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  assumes "x < 1"
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  shows "summable f"
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proof -
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  have "0 \<le> x"
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    by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
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  from `x < 1` obtain z where z: "x < z" "z < 1"
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    by (metis dense)
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  from f `x < z`
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  have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
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    25
    by (rule order_tendstoD)
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  then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
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    using eventually_ge_at_top
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  proof eventually_elim
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    fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
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    from power_strict_mono[OF less, of n] n
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    show "norm (f n) \<le> z ^ n"
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      by simp
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  qed
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  then show "summable f"
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    unfolding eventually_sequentially
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    using z `0 \<le> x` by (auto intro!: summable_comparison_test[OF _  summable_geometric])
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qed
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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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    61
  also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    62
    by (simp only: Suc)
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    63
  also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    64
    by (simp only: mult.left_commute)
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parents: 56181
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    65
  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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    67
  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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3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 55719
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    71
  fixes x :: "'a::{comm_ring,monoid_mult}"
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    72
  shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
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3f5b2745d659 More complex-related lemmas
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parents: 55719
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    73
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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    75
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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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    79
  by (subst nat_diff_setsum_reindex[symmetric]) simp
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cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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lemma power_diff_1_eq:
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    82
  fixes x :: "'a::{comm_ring,monoid_mult}"
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    83
  shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
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cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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parents: 55417
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    84
using lemma_realpow_diff_sumr2 [of x _ 1] 
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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parents: 55417
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    85
  by (cases n) auto
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    86
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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    87
lemma one_diff_power_eq':
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    88
  fixes x :: "'a::{comm_ring,monoid_mult}"
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hoelzl
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    89
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    90
using lemma_realpow_diff_sumr2 [of 1 _ x] 
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    91
  by (cases n) auto
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    92
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    93
lemma one_diff_power_eq:
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    94
  fixes x :: "'a::{comm_ring,monoid_mult}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    95
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    96
by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    97
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
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    99
  x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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   100
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   101
lemma powser_insidea:
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   102
  fixes x z :: "'a::real_normed_div_algebra"
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   103
  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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   104
    and 2: "norm z < norm x"
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   105
  shows "summable (\<lambda>n. norm (f n * z ^ n))"
20849
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   106
proof -
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
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   107
  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
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diff changeset
   108
  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
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huffman
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   109
    by (rule summable_LIMSEQ_zero)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
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   110
  hence "convergent (\<lambda>n. f n * x ^ n)"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   111
    by (rule convergentI)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   112
  hence "Cauchy (\<lambda>n. f n * x ^ n)"
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huffman
parents: 44725
diff changeset
   113
    by (rule convergent_Cauchy)
20849
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huffman
parents: 20692
diff changeset
   114
  hence "Bseq (\<lambda>n. f n * x ^ n)"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   115
    by (rule Cauchy_Bseq)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   116
  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
20849
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huffman
parents: 20692
diff changeset
   117
    by (simp add: Bseq_def, safe)
23082
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huffman
parents: 23069
diff changeset
   118
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   119
                   K * norm (z ^ n) * inverse (norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   120
  proof (intro exI allI impI)
53079
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wenzelm
parents: 53076
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   121
    fix n::nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
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   122
    assume "0 \<le> n"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   123
    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   124
          norm (f n * x ^ n) * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   125
      by (simp add: norm_mult abs_mult)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   126
    also have "\<dots> \<le> K * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   127
      by (simp only: mult_right_mono 4 norm_ge_zero)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   128
    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   129
      by (simp add: x_neq_0)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   130
    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   131
      by (simp only: mult.assoc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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   132
    finally show "norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   133
                  K * norm (z ^ n) * inverse (norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   134
      by (simp add: mult_le_cancel_right x_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   135
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   136
  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   137
  proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   138
    from 2 have "norm (norm (z * inverse x)) < 1"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   139
      using x_neq_0
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   140
      by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   141
    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   142
      by (rule summable_geometric)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   143
    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   144
      by (rule summable_mult)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   145
    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   146
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   147
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   148
                    power_inverse norm_power mult.assoc)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   149
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   150
  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   151
    by (rule summable_comparison_test)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   152
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   153
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   154
lemma powser_inside:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   155
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   156
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   157
    "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   158
      summable (\<lambda>n. f n * (z ^ n))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   159
  by (rule powser_insidea [THEN summable_norm_cancel])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   160
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   161
lemma sum_split_even_odd:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   162
  fixes f :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   163
  shows
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   164
    "(\<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
   165
     (\<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
   166
proof (induct n)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   167
  case 0
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   168
  then show ?case by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   169
next
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   170
  case (Suc n)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   171
  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
   172
    (\<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
   173
    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
   174
  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
   175
    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
  finally show ?case .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   177
qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   178
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   179
lemma sums_if':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   180
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   181
  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
   182
  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
   183
  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
   184
proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   185
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   186
  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
   187
  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
   188
  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
   189
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   190
  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
   191
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   192
    fix m
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   193
    assume "m \<ge> 2 * no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   194
    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
   195
    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
   196
      using sum_split_even_odd by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   197
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   198
      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
   199
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   200
    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
   201
    proof (cases "even m")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   202
      case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   203
      show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   204
        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
   205
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   206
      case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   207
      hence "even (Suc m)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   208
      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   209
        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
   210
      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
   211
      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
   212
      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
   213
      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
   214
      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
   215
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   216
    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
   217
  }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   218
  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
   219
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   220
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   221
lemma sums_if:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   222
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   223
  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
   224
  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
   225
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   226
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   227
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   228
    fix B T E
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   229
    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
   230
      by (cases B) auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   231
  } note if_sum = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   232
  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
   233
    using sums_if'[OF `g sums x`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   234
  {
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 38642
diff changeset
   235
    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
   236
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   237
    have "?s sums y" using sums_if'[OF `f sums y`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   238
    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
   239
    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   240
      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
   241
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   242
  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
   243
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   244
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   245
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
   246
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   247
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
   248
  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
   249
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   250
  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>
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   251
             ((\<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
   252
  (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
   253
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
   254
  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
   255
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   256
  show "\<forall>n. ?f n \<le> ?f (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   257
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   258
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   259
    show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   260
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   261
  show "\<forall>n. ?g (Suc n) \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   262
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   263
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   264
    show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   265
      unfolding One_nat_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   266
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   267
  show "\<forall>n. ?f n \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   268
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   269
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   270
    show "?f n \<le> ?g n" using fg_diff a_pos
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   271
      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
   272
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   273
  show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   274
  proof (rule LIMSEQ_I)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   275
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   276
    assume "0 < r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   277
    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
   278
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   279
    hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   280
    thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   281
  qed
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   282
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   283
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   284
lemma summable_Leibniz':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   285
  fixes a :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   286
  assumes a_zero: "a ----> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   287
    and a_pos: "\<And> n. 0 \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   288
    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
   289
  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
   290
    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
   291
    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
   292
    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
   293
    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
   294
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   295
  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
   296
  let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   297
  let ?f = "\<lambda>n. ?P (2 * n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   298
  let ?g = "\<lambda>n. ?P (2 * n + 1)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   299
  obtain l :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   300
    where below_l: "\<forall> n. ?f n \<le> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   301
      and "?f ----> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   302
      and above_l: "\<forall> n. l \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   303
      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
   304
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   305
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   306
  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
   307
  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
   308
  proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   309
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   310
    assume "0 < r"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   311
    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
   312
    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
   313
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   314
    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
   315
    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
   316
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   317
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   318
      fix n :: nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   319
      assume "n \<ge> (max (2 * f_no) (2 * g_no))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   320
      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
   321
      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
   322
      proof (cases "even n")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   323
        case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   324
        from even_nat_div_two_times_two[OF this]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   325
        have n_eq: "2 * (n div 2) = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   326
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   327
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   328
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   329
        from f[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   330
          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
   331
      next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   332
        case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   333
        hence "even (n - 1)" by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   334
        from even_nat_div_two_times_two[OF this]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   335
        have n_eq: "2 * ((n - 1) div 2) = n - 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   336
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   337
        hence range_eq: "n - 1 + 1 = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   338
          using odd_pos[OF False] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   339
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   340
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   341
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   342
        from g[OF this] show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   343
          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
   344
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   345
    }
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   346
    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
   347
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   348
  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
   349
    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
   350
  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
   351
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   352
  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
   353
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   354
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   355
  show "suminf ?S \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   356
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   357
  show "?f n \<le> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   358
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   359
  show "?g ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   360
    using `?g ----> l` `l = suminf ?S` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   361
  show "?f ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   362
    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
   363
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   364
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   365
theorem summable_Leibniz:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   366
  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
   367
  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
   368
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   369
    and "0 < a 0 \<longrightarrow>
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   370
      (\<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
   371
    and "a 0 < 0 \<longrightarrow>
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   372
      (\<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")
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   373
    and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?f")
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   374
    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
   375
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   376
  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
   377
  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
   378
    case True
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   379
    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
   380
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   381
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   382
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   383
      have "a (Suc n) \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   384
        using ord[where n="Suc n" and m=n] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   385
    } note mono = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   386
    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
   387
    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
   388
    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
   389
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   390
    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
   391
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   392
    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
   393
    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
   394
    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
   395
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   396
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   397
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   398
      have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   399
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   400
    } note monotone = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   401
    note leibniz =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   402
      summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   403
        OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   404
    have "summable (\<lambda> n. (-1)^n * ?a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   405
      using leibniz(1) by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   406
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   407
      unfolding summable_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   408
    from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   409
      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
   410
    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
   411
    moreover
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   412
    have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   413
      unfolding minus_diff_minus by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   414
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   415
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   416
    have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   417
      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
   418
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   419
    have ?pos using `0 \<le> ?a 0` by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   420
    moreover have ?neg
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   421
      using leibniz(2,4)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   422
      unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   423
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   424
    moreover have ?f and ?g
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   425
      using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   426
      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
   427
    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
   428
  qed
54576
e877eec2b698 tidied more proofs
paulson
parents: 54575
diff changeset
   429
  then show ?summable and ?pos and ?neg and ?f and ?g 
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   430
    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
   431
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   432
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   433
subsection {* Term-by-Term Differentiability of Power Series *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   434
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   435
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
   436
  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
   437
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   438
text{*Lemma about distributing negation over it*}
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   439
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   440
  by (simp add: diffs_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   441
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   442
lemma sums_Suc_imp:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   443
  "(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
   444
  using sums_Suc_iff[of f] by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   445
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   446
lemma diffs_equiv:
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   447
  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
   448
  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
   449
      (\<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
   450
  unfolding diffs_def
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   451
  by (simp add: summable_sums sums_Suc_imp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   452
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   453
lemma lemma_termdiff1:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   454
  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
   455
  "(\<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
   456
   (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   457
  by (auto simp add: algebra_simps power_add [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   458
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   459
lemma sumr_diff_mult_const2:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   460
  "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
   461
  by (simp add: setsum_subtractf)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   462
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   463
lemma lemma_termdiff2:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   464
  fixes h :: "'a :: {field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   465
  assumes h: "h \<noteq> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   466
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   467
    "((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
   468
     h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   469
          (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   470
  apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   471
  apply (simp add: right_diff_distrib diff_divide_distrib h)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   472
  apply (simp add: mult.assoc [symmetric])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   473
  apply (cases "n", simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   474
  apply (simp add: lemma_realpow_diff_sumr2 h
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   475
                   right_diff_distrib [symmetric] mult.assoc
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   476
              del: power_Suc setsum_lessThan_Suc of_nat_Suc)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   477
  apply (subst lemma_realpow_rev_sumr)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   478
  apply (subst sumr_diff_mult_const2)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   479
  apply simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   480
  apply (simp only: lemma_termdiff1 setsum_right_distrib)
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   481
  apply (rule setsum.cong [OF refl])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   482
  apply (simp add: less_iff_Suc_add)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   483
  apply (clarify)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   484
  apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 ac_simps
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   485
              del: setsum_lessThan_Suc power_Suc)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   486
  apply (subst mult.assoc [symmetric], subst power_add [symmetric])
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   487
  apply (simp add: ac_simps)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   488
  done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   489
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   490
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
   491
  fixes K :: "'a::linordered_semidom"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   492
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   493
    and K: "0 \<le> K"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   494
  shows "setsum f {..<n-k} \<le> of_nat n * K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   495
  apply (rule order_trans [OF setsum_mono])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   496
  apply (rule f, simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   497
  apply (simp add: mult_right_mono K)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   498
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   499
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   500
lemma lemma_termdiff3:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   501
  fixes h z :: "'a::{real_normed_field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   502
  assumes 1: "h \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   503
    and 2: "norm z \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   504
    and 3: "norm (z + h) \<le> K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   505
  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
   506
          \<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
   507
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   508
  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
   509
        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
   510
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   511
    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
   512
  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
   513
  proof (rule mult_right_mono [OF _ norm_ge_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   514
    from norm_ge_zero 2 have K: "0 \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   515
      by (rule order_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   516
    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
   517
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   518
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   519
      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
   520
      done
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   521
    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
   522
          \<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
   523
      apply (intro
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   524
         order_trans [OF norm_setsum]
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   525
         real_setsum_nat_ivl_bounded2
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   526
         mult_nonneg_nonneg
47489
04e7d09ade7a tuned some proofs;
huffman
parents: 47108
diff changeset
   527
         of_nat_0_le_iff
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   528
         zero_le_power K)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   529
      apply (rule le_Kn, simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   530
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   531
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   532
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   533
    by (simp only: mult.assoc)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   534
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   535
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   536
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   537
lemma lemma_termdiff4:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   538
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   539
  assumes k: "0 < (k::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   540
    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
   541
  shows "f -- 0 --> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   542
proof (rule tendsto_norm_zero_cancel)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   543
  show "(\<lambda>h. norm (f h)) -- 0 --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   544
  proof (rule real_tendsto_sandwich)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   545
    show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   546
      by simp
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   547
    show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   548
      using k by (auto simp add: eventually_at dist_norm le)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   549
    show "(\<lambda>h. 0) -- (0::'a) --> (0::real)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   550
      by (rule tendsto_const)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   551
    have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   552
      by (intro tendsto_intros)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   553
    then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   554
      by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   555
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   556
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   557
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   558
lemma lemma_termdiff5:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   559
  fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   560
  assumes k: "0 < (k::real)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   561
  assumes f: "summable f"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   562
  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
   563
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   564
proof (rule lemma_termdiff4 [OF k])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   565
  fix h::'a
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   566
  assume "h \<noteq> 0" and "norm h < k"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   567
  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
   568
    by (simp add: le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   569
  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
   570
    by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   571
  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
   572
    by (rule summable_mult2)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   573
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   574
    by (rule summable_comparison_test)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   575
  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
   576
    by (rule summable_norm)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   577
  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
   578
    by (rule suminf_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   579
  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
   580
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   581
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   582
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   583
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   584
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   585
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   586
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   587
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   588
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   589
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   590
    and 2: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   591
  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
   592
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   593
proof -
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   594
  from dense [OF 2]
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   595
  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
   596
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   597
    by (rule order_le_less_trans)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   598
  hence r_neq_0: "r \<noteq> 0" by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   599
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   600
  proof (rule lemma_termdiff5)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   601
    show "0 < r - norm x" using r1 by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   602
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   603
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   604
    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
   605
      by (rule powser_insidea)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   606
    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
   607
      using r
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   608
      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
   609
    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
   610
      by (rule diffs_equiv [THEN sums_summable])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   611
    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
   612
      (\<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
   613
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   614
      apply (simp add: diffs_def)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   615
      apply (case_tac n, simp_all add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   616
      done
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   617
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   618
      (\<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
   619
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   620
    also have
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   621
      "(\<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
   622
           r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   623
       (\<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
   624
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   625
      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
   626
      apply (rename_tac nat)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   627
      apply (case_tac "nat", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   628
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   629
      done
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   630
    finally
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   631
    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
   632
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   633
    fix h::'a and n::nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   634
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   635
    assume "norm h < r - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   636
    hence "norm x + norm h < r" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   637
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   638
      by (rule order_le_less_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   639
    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
   640
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   641
      apply (simp only: norm_mult mult.assoc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   642
      apply (rule mult_left_mono [OF _ norm_ge_zero])
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   643
      apply (simp add: mult.assoc [symmetric])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   644
      apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   645
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   646
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   647
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   648
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   649
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   650
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   651
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   652
      and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   653
      and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   654
      and 4: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   655
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   656
  unfolding DERIV_def
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   657
proof (rule LIM_zero_cancel)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   658
  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
   659
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   660
  proof (rule LIM_equal2)
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   661
    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
   662
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   663
    fix h :: 'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   664
    assume "norm (h - 0) < norm K - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   665
    hence "norm x + norm h < norm K" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   666
    hence 5: "norm (x + h) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   667
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   668
    have "summable (\<lambda>n. c n * x ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   669
      and "summable (\<lambda>n. c n * (x + h) ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   670
      and "summable (\<lambda>n. diffs c n * x ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   671
      using 1 2 4 5 by (auto elim: powser_inside)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   672
    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
   673
          (\<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
   674
      by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   675
    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
   676
          (\<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
   677
      by (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   678
  next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   679
    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
   680
      by (rule termdiffs_aux [OF 3 4])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   681
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   682
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   683
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   684
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   685
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
   686
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   687
lemma DERIV_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   688
  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
   689
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   690
    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
   691
    and "summable (f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   692
    and "summable L"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   693
    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
   694
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   695
  unfolding DERIV_def
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   696
proof (rule LIM_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   697
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   698
  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
   699
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   700
  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
   701
    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
   702
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   703
  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
   704
    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
   705
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   706
  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
   707
  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
   708
    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
   709
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   710
  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
   711
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   712
  let ?r = "r / (3 * real ?N)"
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   713
  from `0 < r` have "0 < ?r" by simp
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   714
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   715
  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
   716
  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
   717
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   718
  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
   719
  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
   720
    show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   721
    proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   722
      fix x
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   723
      assume "x \<in> ?s ` {..<?N}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   724
      then obtain n where "x = ?s n" and "n \<in> {..<?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   725
        using image_iff[THEN iffD1] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   726
      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
   727
      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
   728
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   729
      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
   730
      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
   731
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   732
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   733
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   734
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   735
  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
   736
    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
   737
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   738
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   739
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   740
    fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   741
    assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   742
    hence x_in_I: "x0 + x \<in> { a <..< b }"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   743
      using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   744
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   745
    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
   746
    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
   747
    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
   748
    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
   749
    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
   750
    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
   751
    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
   752
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   753
    { fix n
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   754
      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
   755
        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
   756
        unfolding abs_divide .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   757
      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
   758
        using `x \<noteq> 0` by auto }
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   759
    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
   760
    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
   761
      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
   762
    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
   763
      using L_estimate by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   764
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   765
    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
   766
    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
   767
    proof (rule setsum_strict_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   768
      fix n
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   769
      assume "n \<in> {..< ?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   770
      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
   771
      also have "S \<le> S'" using `S \<le> S'` .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   772
      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
   773
      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
   774
        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
   775
          using `n \<in> {..< ?N}` by auto
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   776
        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
   777
      qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   778
      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
   779
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   780
      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
   781
      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
   782
      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
   783
        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
   784
    qed auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   785
    also have "\<dots> = of_nat (card {..<?N}) * ?r"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   786
      by (rule setsum_constant)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   787
    also have "\<dots> = real ?N * ?r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   788
      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
   789
    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
   790
    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
   791
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   792
    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
   793
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   794
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   795
      unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   796
      using suminf_divide[OF diff_smbl, symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   797
    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
   798
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   799
      unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   800
      apply (subst (5) add.commute)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   801
      by (rule abs_triangle_ineq)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   802
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   803
      using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   804
    also have "\<dots> < r /3 + r/3 + r/3"
36842
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
   805
      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
   806
      by (rule add_strict_mono [OF add_less_le_mono])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   807
    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
   808
      by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   809
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   810
  thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   811
      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
   812
    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
   813
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   814
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   815
lemma DERIV_power_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   816
  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
   817
  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
   818
    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
   819
  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
   820
  (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
   821
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   822
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   823
    fix R'
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   824
    assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   825
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   826
      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
   827
    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
   828
    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
   829
      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
   830
      proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   831
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   832
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   833
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   834
          using `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   835
        have "norm R' < norm ((R' + R) / 2)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   836
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   837
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   838
          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
   839
      qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   840
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   841
        fix n x y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   842
        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
   843
        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
   844
        proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   845
          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
   846
            (\<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
   847
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   848
            by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   849
          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
   850
          proof (rule mult_left_mono)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   851
            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
   852
              by (rule setsum_abs)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   853
            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
   854
            proof (rule setsum_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   855
              fix p
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   856
              assume "p \<in> {..<Suc n}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   857
              hence "p \<le> n" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   858
              {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   859
                fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   860
                fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   861
                assume "x \<in> {-R'<..<R'}"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   862
                hence "\<bar>x\<bar> \<le> R'"  by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   863
                hence "\<bar>x^n\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   864
                  unfolding power_abs by (rule power_mono, auto)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   865
              }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   866
              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
   867
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   868
                unfolding abs_mult by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   869
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   870
                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
   871
            qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   872
            also have "\<dots> = real (Suc n) * R' ^ n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   873
              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
   874
            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
   875
              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
   876
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   877
              unfolding abs_mult[symmetric] by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   878
          qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   879
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   880
            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
   881
          finally show ?thesis .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   882
        qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   883
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   884
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   885
        fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   886
        show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   887
          by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   888
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   889
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   890
        fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   891
        assume "x \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   892
        hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   893
          using assms `R' < R` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   894
        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
   895
        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
   896
          fix n
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   897
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   898
            by (rule mult_left_mono) auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   899
          show "norm (f n * x ^ n) \<le> norm (f n * real (Suc n) * x ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   900
            unfolding real_norm_def abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   901
            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
   902
        qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`])
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   903
        from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   904
        show "summable (?f x)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   905
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   906
      show "summable (?f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   907
        using converges[OF `x0 \<in> {-R <..< R}`] .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   908
      show "x0 \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   909
        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
   910
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   911
  } 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
   912
  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
   913
  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
   914
  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
   915
  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
   916
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   917
    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
   918
    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
   919
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   920
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   921
    have "- ?R < 0" using assms by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   922
    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
   923
    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
   924
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   925
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   926
    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
   927
  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
   928
  show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   929
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   930
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   931
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   932
subsection {* Exponential Function *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   933
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
   934
definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   935
  where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   936
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   937
lemma summable_exp_generic:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   938
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   939
  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
   940
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   941
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   942
  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
   943
    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
   944
  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
   945
    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
   946
  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
   947
    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
   948
  from r1 show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   949
  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
   950
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   951
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   952
    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
   953
      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
   954
    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
   955
      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
   956
    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
   957
      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
   958
    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
   959
      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
   960
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   961
      by (simp add: pos_divide_le_eq ac_simps)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   962
    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
   963
      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
   964
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   965
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   966
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   967
lemma summable_norm_exp:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   968
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   969
  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
   970
proof (rule summable_norm_comparison_test [OF exI, rule_format])
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   971
  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
   972
    by (rule summable_exp_generic)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   973
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   974
  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
   975
    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
   976
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   977
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   978
lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   979
  using summable_exp_generic [where x=x] by simp
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   980
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   981
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   982
  unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   983
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   984
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   985
lemma exp_fdiffs:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   986
      "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   987
  by (simp add: diffs_def mult.assoc [symmetric] real_of_nat_def of_nat_mult
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   988
        del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   989
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   990
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
   991
  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
   992
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   993
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   994
  unfolding exp_def scaleR_conv_of_real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   995
  apply (rule DERIV_cong)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   996
  apply (rule termdiffs [where K="of_real (1 + norm x)"])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   997
  apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   998
  apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   999
  apply (simp del: of_real_add)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1000
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1001
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  1002
declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  1003
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1004
lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1005
proof -
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1006
  from summable_norm[OF summable_norm_exp, of x]
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1007
  have "norm (exp x) \<le> (\<Sum>n. inverse (real (fact n)) * norm (x ^ n))"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1008
    by (simp add: exp_def)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1009
  also have "\<dots> \<le> exp (norm x)"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1010
    using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1011
    by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1012
  finally show ?thesis .
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1013
qed
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1014
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1015
lemma isCont_exp:
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1016
  fixes x::"'a::{real_normed_field,banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1017
  shows "isCont exp x"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1018
  by (rule DERIV_exp [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1019
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1020
lemma isCont_exp' [simp]:
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1021
  fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1022
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1023
  by (rule isCont_o2 [OF _ isCont_exp])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1024
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1025
lemma tendsto_exp [tendsto_intros]:
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1026
  fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1027
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1028
  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
  1029
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1030
lemma continuous_exp [continuous_intros]:
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1031
  fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1032
  shows "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
  1033
  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
  1034
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  1035
lemma continuous_on_exp [continuous_intros]:
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1036
  fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1037
  shows "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
  1038
  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
  1039
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1040
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1041
subsubsection {* Properties of the Exponential Function *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1042
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1043
lemma powser_zero:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1044
  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
  1045
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1046
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1047
  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
  1048
    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
  1049
  thus ?thesis unfolding One_nat_def by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1050
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1051
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1052
lemma exp_zero [simp]: "exp 0 = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1053
  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
  1054
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1055
lemma exp_series_add_commuting:
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1056
  fixes x y :: "'a::{real_normed_algebra_1, banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1057
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1058
  assumes comm: "x * y = y * x"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1059
  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
  1060
proof (induct n)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1061
  case 0
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1062
  show ?case
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1063
    unfolding S_def by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1064
next
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1065
  case (Suc n)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1066
  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
  1067
    unfolding S_def by (simp del: mult_Suc)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1068
  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
  1069
    by simp
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1070
  have S_comm: "\<And>n. S x n * y = y * S x n"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1071
    by (simp add: power_commuting_commutes comm S_def)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1072
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1073
  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
  1074
    by (simp only: times_S)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1075
  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
  1076
    by (simp only: Suc)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1077
  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
  1078
                + 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
  1079
    by (rule distrib_right)
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1080
  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1081
                + (\<Sum>i\<le>n. S x i * y * S y (n-i))"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1082
    by (simp add: setsum_right_distrib ac_simps S_comm)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1083
  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1084
                + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1085
    by (simp add: ac_simps)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1086
  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
  1087
                + (\<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
  1088
    by (simp add: times_S Suc_diff_le)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1089
  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
  1090
             (\<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
  1091
    by (subst setsum_atMost_Suc_shift) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1092
  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
  1093
             (\<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
  1094
    by simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1095
  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
  1096
             (\<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
  1097
             (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
  1098
    by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric]
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1099
                   real_of_nat_add [symmetric]) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1100
  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
  1101
    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
  1102
  finally show
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1103
    "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
  1104
    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
  1105
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1106
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1107
lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1108
  unfolding exp_def
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1109
  by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1110
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1111
lemma exp_add:
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1112
  fixes x y::"'a::{real_normed_field,banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1113
  shows "exp (x + y) = exp x * exp y"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1114
  by (rule exp_add_commuting) (simp add: ac_simps)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1115
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1116
lemmas mult_exp_exp = exp_add [symmetric]
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1117
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1118
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1119
  unfolding exp_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1120
  apply (subst suminf_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1121
  apply (rule summable_exp_generic)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1122
  apply (simp add: scaleR_conv_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1123
  done
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1124
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1125
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1126
proof
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1127
  have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric])
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1128
  also assume "exp x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1129
  finally show "False" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1130
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1131
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1132
lemma exp_minus_inverse:
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1133
  shows "exp x * exp (- x) = 1"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1134
  by (simp add: exp_add_commuting[symmetric])
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1135
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1136
lemma exp_minus:
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1137
  fixes x :: "'a::{real_normed_field, banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1138
  shows "exp (- x) = inverse (exp x)"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1139
  by (intro inverse_unique [symmetric] exp_minus_inverse)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1140
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1141
lemma exp_diff:
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1142
  fixes x :: "'a::{real_normed_field, banach}"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1143
  shows "exp (x - y) = exp x / exp y"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
  1144
  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
  1145
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1146
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1147
subsubsection {* Properties of the Exponential Function on Reals *}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1148
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1149
text {* Comparisons of @{term "exp x"} with zero. *}
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1150
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1151
text{*Proof: because every exponential can be seen as a square.*}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1152
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1153
proof -
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1154
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1155
  thus ?thesis by (simp add: exp_add [symmetric])
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1156
qed
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1157
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1158
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1159
  by (simp add: order_less_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1160
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1161
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1162
  by (simp add: not_less)
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1163
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1164
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1165
  by (simp add: not_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1166
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1167
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1168
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1169
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1170
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1171
  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
  1172
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1173
text {* Strict monotonicity of exponential. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1174
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1175
lemma exp_ge_add_one_self_aux: 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1176
  assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1177
using order_le_imp_less_or_eq [OF assms]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1178
proof 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1179
  assume "0 < x"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1180
  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
  1181
    by (auto simp add: numeral_2_eq_2)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1182
  also have "... \<le> (\<Sum>n. inverse (real (fact n)) * x ^ n)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1183
    apply (rule setsum_le_suminf [OF summable_exp])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1184
    using `0 < x`
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1185
    apply (auto  simp add:  zero_le_mult_iff)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1186
    done
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1187
  finally show "1+x \<le> exp x" 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1188
    by (simp add: exp_def)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1189
next
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1190
  assume "0 = x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1191
  then show "1 + x \<le> exp x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1192
    by auto
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1193
qed
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1194
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1195
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1196
proof -
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1197
  assume x: "0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1198
  hence "1 < 1 + x" by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1199
  also from x have "1 + x \<le> exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1200
    by (simp add: exp_ge_add_one_self_aux)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1201
  finally show ?thesis .
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1202
qed
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1203
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1204
lemma exp_less_mono:
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1205
  fixes x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1206
  assumes "x < y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1207
  shows "exp x < exp y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1208
proof -
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1209
  from `x < y` have "0 < y - x" by simp
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1210
  hence "1 < exp (y - x)" by (rule exp_gt_one)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1211
  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
  1212
  thus "exp x < exp y" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1213
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1214
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1215
lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1216
  unfolding linorder_not_le [symmetric]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1217
  by (auto simp add: order_le_less exp_less_mono)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1218
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1219
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1220
  by (auto intro: exp_less_mono exp_less_cancel)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1221
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1222
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"