src/HOL/Transcendental.thy
author paulson <lp15@cam.ac.uk>
Mon, 24 Feb 2014 16:56:04 +0000
changeset 55719 cdddd073bff8
parent 55417 01fbfb60c33e
child 55734 3f5b2745d659
permissions -rw-r--r--
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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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*)
a3be6b3a9c0b new theories from Jacques Fleuriot
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header{*Power Series, Transcendental Functions etc.*}
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theory Transcendental
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imports Fact Series Deriv NthRoot
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begin
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subsection {* Properties of Power Series *}
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lemma lemma_realpow_diff:
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  fixes y :: "'a::monoid_mult"
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  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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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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ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
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  thus ?thesis by (simp add: power_commutes)
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qed
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lemma lemma_realpow_diff_sumr:
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  fixes y :: "'a::{comm_semiring_0,monoid_mult}"
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  shows
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    "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
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      y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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  by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac del: setsum_op_ivl_Suc)
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lemma lemma_realpow_diff_sumr2:
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  fixes y :: "'a::{comm_ring,monoid_mult}"
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  shows
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    "x ^ (Suc n) - y ^ (Suc n) =
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      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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proof (induct n)
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  case 0 show ?case
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    by simp
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next
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  case (Suc n)
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  have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x ^ n) - y * (y * y ^ n)"
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    by simp
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  also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x ^ n)"
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    by (simp add: algebra_simps)
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  also have "... = y * ((x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    by (simp only: Suc)
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  also have "... = (x - y) * (y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    by (simp only: mult_left_commute)
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  also have "... = (x - y) * (\<Sum>p = 0..<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
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    by (simp add: setsum_op_ivl_Suc [where n = "Suc n"] distrib_left lemma_realpow_diff_sumr
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             del: setsum_op_ivl_Suc)
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  finally show ?case .
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qed
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lemma lemma_realpow_rev_sumr:
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   "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
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    (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
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  apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
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  apply (rule inj_onI, auto)
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  apply (metis atLeastLessThan_iff diff_diff_cancel diff_less_Suc imageI le0 less_Suc_eq_le)
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  done
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lemma power_diff_1_eq:
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  fixes x :: "'a::{comm_ring,monoid_mult}"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
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    65
  shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i=0..<n. (x^i))"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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using lemma_realpow_diff_sumr2 [of x _ 1] 
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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  by (cases n) auto
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cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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lemma one_diff_power_eq':
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  fixes x :: "'a::{comm_ring,monoid_mult}"
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paulson <lp15@cam.ac.uk>
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    71
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i=0..<n. x^(n - Suc i))"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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using lemma_realpow_diff_sumr2 [of 1 _ x] 
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  by (cases n) auto
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cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
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lemma one_diff_power_eq:
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  fixes x :: "'a::{comm_ring,monoid_mult}"
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paulson <lp15@cam.ac.uk>
parents: 55417
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    77
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i=0..<n. x^i)"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
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    78
by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
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    79
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
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  x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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lemma powser_insidea:
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  fixes x z :: "'a::real_normed_div_algebra"
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    85
  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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    and 2: "norm z < norm x"
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  shows "summable (\<lambda>n. norm (f n * z ^ n))"
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    88
proof -
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  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
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    90
  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
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    91
    by (rule summable_LIMSEQ_zero)
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    92
  hence "convergent (\<lambda>n. f n * x ^ n)"
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huffman
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    93
    by (rule convergentI)
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    94
  hence "Cauchy (\<lambda>n. f n * x ^ n)"
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    95
    by (rule convergent_Cauchy)
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huffman
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    96
  hence "Bseq (\<lambda>n. f n * x ^ n)"
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huffman
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    97
    by (rule Cauchy_Bseq)
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    98
  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
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    99
    by (simp add: Bseq_def, safe)
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   100
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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                   K * norm (z ^ n) * inverse (norm (x ^ n))"
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   102
  proof (intro exI allI impI)
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    fix n::nat
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   104
    assume "0 \<le> n"
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ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   105
    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
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huffman
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   106
          norm (f n * x ^ n) * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
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   107
      by (simp add: norm_mult abs_mult)
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huffman
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   108
    also have "\<dots> \<le> K * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
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   109
      by (simp only: mult_right_mono 4 norm_ge_zero)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
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   110
    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
20849
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huffman
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   111
      by (simp add: x_neq_0)
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   112
    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
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   113
      by (simp only: mult_assoc)
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   114
    finally show "norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
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   115
                  K * norm (z ^ n) * inverse (norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
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   116
      by (simp add: mult_le_cancel_right x_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
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parents: 20692
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   117
  qed
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   118
  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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huffman
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   119
  proof -
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   120
    from 2 have "norm (norm (z * inverse x)) < 1"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   121
      using x_neq_0
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78ea983f7987 generalize lemmas
huffman
parents: 53079
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   122
      by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   123
    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
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parents: 20692
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   124
      by (rule summable_geometric)
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ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   125
    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
20849
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parents: 20692
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   126
      by (rule summable_mult)
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parents: 23069
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   127
    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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   128
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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   129
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
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   130
                    power_inverse norm_power mult_assoc)
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   131
  qed
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   132
  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
20849
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   133
    by (rule summable_comparison_test)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   134
qed
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paulson
parents: 15013
diff changeset
   135
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   136
lemma powser_inside:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   137
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   138
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   139
    "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   140
      summable (\<lambda>n. f n * (z ^ n))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   141
  by (rule powser_insidea [THEN summable_norm_cancel])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   142
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   143
lemma sum_split_even_odd:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   144
  fixes f :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   145
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   146
    "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   147
     (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< 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
   148
proof (induct n)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   149
  case 0
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   150
  then show ?case by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   151
next
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   152
  case (Suc n)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   153
  have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   154
    (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< 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
   155
    using Suc.hyps unfolding One_nat_def by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   156
  also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   157
    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
   158
  finally show ?case .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   159
qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   160
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   161
lemma sums_if':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   162
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   163
  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
   164
  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
   165
  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
   166
proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   167
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   168
  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
   169
  from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   170
  obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   171
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   172
  let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   173
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   174
    fix m
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   175
    assume "m \<ge> 2 * no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   176
    hence "m div 2 \<ge> no" by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   177
    have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   178
      using sum_split_even_odd by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   179
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   180
      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
   181
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   182
    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
   183
    proof (cases "even m")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   184
      case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   185
      show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   186
        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
   187
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   188
      case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   189
      hence "even (Suc m)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   190
      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   191
        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
   192
      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
   193
      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
   194
      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
   195
      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
   196
      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
   197
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   198
    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
   199
  }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   200
  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
   201
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   202
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   203
lemma sums_if:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   204
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   205
  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
   206
  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
   207
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   208
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   209
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   210
    fix B T E
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   211
    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
   212
      by (cases B) auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   213
  } note if_sum = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   214
  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
   215
    using sums_if'[OF `g sums x`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   216
  {
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   217
    have "?s 0 = 0" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   218
    have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 38642
diff changeset
   219
    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
   220
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   221
    have "?s sums y" using sums_if'[OF `f sums y`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   222
    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
   223
    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   224
      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   225
                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
31148
7ba7c1f8bc22 Cleaned up Parity a little
nipkow
parents: 31017
diff changeset
   226
                even_Suc Suc_m1 if_eq .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   227
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   228
  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
   229
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   230
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   231
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
   232
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   233
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
   234
  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
   235
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   236
  shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   237
             ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   238
  (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
   239
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
   240
  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
   241
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   242
  show "\<forall>n. ?f n \<le> ?f (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   243
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   244
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   245
    show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   246
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   247
  show "\<forall>n. ?g (Suc n) \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   248
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   249
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   250
    show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   251
      unfolding One_nat_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   252
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   253
  show "\<forall>n. ?f n \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   254
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   255
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   256
    show "?f n \<le> ?g n" using fg_diff a_pos
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   257
      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
   258
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   259
  show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   260
  proof (rule LIMSEQ_I)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   261
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   262
    assume "0 < r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   263
    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
   264
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   265
    hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   266
    thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   267
  qed
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   268
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   269
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   270
lemma summable_Leibniz':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   271
  fixes a :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   272
  assumes a_zero: "a ----> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   273
    and a_pos: "\<And> n. 0 \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   274
    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
   275
  shows summable: "summable (\<lambda> n. (-1)^n * a n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   276
    and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   277
    and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   278
    and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   279
    and "(\<lambda>n. \<Sum>i=0..<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
   280
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   281
  let ?S = "\<lambda>n. (-1)^n * a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   282
  let ?P = "\<lambda>n. \<Sum>i=0..<n. ?S i"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   283
  let ?f = "\<lambda>n. ?P (2 * n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   284
  let ?g = "\<lambda>n. ?P (2 * n + 1)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   285
  obtain l :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   286
    where below_l: "\<forall> n. ?f n \<le> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   287
      and "?f ----> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   288
      and above_l: "\<forall> n. l \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   289
      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
   290
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   291
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   292
  let ?Sa = "\<lambda>m. \<Sum> n = 0..<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
   293
  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
   294
  proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   295
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   296
    assume "0 < r"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   297
    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
   298
    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
   299
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   300
    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
   301
    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
   302
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   303
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   304
      fix n :: nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   305
      assume "n \<ge> (max (2 * f_no) (2 * g_no))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   306
      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
   307
      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
   308
      proof (cases "even n")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   309
        case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   310
        from even_nat_div_two_times_two[OF this]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   311
        have n_eq: "2 * (n div 2) = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   312
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   313
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   314
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   315
        from f[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   316
          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
   317
      next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   318
        case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   319
        hence "even (n - 1)" by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   320
        from even_nat_div_two_times_two[OF this]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   321
        have n_eq: "2 * ((n - 1) div 2) = n - 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   322
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   323
        hence range_eq: "n - 1 + 1 = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   324
          using odd_pos[OF False] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   325
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   326
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   327
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   328
        from g[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   329
          unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   330
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   331
    }
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   332
    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
   333
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   334
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   335
    unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   336
  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
   337
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   338
  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
   339
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   340
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   341
  show "suminf ?S \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   342
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   343
  show "?f n \<le> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   344
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   345
  show "?g ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   346
    using `?g ----> l` `l = suminf ?S` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   347
  show "?f ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   348
    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
   349
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   350
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   351
theorem summable_Leibniz:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   352
  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
   353
  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
   354
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   355
    and "0 < a 0 \<longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   356
      (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   357
    and "a 0 < 0 \<longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   358
      (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   359
    and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   360
    and "(\<lambda>n. \<Sum>i=0..<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
   361
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   362
  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
   363
  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
   364
    case True
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   365
    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
   366
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   367
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   368
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   369
      have "a (Suc n) \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   370
        using ord[where n="Suc n" and m=n] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   371
    } note mono = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   372
    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
   373
    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
   374
    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
   375
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   376
    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
   377
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   378
    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
   379
    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
   380
    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
   381
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   382
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   383
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   384
      have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   385
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   386
    } note monotone = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   387
    note leibniz =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   388
      summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   389
        OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   390
    have "summable (\<lambda> n. (-1)^n * ?a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   391
      using leibniz(1) by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   392
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   393
      unfolding summable_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   394
    from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   395
      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
   396
    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
   397
    moreover
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   398
    have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   399
      unfolding minus_diff_minus by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   400
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   401
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   402
    have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   403
      by auto
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   404
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   405
    have ?pos using `0 \<le> ?a 0` by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   406
    moreover have ?neg
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   407
      using leibniz(2,4)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   408
      unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   409
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   410
    moreover have ?f and ?g
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   411
      using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   412
      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
   413
    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
   414
  qed
54576
e877eec2b698 tidied more proofs
paulson
parents: 54575
diff changeset
   415
  then show ?summable and ?pos and ?neg and ?f and ?g 
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   416
    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
   417
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   418
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   419
subsection {* Term-by-Term Differentiability of Power Series *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   420
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   421
definition diffs :: "(nat => 'a::ring_1) => nat => 'a"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   422
  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
   423
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   424
text{*Lemma about distributing negation over it*}
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   425
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   426
  by (simp add: diffs_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   427
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   428
lemma sums_Suc_imp:
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   429
  assumes f: "f 0 = 0"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   430
  shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   431
  unfolding sums_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   432
  apply (rule LIMSEQ_imp_Suc)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   433
  apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   434
  apply (simp only: setsum_shift_bounds_Suc_ivl)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   435
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   436
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   437
lemma diffs_equiv:
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   438
  fixes x :: "'a::{real_normed_vector, ring_1}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   439
  shows "summable (\<lambda>n. (diffs c)(n) * (x ^ n)) \<Longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   440
      (\<lambda>n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   441
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   442
  unfolding diffs_def
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   443
  by (simp add: summable_sums sums_Suc_imp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   444
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   445
lemma lemma_termdiff1:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   446
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   447
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   448
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   449
  by (auto simp add: algebra_simps power_add [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   450
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   451
lemma sumr_diff_mult_const2:
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   452
  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   453
  by (simp add: setsum_subtractf)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   454
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   455
lemma lemma_termdiff2:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   456
  fixes h :: "'a :: {field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   457
  assumes h: "h \<noteq> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   458
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   459
    "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   460
     h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   461
          (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   462
  apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   463
  apply (simp add: right_diff_distrib diff_divide_distrib h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   464
  apply (simp add: mult_assoc [symmetric])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   465
  apply (cases "n", simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   466
  apply (simp add: lemma_realpow_diff_sumr2 h
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   467
                   right_diff_distrib [symmetric] mult_assoc
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   468
              del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   469
  apply (subst lemma_realpow_rev_sumr)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   470
  apply (subst sumr_diff_mult_const2)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   471
  apply simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   472
  apply (simp only: lemma_termdiff1 setsum_right_distrib)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   473
  apply (rule setsum_cong [OF refl])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   474
  apply (simp add: less_iff_Suc_add)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   475
  apply (clarify)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   476
  apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   477
              del: setsum_op_ivl_Suc power_Suc)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   478
  apply (subst mult_assoc [symmetric], subst power_add [symmetric])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   479
  apply (simp add: mult_ac)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   480
  done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   481
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   482
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
   483
  fixes K :: "'a::linordered_semidom"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   484
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   485
    and K: "0 \<le> K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   486
  shows "setsum f {0..<n-k} \<le> of_nat n * K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   487
  apply (rule order_trans [OF setsum_mono])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   488
  apply (rule f, simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   489
  apply (simp add: mult_right_mono K)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   490
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   491
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   492
lemma lemma_termdiff3:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   493
  fixes h z :: "'a::{real_normed_field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   494
  assumes 1: "h \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   495
    and 2: "norm z \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   496
    and 3: "norm (z + h) \<le> K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   497
  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
   498
          \<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
   499
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   500
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   501
        norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   502
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   503
    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
   504
  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
   505
  proof (rule mult_right_mono [OF _ norm_ge_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   506
    from norm_ge_zero 2 have K: "0 \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   507
      by (rule order_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   508
    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
   509
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   510
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   511
      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
   512
      done
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   513
    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<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
   514
          \<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
   515
      apply (intro
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   516
         order_trans [OF norm_setsum]
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   517
         real_setsum_nat_ivl_bounded2
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   518
         mult_nonneg_nonneg
47489
04e7d09ade7a tuned some proofs;
huffman
parents: 47108
diff changeset
   519
         of_nat_0_le_iff
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   520
         zero_le_power K)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   521
      apply (rule le_Kn, simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   522
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   523
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   524
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   525
    by (simp only: mult_assoc)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   526
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   527
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   528
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   529
lemma lemma_termdiff4:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   530
  fixes f :: "'a::{real_normed_field} \<Rightarrow>
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   531
              'b::real_normed_vector"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   532
  assumes k: "0 < (k::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   533
    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
   534
  shows "f -- 0 --> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   535
  unfolding LIM_eq diff_0_right
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   536
proof safe
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   537
  let ?h = "of_real (k / 2)::'a"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   538
  have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   539
  hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   540
  hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   541
  hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   542
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   543
  fix r::real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   544
  assume r: "0 < r"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   545
  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   546
  proof cases
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   547
    assume "K = 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   548
    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   549
      by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   550
    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   551
  next
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   552
    assume K_neq_zero: "K \<noteq> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   553
    with zero_le_K have K: "0 < K" by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   554
    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   555
    proof (rule exI, safe)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   556
      from k r K
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   557
      show "0 < min k (r * inverse K / 2)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   558
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   559
    next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   560
      fix x::'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   561
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   562
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   563
        by simp_all
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   564
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   565
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   566
        by (rule mult_strict_left_mono)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   567
      also have "\<dots> = r / 2"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   568
        using K_neq_zero by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   569
      also have "r / 2 < r"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   570
        using r by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   571
      finally show "norm (f x) < r" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   572
    qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   573
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   574
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   575
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   576
lemma lemma_termdiff5:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   577
  fixes g :: "'a::real_normed_field \<Rightarrow> nat \<Rightarrow> 'b::banach"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   578
  assumes k: "0 < (k::real)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   579
  assumes f: "summable f"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   580
  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
   581
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   582
proof (rule lemma_termdiff4 [OF k])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   583
  fix h::'a
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   584
  assume "h \<noteq> 0" and "norm h < k"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   585
  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
   586
    by (simp add: le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   587
  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
   588
    by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   589
  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
   590
    by (rule summable_mult2)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   591
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   592
    by (rule summable_comparison_test)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   593
  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
   594
    by (rule summable_norm)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   595
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   596
    by (rule summable_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   597
  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
   598
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   599
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   600
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   601
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   602
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   603
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   604
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   605
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   606
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   607
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   608
    and 2: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   609
  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
   610
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   611
proof -
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   612
  from dense [OF 2]
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   613
  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
   614
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   615
    by (rule order_le_less_trans)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   616
  hence r_neq_0: "r \<noteq> 0" by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   617
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   618
  proof (rule lemma_termdiff5)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   619
    show "0 < r - norm x" using r1 by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   620
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   621
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   622
    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
   623
      by (rule powser_insidea)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   624
    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
   625
      using r
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   626
      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
   627
    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
   628
      by (rule diffs_equiv [THEN sums_summable])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   629
    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
   630
      (\<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
   631
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   632
      apply (simp add: diffs_def)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   633
      apply (case_tac n, simp_all add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   634
      done
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   635
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   636
      (\<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
   637
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   638
    also have
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   639
      "(\<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
   640
           r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   641
       (\<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
   642
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   643
      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
   644
      apply (rename_tac nat)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   645
      apply (case_tac "nat", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   646
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   647
      done
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   648
    finally
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   649
    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
   650
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   651
    fix h::'a and n::nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   652
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   653
    assume "norm h < r - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   654
    hence "norm x + norm h < r" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   655
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   656
      by (rule order_le_less_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   657
    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
   658
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   659
      apply (simp only: norm_mult mult_assoc)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   660
      apply (rule mult_left_mono [OF _ norm_ge_zero])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   661
      apply (simp add: mult_assoc [symmetric])
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   662
      apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   663
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   664
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   665
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   666
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   667
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   668
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   669
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   670
      and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   671
      and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   672
      and 4: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   673
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   674
  unfolding deriv_def
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   675
proof (rule LIM_zero_cancel)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   676
  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
   677
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   678
  proof (rule LIM_equal2)
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   679
    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
   680
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   681
    fix h :: 'a
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   682
    assume "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   683
    assume "norm (h - 0) < norm K - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   684
    hence "norm x + norm h < norm K" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   685
    hence 5: "norm (x + h) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   686
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   687
    have A: "summable (\<lambda>n. c n * x ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   688
      by (rule powser_inside [OF 1 4])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   689
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   690
      by (rule powser_inside [OF 1 5])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   691
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   692
      by (rule powser_inside [OF 2 4])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   693
    let ?dp = "(\<Sum>n. of_nat n * c n * x ^ (n - Suc 0))"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   694
    have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   695
          ((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - ?dp"  
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   696
      by (metis sums_unique [OF diffs_equiv [OF C]])
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   697
    also have "... = (\<Sum>n. c n * (x + h) ^ n - c n * x ^ n) / h - ?dp"  
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   698
      by (metis suminf_diff [OF B A])
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   699
    also have "... = (\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h)  - ?dp"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   700
      by (metis suminf_divide [OF summable_diff [OF B A]] )  
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   701
    also have "... = (\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h - of_nat n * c n * x ^ (n - Suc 0))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   702
      apply (subst suminf_diff)
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   703
      apply (auto intro: summable_divide summable_diff [OF B A] sums_summable [OF diffs_equiv [OF C]])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   704
      done
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   705
    also have "... = (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   706
      by (simp add: algebra_simps)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   707
    finally show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   708
                   - (\<Sum>n. diffs c n * x ^ n) =
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   709
                  (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))" . 
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   710
  next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   711
    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
   712
      by (rule termdiffs_aux [OF 3 4])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   713
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   714
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   715
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   716
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   717
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
   718
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   719
lemma DERIV_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   720
  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
   721
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   722
    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
   723
    and "summable (f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   724
    and "summable L"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   725
    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
   726
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   727
  unfolding deriv_def
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   728
proof (rule LIM_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   729
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   730
  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
   731
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   732
  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
   733
    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
   734
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   735
  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
   736
    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
   737
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   738
  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
   739
  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
   740
    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
   741
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   742
  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
   743
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   744
  let ?r = "r / (3 * real ?N)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   745
  have "0 < 3 * real ?N" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   746
  from divide_pos_pos[OF `0 < r` this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   747
  have "0 < ?r" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   748
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   749
  let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   750
  def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   751
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   752
  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
   753
  proof (rule iffD2[OF Min_gr_iff])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   754
    show "\<forall>x \<in> (?s ` { 0 ..< ?N }). 0 < x"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   755
    proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   756
      fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   757
      assume "x \<in> ?s ` {0..<?N}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   758
      then obtain n where "x = ?s n" and "n \<in> {0..<?N}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   759
        using image_iff[THEN iffD1] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   760
      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
   761
      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
   762
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   763
      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
   764
      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
   765
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   766
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   767
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   768
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   769
  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
   770
    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
   771
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   772
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   773
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   774
    fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   775
    assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   776
    hence x_in_I: "x0 + x \<in> { a <..< b }"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   777
      using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   778
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   779
    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
   780
    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
   781
    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
   782
    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
   783
    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
   784
    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
   785
    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
   786
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   787
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   788
      fix n
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   789
      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
   790
        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
   791
        unfolding abs_divide .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   792
      hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   793
        using `x \<noteq> 0` 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
   794
    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   795
    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   796
    have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   797
    hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   798
      using L_estimate by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   799
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   800
    have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   801
      (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   802
    also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   803
    proof (rule setsum_strict_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   804
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   805
      assume "n \<in> { 0 ..< ?N}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   806
      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
   807
      also have "S \<le> S'" using `S \<le> S'` .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   808
      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
   809
      proof (rule Min_le_iff[THEN iffD2])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   810
        have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   811
          using `n \<in> { 0 ..< ?N}` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   812
        thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   813
      qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   814
      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
   815
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   816
      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
   817
      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
   818
      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
   819
        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
   820
    qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   821
    also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   822
      by (rule setsum_constant)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   823
    also have "\<dots> = real ?N * ?r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   824
      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
   825
    also have "\<dots> = r/3" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   826
    finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   827
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   828
    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
   829
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   830
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   831
      unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   832
      using suminf_divide[OF diff_smbl, symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   833
    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
   834
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   835
      unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   836
      by (rule abs_triangle_ineq)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   837
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   838
      using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   839
    also have "\<dots> < r /3 + r/3 + r/3"
36842
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
   840
      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
   841
      by (rule add_strict_mono [OF add_less_le_mono])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   842
    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
   843
      by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   844
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   845
  thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   846
      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
   847
    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
   848
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   849
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   850
lemma DERIV_power_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   851
  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
   852
  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
   853
    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
   854
  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
   855
  (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
   856
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   857
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   858
    fix R'
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   859
    assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   860
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   861
      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
   862
    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
   863
    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
   864
      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
   865
      proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   866
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   867
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   868
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   869
          using `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   870
        have "norm R' < norm ((R' + R) / 2)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   871
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   872
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   873
          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
   874
      qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   875
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   876
        fix n x y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   877
        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
   878
        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
   879
        proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   880
          have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   881
            (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   882
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   883
            by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   884
          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
   885
          proof (rule mult_left_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   886
            have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   887
              by (rule setsum_abs)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   888
            also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   889
            proof (rule setsum_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   890
              fix p
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   891
              assume "p \<in> {0..<Suc n}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   892
              hence "p \<le> n" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   893
              {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   894
                fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   895
                fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   896
                assume "x \<in> {-R'<..<R'}"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   897
                hence "\<bar>x\<bar> \<le> R'"  by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   898
                hence "\<bar>x^n\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   899
                  unfolding power_abs by (rule power_mono, auto)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   900
              }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   901
              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
   902
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   903
                unfolding abs_mult by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   904
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   905
                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
   906
            qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   907
            also have "\<dots> = real (Suc n) * R' ^ n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   908
              unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   909
            finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   910
              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
   911
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   912
              unfolding abs_mult[symmetric] by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   913
          qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   914
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   915
            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
   916
          finally show ?thesis .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   917
        qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   918
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   919
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   920
        fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   921
        show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   922
          by (auto intro!: DERIV_intros simp del: power_Suc)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   923
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   924
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   925
        fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   926
        assume "x \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   927
        hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   928
          using assms `R' < R` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   929
        have "summable (\<lambda> n. f n * x^n)"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   930
        proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   931
          fix n
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   932
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   933
            by (rule mult_left_mono) auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   934
          show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   935
            unfolding real_norm_def abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   936
            by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   937
        qed
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 36776
diff changeset
   938
        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   939
        show "summable (?f x)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   940
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   941
      show "summable (?f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   942
        using converges[OF `x0 \<in> {-R <..< R}`] .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   943
      show "x0 \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   944
        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
   945
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   946
  } 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
   947
  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
   948
  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
   949
  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
   950
  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
   951
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   952
    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
   953
    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
   954
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   955
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   956
    have "- ?R < 0" using assms by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   957
    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
   958
    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
   959
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   960
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   961
    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
   962
  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
   963
  show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   964
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   965
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   966
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   967
subsection {* Exponential Function *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   968
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   969
definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   970
  where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   971
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   972
lemma summable_exp_generic:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   973
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   974
  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
   975
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   976
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   977
  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
   978
    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
   979
  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
   980
    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
   981
  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
   982
    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
   983
  from r1 show ?thesis
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   984
  proof (rule ratio_test [rule_format])
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   985
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   986
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   987
    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
   988
      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
   989
    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
   990
      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
   991
    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
   992
      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
   993
    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
   994
      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
   995
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   996
      by (simp add: pos_divide_le_eq mult_ac)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   997
    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
   998
      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
   999
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1000
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1001
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1002
lemma summable_norm_exp:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1003
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1004
  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
  1005
proof (rule summable_norm_comparison_test [OF exI, rule_format])
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1006
  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
  1007
    by (rule summable_exp_generic)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1008
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1009
  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
  1010
    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
  1011
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1012
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1013
lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1014
  using summable_exp_generic [where x=x] by simp
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1015
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1016
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1017
  unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1018
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1019
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1020
lemma exp_fdiffs:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1021
      "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1022
  by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1023
        del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1024
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1025
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
  1026
  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
  1027
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1028
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1029
  unfolding exp_def scaleR_conv_of_real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1030
  apply (rule DERIV_cong)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1031
  apply (rule termdiffs [where K="of_real (1 + norm x)"])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1032
  apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1033
  apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1034
  apply (simp del: of_real_add)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1035
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1036
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  1037
declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  1038
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1039
lemma isCont_exp: "isCont exp x"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1040
  by (rule DERIV_exp [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1041
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1042
lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1043
  by (rule isCont_o2 [OF _ isCont_exp])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1044
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1045
lemma tendsto_exp [tendsto_intros]:
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1046
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1047
  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
  1048
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1049
lemma continuous_exp [continuous_intros]:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1050
  "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
  1051
  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
  1052
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1053
lemma continuous_on_exp [continuous_on_intros]:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1054
  "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
  1055
  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
  1056
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1057
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1058
subsubsection {* Properties of the Exponential Function *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1059
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1060
lemma powser_zero:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1061
  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
  1062
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1063
proof -
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1064
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1065
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
  1066
  thus ?thesis unfolding One_nat_def by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1067
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1068
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1069
lemma exp_zero [simp]: "exp 0 = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1070
  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
  1071
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1072
lemma setsum_cl_ivl_Suc2:
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1073
  "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1074
  by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1075
           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
  1076
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1077
lemma exp_series_add:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1078
  fixes x y :: "'a::{real_field}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1079
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1080
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1081
proof (induct n)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1082
  case 0
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1083
  show ?case
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1084
    unfolding S_def by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1085
next
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1086
  case (Suc n)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1087
  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
  1088
    unfolding S_def by (simp del: mult_Suc)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1089
  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
  1090
    by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1091
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1092
  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
  1093
    by (simp only: times_S)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1094
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1095
    by (simp only: Suc)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1096
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1097
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47489
diff changeset
  1098
    by (rule distrib_right)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1099
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1100
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1101
    by (simp only: setsum_right_distrib mult_ac)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1102
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1103
                + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1104
    by (simp add: times_S Suc_diff_le)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1105
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1106
             (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1107
    by (subst setsum_cl_ivl_Suc2, simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1108
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1109
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1110
    by (subst setsum_cl_ivl_Suc, simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1111
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1112
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1113
             (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1114
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1115
              real_of_nat_add [symmetric], simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1116
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23115
diff changeset
  1117
    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
  1118
  finally show
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1119
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
  1120
    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
  1121
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1122
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1123
lemma exp_add: "exp (x + y) = exp x * exp y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1124
  unfolding exp_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1125
  by (simp only: Cauchy_product summable_norm_exp exp_series_add)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1126
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1127
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1128
  by (rule exp_add [symmetric])
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1129
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1130
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1131
  unfolding exp_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1132
  apply (subst suminf_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1133
  apply (rule summable_exp_generic)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1134
  apply (simp add: scaleR_conv_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1135
  done
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1136
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1137
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1138
proof
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1139
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1140
  also assume "exp x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1141
  finally show "False" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1142
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1143
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1144
lemma exp_minus: "exp (- x) = inverse (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1145
  by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1146
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1147
lemma exp_diff: "exp (x - y) = exp x / exp y"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
  1148
  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
  1149
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
subsubsection {* Properties of the Exponential Function on Reals *}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1152
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1153
text {* Comparisons of @{term "exp x"} with zero. *}
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1154
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1155
text{*Proof: because every exponential can be seen as a square.*}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1156
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1157
proof -
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1158
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1159
  thus ?thesis by (simp add: exp_add [symmetric])
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1160
qed
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1161
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1162
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1163
  by (simp add: order_less_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1164
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1165
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1166
  by (simp add: not_less)
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1167
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1168
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1169
  by (simp add: not_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1170
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1171
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1172
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1173
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1174
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1175
  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
  1176
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1177
text {* Strict monotonicity of exponential. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1178
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1179
lemma exp_ge_add_one_self_aux: 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1180
  assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1181
using order_le_imp_less_or_eq [OF assms]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1182
proof 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1183
  assume "0 < x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1184
  have "1+x \<le> (\<Sum>n = 0..<2. inverse (real (fact n)) * x ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1185
    by (auto simp add: numeral_2_eq_2)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1186
  also have "... \<le> (\<Sum>n. inverse (real (fact n)) * x ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1187
    apply (rule series_pos_le [OF summable_exp])
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1188
    using `0 < x`
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1189
    apply (auto  simp add:  zero_le_mult_iff)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1190
    done
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1191
  finally show "1+x \<le> exp x" 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1192
    by (simp add: exp_def)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1193
next
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1194
  assume "0 = x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1195
  then show "1 + x \<le> exp x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1196
    by auto
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1197
qed
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1198
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1199
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1200
proof -
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1201
  assume x: "0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1202
  hence "1 < 1 + x" by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1203
  also from x have "1 + x \<le> exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1204
    by (simp add: exp_ge_add_one_self_aux)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1205
  finally show ?thesis .
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1206
qed
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1207
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1208
lemma exp_less_mono:
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1209
  fixes x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1210
  assumes "x < y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1211
  shows "exp x < exp y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1212
proof -
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1213
  from `x < y` have "0 < y - x" by simp
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1214
  hence "1 < exp (y - x)" by (rule exp_gt_one)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1215
  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
  1216
  thus "exp x < exp y" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1217
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1218
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1219
lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1220
  unfolding linorder_not_le [symmetric]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1221
  by (auto simp add: order_le_less exp_less_mono)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1222
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1223
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1224
  by (auto intro: exp_less_mono exp_less_cancel)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1225
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1226
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1227
  by (auto simp add: linorder_not_less [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1228
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1229
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
53079