src/HOL/Transcendental.thy
author hoelzl
Tue, 20 May 2014 19:24:39 +0200
changeset 57025 e7fd64f82876
parent 56952 efa2a83d548b
child 57129 7edb7550663e
permissions -rw-r--r--
add various lemmas
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
     1
(*  Title:      HOL/Transcendental.thy
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
     2
    Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
     3
    Author:     Lawrence C Paulson
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
     4
    Author:     Jeremy Avigad
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     5
*)
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     6
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
     7
header{*Power Series, Transcendental Functions etc.*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
     8
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15086
diff changeset
     9
theory Transcendental
25600
73431bd8c4c4 joined EvenOdd theory with Parity
haftmann
parents: 25153
diff changeset
    10
imports Fact Series Deriv NthRoot
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15086
diff changeset
    11
begin
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    12
57025
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    13
lemma root_test_convergence:
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    14
  fixes f :: "nat \<Rightarrow> 'a::banach"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    15
  assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    16
  assumes "x < 1"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    17
  shows "summable f"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    18
proof -
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    19
  have "0 \<le> x"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    20
    by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    21
  from `x < 1` obtain z where z: "x < z" "z < 1"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    22
    by (metis dense)
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    23
  from f `x < z`
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    24
  have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    25
    by (rule order_tendstoD)
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    26
  then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    27
    using eventually_ge_at_top
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    28
  proof eventually_elim
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    29
    fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    30
    from power_strict_mono[OF less, of n] n
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    31
    show "norm (f n) \<le> z ^ n"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    32
      by simp
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    33
  qed
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    34
  then show "summable f"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    35
    unfolding eventually_sequentially
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    36
    using z `0 \<le> x` by (auto intro!: summable_comparison_test[OF _  summable_geometric])
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    37
qed
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
diff changeset
    38
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
    39
subsection {* Properties of Power Series *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    40
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    41
lemma lemma_realpow_diff:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
    42
  fixes y :: "'a::monoid_mult"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    43
  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    44
proof -
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    45
  assume "p \<le> n"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    46
  hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30082
diff changeset
    47
  thus ?thesis by (simp add: power_commutes)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    48
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    49
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
    50
lemma lemma_realpow_diff_sumr2:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
    51
  fixes y :: "'a::{comm_ring,monoid_mult}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
    52
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
    53
    "x ^ (Suc n) - y ^ (Suc n) =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    54
      (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    55
proof (induct n)
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    56
  case (Suc n)
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    57
  have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x ^ n) - y * (y * y ^ n)"
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    58
    by simp
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    59
  also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x ^ n)"
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    60
    by (simp add: algebra_simps)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    61
  also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    62
    by (simp only: Suc)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    63
  also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    64
    by (simp only: mult_left_commute)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    65
  also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    66
    by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    67
  finally show ?case .
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    68
qed simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    69
55832
8dd16f8dfe99 repaired document;
wenzelm
parents: 55734
diff changeset
    70
corollary power_diff_sumr2: --{* @{text COMPLEX_POLYFUN} in HOL Light *}
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 55719
diff changeset
    71
  fixes x :: "'a::{comm_ring,monoid_mult}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    72
  shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 55719
diff changeset
    73
using lemma_realpow_diff_sumr2[of x "n - 1" y]
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 55719
diff changeset
    74
by (cases "n = 0") (simp_all add: field_simps)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 55719
diff changeset
    75
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
    76
lemma lemma_realpow_rev_sumr:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    77
   "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    78
    (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
    79
  apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    80
  apply (auto simp: image_iff Bex_def intro!: inj_onI)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    81
  apply arith
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
    82
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    83
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    84
lemma power_diff_1_eq:
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    85
  fixes x :: "'a::{comm_ring,monoid_mult}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    86
  shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    87
using lemma_realpow_diff_sumr2 [of x _ 1] 
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    88
  by (cases n) auto
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    89
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    90
lemma one_diff_power_eq':
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    91
  fixes x :: "'a::{comm_ring,monoid_mult}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    92
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    93
using lemma_realpow_diff_sumr2 [of 1 _ x] 
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    94
  by (cases n) auto
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    95
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    96
lemma one_diff_power_eq:
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    97
  fixes x :: "'a::{comm_ring,monoid_mult}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    98
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    99
by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
   100
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   101
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   102
  x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   103
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   104
lemma powser_insidea:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   105
  fixes x z :: "'a::real_normed_div_algebra"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   106
  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   107
    and 2: "norm z < norm x"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   108
  shows "summable (\<lambda>n. norm (f n * z ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   109
proof -
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   110
  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   111
  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   112
    by (rule summable_LIMSEQ_zero)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   113
  hence "convergent (\<lambda>n. f n * x ^ n)"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   114
    by (rule convergentI)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   115
  hence "Cauchy (\<lambda>n. f n * x ^ n)"
44726
8478eab380e9 generalize some lemmas
huffman
parents: 44725
diff changeset
   116
    by (rule convergent_Cauchy)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   117
  hence "Bseq (\<lambda>n. f n * x ^ n)"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   118
    by (rule Cauchy_Bseq)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   119
  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   120
    by (simp add: Bseq_def, safe)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   121
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   122
                   K * norm (z ^ n) * inverse (norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   123
  proof (intro exI allI impI)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   124
    fix n::nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   125
    assume "0 \<le> n"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   126
    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   127
          norm (f n * x ^ n) * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   128
      by (simp add: norm_mult abs_mult)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   129
    also have "\<dots> \<le> K * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   130
      by (simp only: mult_right_mono 4 norm_ge_zero)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   131
    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   132
      by (simp add: x_neq_0)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   133
    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   134
      by (simp only: mult_assoc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   135
    finally show "norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   136
                  K * norm (z ^ n) * inverse (norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   137
      by (simp add: mult_le_cancel_right x_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   138
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   139
  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   140
  proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   141
    from 2 have "norm (norm (z * inverse x)) < 1"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   142
      using x_neq_0
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   143
      by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   144
    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   145
      by (rule summable_geometric)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   146
    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   147
      by (rule summable_mult)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   148
    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   149
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   150
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   151
                    power_inverse norm_power mult_assoc)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   152
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   153
  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   154
    by (rule summable_comparison_test)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   155
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   156
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   157
lemma powser_inside:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   158
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   159
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   160
    "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   161
      summable (\<lambda>n. f n * (z ^ n))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   162
  by (rule powser_insidea [THEN summable_norm_cancel])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   163
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   164
lemma sum_split_even_odd:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   165
  fixes f :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   166
  shows
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   167
    "(\<Sum>i<2 * n. if even i then f i else g i) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   168
     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   169
proof (induct n)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   170
  case 0
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   171
  then show ?case by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   172
next
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   173
  case (Suc n)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   174
  have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   175
    (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   176
    using Suc.hyps unfolding One_nat_def by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   177
  also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   178
    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
   179
  finally show ?case .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   180
qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   181
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   182
lemma sums_if':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   183
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   184
  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
   185
  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
   186
  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
   187
proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   188
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   189
  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
   190
  from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   191
  obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   192
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   193
  let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   194
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   195
    fix m
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   196
    assume "m \<ge> 2 * no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   197
    hence "m div 2 \<ge> no" by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   198
    have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   199
      using sum_split_even_odd by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   200
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   201
      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
   202
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   203
    have "?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
   204
    proof (cases "even m")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   205
      case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   206
      show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   207
        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
   208
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   209
      case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   210
      hence "even (Suc m)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   211
      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   212
        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
   213
      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
   214
      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
   215
      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
   216
      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
   217
      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
   218
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   219
    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
   220
  }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   221
  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
   222
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   223
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   224
lemma sums_if:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   225
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   226
  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
   227
  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
   228
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   229
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   230
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   231
    fix B T E
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   232
    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
   233
      by (cases B) auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   234
  } note if_sum = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   235
  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
   236
    using sums_if'[OF `g sums x`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   237
  {
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 38642
diff changeset
   238
    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
   239
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   240
    have "?s sums y" using sums_if'[OF `f sums y`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   241
    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
   242
    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   243
      by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum_reindex if_eq sums_def cong del: if_cong)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   244
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   245
  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
   246
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   247
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   248
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
   249
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   250
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
   251
  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
   252
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   253
  shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. -1^i*a i) ----> l) \<and>
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   254
             ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. -1^i*a i) ----> l)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   255
  (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
   256
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
   257
  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
   258
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   259
  show "\<forall>n. ?f n \<le> ?f (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   260
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   261
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   262
    show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   263
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   264
  show "\<forall>n. ?g (Suc n) \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   265
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   266
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   267
    show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   268
      unfolding One_nat_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   269
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   270
  show "\<forall>n. ?f n \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   271
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   272
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   273
    show "?f n \<le> ?g n" using fg_diff a_pos
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   274
      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
   275
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   276
  show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   277
  proof (rule LIMSEQ_I)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   278
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   279
    assume "0 < r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   280
    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
   281
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   282
    hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   283
    thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   284
  qed
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   285
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   286
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   287
lemma summable_Leibniz':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   288
  fixes a :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   289
  assumes a_zero: "a ----> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   290
    and a_pos: "\<And> n. 0 \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   291
    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
   292
  shows summable: "summable (\<lambda> n. (-1)^n * a n)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   293
    and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   294
    and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   295
    and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   296
    and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   297
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   298
  let ?S = "\<lambda>n. (-1)^n * a n"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   299
  let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   300
  let ?f = "\<lambda>n. ?P (2 * n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   301
  let ?g = "\<lambda>n. ?P (2 * n + 1)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   302
  obtain l :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   303
    where below_l: "\<forall> n. ?f n \<le> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   304
      and "?f ----> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   305
      and above_l: "\<forall> n. l \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   306
      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
   307
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   308
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   309
  let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   310
  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
   311
  proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   312
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   313
    assume "0 < r"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   314
    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
   315
    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
   316
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   317
    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
   318
    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
   319
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   320
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   321
      fix n :: nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   322
      assume "n \<ge> (max (2 * f_no) (2 * g_no))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   323
      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
   324
      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
   325
      proof (cases "even n")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   326
        case True
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   327
        from even_nat_div_two_times_two[OF this]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   328
        have n_eq: "2 * (n div 2) = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   329
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   330
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   331
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   332
        from f[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   333
          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
   334
      next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   335
        case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   336
        hence "even (n - 1)" by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   337
        from even_nat_div_two_times_two[OF this]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   338
        have n_eq: "2 * ((n - 1) div 2) = n - 1"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   339
          unfolding numeral_2_eq_2[symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   340
        hence range_eq: "n - 1 + 1 = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   341
          using odd_pos[OF False] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   342
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   343
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   344
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   345
        from g[OF this] show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   346
          unfolding n_eq range_eq .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   347
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   348
    }
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   349
    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
   350
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   351
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   352
    unfolding sums_def .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   353
  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
   354
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   355
  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
   356
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   357
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   358
  show "suminf ?S \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   359
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   360
  show "?f n \<le> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   361
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   362
  show "?g ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   363
    using `?g ----> l` `l = suminf ?S` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   364
  show "?f ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   365
    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
   366
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   367
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   368
theorem summable_Leibniz:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   369
  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
   370
  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
   371
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   372
    and "0 < a 0 \<longrightarrow>
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   373
      (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i<2*n. -1^i * a i .. \<Sum>i<2*n+1. -1^i * a i})" (is "?pos")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   374
    and "a 0 < 0 \<longrightarrow>
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   375
      (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i<2*n+1. -1^i * a i .. \<Sum>i<2*n. -1^i * a i})" (is "?neg")
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   376
    and "(\<lambda>n. \<Sum>i<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   377
    and "(\<lambda>n. \<Sum>i<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   378
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   379
  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
   380
  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
   381
    case True
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   382
    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
   383
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   384
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   385
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   386
      have "a (Suc n) \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   387
        using ord[where n="Suc n" and m=n] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   388
    } note mono = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   389
    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
   390
    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
   391
    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
   392
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   393
    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
   394
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   395
    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
   396
    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
   397
    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
   398
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   399
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   400
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   401
      have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   402
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   403
    } note monotone = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   404
    note leibniz =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   405
      summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   406
        OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   407
    have "summable (\<lambda> n. (-1)^n * ?a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   408
      using leibniz(1) by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   409
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   410
      unfolding summable_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   411
    from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
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
    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
   414
    moreover
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   415
    have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   416
      unfolding minus_diff_minus by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   417
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   418
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   419
    have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   420
      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
   421
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   422
    have ?pos using `0 \<le> ?a 0` by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   423
    moreover have ?neg
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   424
      using leibniz(2,4)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   425
      unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   426
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   427
    moreover have ?f and ?g
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   428
      using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   429
      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
   430
    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
   431
  qed
54576
e877eec2b698 tidied more proofs
paulson
parents: 54575
diff changeset
   432
  then show ?summable and ?pos and ?neg and ?f and ?g 
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   433
    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
   434
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   435
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   436
subsection {* Term-by-Term Differentiability of Power Series *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   437
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   438
definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   439
  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
   440
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   441
text{*Lemma about distributing negation over it*}
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   442
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   443
  by (simp add: diffs_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   444
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   445
lemma sums_Suc_imp:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   446
  "(f::nat \<Rightarrow> 'a::real_normed_vector) 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   447
  using sums_Suc_iff[of f] by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   448
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   449
lemma diffs_equiv:
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   450
  fixes x :: "'a::{real_normed_vector, ring_1}"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   451
  shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   452
      (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   453
  unfolding diffs_def
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   454
  by (simp add: summable_sums sums_Suc_imp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   455
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   456
lemma lemma_termdiff1:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   457
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   458
  "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   459
   (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   460
  by (auto simp add: algebra_simps power_add [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   461
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   462
lemma sumr_diff_mult_const2:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   463
  "setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   464
  by (simp add: setsum_subtractf)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   465
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   466
lemma lemma_termdiff2:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   467
  fixes h :: "'a :: {field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   468
  assumes h: "h \<noteq> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   469
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   470
    "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   471
     h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   472
          (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   473
  apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   474
  apply (simp add: right_diff_distrib diff_divide_distrib h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   475
  apply (simp add: mult_assoc [symmetric])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   476
  apply (cases "n", simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   477
  apply (simp add: lemma_realpow_diff_sumr2 h
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   478
                   right_diff_distrib [symmetric] mult_assoc
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   479
              del: power_Suc setsum_lessThan_Suc of_nat_Suc)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   480
  apply (subst lemma_realpow_rev_sumr)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   481
  apply (subst sumr_diff_mult_const2)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   482
  apply simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   483
  apply (simp only: lemma_termdiff1 setsum_right_distrib)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   484
  apply (rule setsum_cong [OF refl])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   485
  apply (simp add: less_iff_Suc_add)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   486
  apply (clarify)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   487
  apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   488
              del: setsum_lessThan_Suc power_Suc)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   489
  apply (subst mult_assoc [symmetric], subst power_add [symmetric])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   490
  apply (simp add: mult_ac)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   491
  done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   492
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   493
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
   494
  fixes K :: "'a::linordered_semidom"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   495
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   496
    and K: "0 \<le> K"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   497
  shows "setsum f {..<n-k} \<le> of_nat n * K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   498
  apply (rule order_trans [OF setsum_mono])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   499
  apply (rule f, simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   500
  apply (simp add: mult_right_mono K)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   501
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   502
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   503
lemma lemma_termdiff3:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   504
  fixes h z :: "'a::{real_normed_field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   505
  assumes 1: "h \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   506
    and 2: "norm z \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   507
    and 3: "norm (z + h) \<le> K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   508
  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
   509
          \<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
   510
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   511
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   512
        norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   513
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   514
    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
   515
  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
   516
  proof (rule mult_right_mono [OF _ norm_ge_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   517
    from norm_ge_zero 2 have K: "0 \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   518
      by (rule order_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   519
    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
   520
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   521
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   522
      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
   523
      done
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   524
    show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   525
          \<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
   526
      apply (intro
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   527
         order_trans [OF norm_setsum]
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   528
         real_setsum_nat_ivl_bounded2
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   529
         mult_nonneg_nonneg
47489
04e7d09ade7a tuned some proofs;
huffman
parents: 47108
diff changeset
   530
         of_nat_0_le_iff
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   531
         zero_le_power K)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   532
      apply (rule le_Kn, simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   533
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   534
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   535
  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
   536
    by (simp only: mult_assoc)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   537
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   538
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   539
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   540
lemma lemma_termdiff4:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   541
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   542
  assumes k: "0 < (k::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   543
    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
   544
  shows "f -- 0 --> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   545
proof (rule tendsto_norm_zero_cancel)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   546
  show "(\<lambda>h. norm (f h)) -- 0 --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   547
  proof (rule real_tendsto_sandwich)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   548
    show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   549
      by simp
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   550
    show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   551
      using k by (auto simp add: eventually_at dist_norm le)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   552
    show "(\<lambda>h. 0) -- (0::'a) --> (0::real)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   553
      by (rule tendsto_const)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   554
    have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   555
      by (intro tendsto_intros)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   556
    then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   557
      by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   558
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   559
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   560
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   561
lemma lemma_termdiff5:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   562
  fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   563
  assumes k: "0 < (k::real)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   564
  assumes f: "summable f"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   565
  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
   566
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   567
proof (rule lemma_termdiff4 [OF k])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   568
  fix h::'a
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   569
  assume "h \<noteq> 0" and "norm h < k"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   570
  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
   571
    by (simp add: le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   572
  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
   573
    by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   574
  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
   575
    by (rule summable_mult2)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   576
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   577
    by (rule summable_comparison_test)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   578
  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
   579
    by (rule summable_norm)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   580
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
   581
    by (rule suminf_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   582
  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
   583
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   584
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   585
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   586
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   587
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   588
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   589
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   590
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   591
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   592
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   593
    and 2: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   594
  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
   595
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   596
proof -
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   597
  from dense [OF 2]
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   598
  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
   599
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   600
    by (rule order_le_less_trans)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   601
  hence r_neq_0: "r \<noteq> 0" by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   602
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   603
  proof (rule lemma_termdiff5)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   604
    show "0 < r - norm x" using r1 by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   605
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   606
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   607
    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
   608
      by (rule powser_insidea)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   609
    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
   610
      using r
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   611
      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
   612
    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
   613
      by (rule diffs_equiv [THEN sums_summable])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   614
    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
   615
      (\<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
   616
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   617
      apply (simp add: diffs_def)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   618
      apply (case_tac n, simp_all add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   619
      done
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   620
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   621
      (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   622
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   623
    also have
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   624
      "(\<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
   625
           r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   626
       (\<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
   627
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   628
      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
   629
      apply (rename_tac nat)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   630
      apply (case_tac "nat", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   631
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   632
      done
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   633
    finally
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   634
    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
   635
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   636
    fix h::'a and n::nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   637
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   638
    assume "norm h < r - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   639
    hence "norm x + norm h < r" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   640
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   641
      by (rule order_le_less_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   642
    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
   643
          \<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
   644
      apply (simp only: norm_mult mult_assoc)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   645
      apply (rule mult_left_mono [OF _ norm_ge_zero])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   646
      apply (simp add: mult_assoc [symmetric])
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   647
      apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   648
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   649
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   650
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   651
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   652
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   653
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   654
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   655
      and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   656
      and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   657
      and 4: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   658
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   659
  unfolding DERIV_def
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   660
proof (rule LIM_zero_cancel)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   661
  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
   662
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   663
  proof (rule LIM_equal2)
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   664
    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
   665
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   666
    fix h :: 'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   667
    assume "norm (h - 0) < norm K - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   668
    hence "norm x + norm h < norm K" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   669
    hence 5: "norm (x + h) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   670
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   671
    have "summable (\<lambda>n. c n * x ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   672
      and "summable (\<lambda>n. c n * (x + h) ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   673
      and "summable (\<lambda>n. diffs c n * x ^ n)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   674
      using 1 2 4 5 by (auto elim: powser_inside)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   675
    then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   676
          (\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h - of_nat n * c n * x ^ (n - Suc 0))"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   677
      by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   678
    then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   679
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   680
      by (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   681
  next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   682
    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
   683
      by (rule termdiffs_aux [OF 3 4])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   684
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   685
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   686
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   687
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   688
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
   689
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   690
lemma DERIV_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   691
  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
   692
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   693
    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
   694
    and "summable (f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   695
    and "summable L"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   696
    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
   697
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   698
  unfolding DERIV_def
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   699
proof (rule LIM_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   700
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   701
  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
   702
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   703
  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
   704
    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
   705
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   706
  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
   707
    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
   708
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   709
  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
   710
  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
   711
    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
   712
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   713
  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
   714
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   715
  let ?r = "r / (3 * real ?N)"
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   716
  from `0 < r` have "0 < ?r" by simp
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   717
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   718
  let ?s = "\<lambda>n. SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   719
  def S' \<equiv> "Min (?s ` {..< ?N })"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   720
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   721
  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
   722
  proof (rule iffD2[OF Min_gr_iff])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   723
    show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   724
    proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   725
      fix x
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   726
      assume "x \<in> ?s ` {..<?N}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   727
      then obtain n where "x = ?s n" and "n \<in> {..<?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   728
        using image_iff[THEN iffD1] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   729
      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
   730
      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
   731
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   732
      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
   733
      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
   734
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   735
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   736
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   737
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   738
  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
   739
    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
   740
    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
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   743
    fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   744
    assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   745
    hence x_in_I: "x0 + x \<in> { a <..< b }"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   746
      using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   747
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   748
    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
   749
    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
   750
    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
   751
    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
   752
    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
   753
    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
   754
    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
   755
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   756
    { fix n
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   757
      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
   758
        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
   759
        unfolding abs_divide .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   760
      hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   761
        using `x \<noteq> 0` by auto }
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   762
    note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF `summable L`]]
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   763
    then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
   764
      by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF `summable L`]]])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   765
    then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   766
      using L_estimate by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   767
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   768
    have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" ..
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   769
    also have "\<dots> < (\<Sum>n<?N. ?r)"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   770
    proof (rule setsum_strict_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   771
      fix n
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   772
      assume "n \<in> {..< ?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   773
      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
   774
      also have "S \<le> S'" using `S \<le> S'` .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   775
      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
   776
      proof (rule Min_le_iff[THEN iffD2])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   777
        have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   778
          using `n \<in> {..< ?N}` by auto
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   779
        thus "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" by blast
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   780
      qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   781
      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
   782
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   783
      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
   784
      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
   785
      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
   786
        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
   787
    qed auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   788
    also have "\<dots> = of_nat (card {..<?N}) * ?r"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   789
      by (rule setsum_constant)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   790
    also have "\<dots> = real ?N * ?r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   791
      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
   792
    also have "\<dots> = r/3" by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   793
    finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   794
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   795
    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
   796
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   797
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   798
      unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   799
      using suminf_divide[OF diff_smbl, symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   800
    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
   801
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   802
      unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   803
      apply (subst (5) add_commute)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   804
      by (rule abs_triangle_ineq)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   805
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   806
      using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   807
    also have "\<dots> < r /3 + r/3 + r/3"
36842
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
   808
      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
   809
      by (rule add_strict_mono [OF add_less_le_mono])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   810
    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
   811
      by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   812
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   813
  thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   814
      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
   815
    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
   816
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   817
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   818
lemma DERIV_power_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   819
  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
   820
  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
   821
    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
   822
  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
   823
  (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
   824
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   825
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   826
    fix R'
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   827
    assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   828
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   829
      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
   830
    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
   831
    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
   832
      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
   833
      proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   834
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   835
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   836
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   837
          using `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   838
        have "norm R' < norm ((R' + R) / 2)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   839
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   840
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   841
          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
   842
      qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   843
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   844
        fix n x y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   845
        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
   846
        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
   847
        proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   848
          have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   849
            (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   850
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   851
            by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   852
          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
   853
          proof (rule mult_left_mono)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   854
            have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   855
              by (rule setsum_abs)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   856
            also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   857
            proof (rule setsum_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   858
              fix p
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   859
              assume "p \<in> {..<Suc n}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   860
              hence "p \<le> n" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   861
              {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   862
                fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   863
                fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   864
                assume "x \<in> {-R'<..<R'}"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   865
                hence "\<bar>x\<bar> \<le> R'"  by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   866
                hence "\<bar>x^n\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   867
                  unfolding power_abs by (rule power_mono, auto)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   868
              }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   869
              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
   870
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   871
                unfolding abs_mult by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   872
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   873
                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
   874
            qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   875
            also have "\<dots> = real (Suc n) * R' ^ n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   876
              unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   877
            finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   878
              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
   879
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   880
              unfolding abs_mult[symmetric] by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   881
          qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   882
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   883
            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
   884
          finally show ?thesis .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   885
        qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   886
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   887
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   888
        fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   889
        show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   890
          by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   891
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   892
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   893
        fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   894
        assume "x \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   895
        hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   896
          using assms `R' < R` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   897
        have "summable (\<lambda> n. f n * x^n)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   898
        proof (rule summable_comparison_test, intro exI allI impI)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
   899
          fix n
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   900
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   901
            by (rule mult_left_mono) auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   902
          show "norm (f n * x ^ n) \<le> norm (f n * real (Suc n) * x ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   903
            unfolding real_norm_def abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   904
            by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   905
        qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`])
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 36776
diff changeset
   906
        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   907
        show "summable (?f x)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   908
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   909
      show "summable (?f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   910
        using converges[OF `x0 \<in> {-R <..< R}`] .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   911
      show "x0 \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   912
        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
   913
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   914
  } 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
   915
  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
   916
  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
   917
  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
   918
  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
   919
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   920
    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
   921
    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
   922
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   923
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   924
    have "- ?R < 0" using assms by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   925
    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
   926
    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
   927
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   928
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   929
    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
   930
  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
   931
  show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   932
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   933
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   934
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   935
subsection {* Exponential Function *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   936
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   937
definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   938
  where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   939
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   940
lemma summable_exp_generic:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   941
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   942
  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
   943
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   944
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   945
  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
   946
    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
   947
  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
   948
    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
   949
  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
   950
    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
   951
  from r1 show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   952
  proof (rule summable_ratio_test [rule_format])
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   953
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   954
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   955
    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
   956
      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
   957
    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
   958
      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
   959
    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
   960
      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
   961
    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
   962
      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
   963
    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
   964
      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
   965
    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
   966
      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
   967
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   968
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   969
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   970
lemma summable_norm_exp:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   971
  fixes x :: "'a::{real_normed_algebra_1,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   972
  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
   973
proof (rule summable_norm_comparison_test [OF exI, rule_format])
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   974
  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
   975
    by (rule summable_exp_generic)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   976
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   977
  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
   978
    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
   979
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   980
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   981
lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   982
  using summable_exp_generic [where x=x] by simp
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   983
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   984
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   985
  unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   986
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   987
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   988
lemma exp_fdiffs:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   989
      "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   990
  by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   991
        del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   992
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   993
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
   994
  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
   995
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   996
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   997
  unfolding exp_def scaleR_conv_of_real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   998
  apply (rule DERIV_cong)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   999
  apply (rule termdiffs [where K="of_real (1 + norm x)"])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1000
  apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1001
  apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1002
  apply (simp del: of_real_add)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1003
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1004
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  1005
declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  1006
44311
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1007
lemma isCont_exp: "isCont exp x"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1008
  by (rule DERIV_exp [THEN DERIV_isCont])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1009
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1010
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
  1011
  by (rule isCont_o2 [OF _ isCont_exp])
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1012
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1013
lemma tendsto_exp [tendsto_intros]:
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1014
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
42c5cbf68052 Transcendental.thy: add tendsto_intros lemmas;
huffman
parents: 44308
diff changeset
  1015
  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
  1016
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1017
lemma continuous_exp [continuous_intros]:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1018
  "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
  1019
  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
  1020
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56261
diff changeset
  1021
lemma continuous_on_exp [continuous_intros]:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1022
  "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
  1023
  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
  1024
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1025
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1026
subsubsection {* Properties of the Exponential Function *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1027
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1028
lemma powser_zero:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1029
  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
  1030
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1031
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1032
  have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1033
    by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
  1034
  thus ?thesis unfolding One_nat_def by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1035
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1036
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
  1037
lemma exp_zero [simp]: "exp 0 = 1"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1038
  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
  1039
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1040
lemma exp_series_add:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1041
  fixes x y :: "'a::{real_field}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1042
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1043
  shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1044
proof (induct n)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1045
  case 0
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1046
  show ?case
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1047
    unfolding S_def by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1048
next
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1049
  case (Suc n)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1050
  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
  1051
    unfolding S_def by (simp del: mult_Suc)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1052
  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
  1053
    by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1054
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
  1055
  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
  1056
    by (simp only: times_S)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1057
  also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1058
    by (simp only: Suc)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1059
  also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i))
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1060
                + y * (\<Sum>i\<le>n. S x i * S y (n-i))"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47489
diff changeset
  1061
    by (rule distrib_right)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1062
  also have "\<dots> = (\<Sum>i\<le>n. (x * S x i) * S y (n-i))
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1063
                + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1064
    by (simp only: setsum_right_distrib mult_ac)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1065
  also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1066
                + (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1067
    by (simp add: times_S Suc_diff_le)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1068
  also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1069
             (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1070
    by (subst setsum_atMost_Suc_shift) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1071
  also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1072
             (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1073
    by simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1074
  also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1075
             (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1076
             (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1077
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1078
                   real_of_nat_add [symmetric]) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1079
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23115
diff changeset
  1080
    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
  1081
  finally show
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1082
    "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
  1083
    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
  1084
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1085
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1086
lemma exp_add: "exp (x + y) = exp x * exp y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1087
  unfolding exp_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1088
  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
  1089
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1090
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1091
  by (rule exp_add [symmetric])
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1092
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1093
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1094
  unfolding exp_def
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1095
  apply (subst suminf_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1096
  apply (rule summable_exp_generic)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1097
  apply (simp add: scaleR_conv_of_real)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1098
  done
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
  1099
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1100
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1101
proof
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1102
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1103
  also assume "exp x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1104
  finally show "False" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1105
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1106
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1107
lemma exp_minus: "exp (- x) = inverse (exp x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1108
  by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1109
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1110
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
  1111
  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
  1112
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1113
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1114
subsubsection {* Properties of the Exponential Function on Reals *}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1115
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1116
text {* Comparisons of @{term "exp x"} with zero. *}
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1117
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1118
text{*Proof: because every exponential can be seen as a square.*}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1119
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1120
proof -
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1121
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1122
  thus ?thesis by (simp add: exp_add [symmetric])
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1123
qed
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1124
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1125
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1126
  by (simp add: order_less_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1127
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1128
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1129
  by (simp add: not_less)
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1130
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1131
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1132
  by (simp add: not_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1133
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1134
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1135
  by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1136
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1137
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1138
  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
  1139
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1140
text {* Strict monotonicity of exponential. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1141
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1142
lemma exp_ge_add_one_self_aux: 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1143
  assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1144
using order_le_imp_less_or_eq [OF assms]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1145
proof 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1146
  assume "0 < x"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1147
  have "1+x \<le> (\<Sum>n<2. inverse (real (fact n)) * x ^ n)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1148
    by (auto simp add: numeral_2_eq_2)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1149
  also have "... \<le> (\<Sum>n. inverse (real (fact n)) * x ^ n)"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56193
diff changeset
  1150
    apply (rule setsum_le_suminf [OF summable_exp])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1151
    using `0 < x`
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1152
    apply (auto  simp add:  zero_le_mult_iff)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1153
    done
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1154
  finally show "1+x \<le> exp x" 
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1155
    by (simp add: exp_def)
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1156
next
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1157
  assume "0 = x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1158
  then show "1 + x \<le> exp x"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1159
    by auto
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1160
qed
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1161
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1162
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1163
proof -
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1164
  assume x: "0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1165
  hence "1 < 1 + x" by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1166
  also from x have "1 + x \<le> exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1167
    by (simp add: exp_ge_add_one_self_aux)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1168
  finally show ?thesis .
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1169
qed
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1170
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1171
lemma exp_less_mono:
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1172
  fixes x y :: real
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1173
  assumes "x < y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1174
  shows "exp x < exp y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1175
proof -
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1176
  from `x < y` have "0 < y - x" by simp
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1177
  hence "1 < exp (y - x)" by (rule exp_gt_one)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1178
  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
  1179
  thus "exp x < exp y" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1180
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1181
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1182
lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1183
  unfolding linorder_not_le [symmetric]
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
  1184
  by (auto simp add: order_le_less exp_less_mono)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1185
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1186
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1187
  by (auto intro: exp_less_mono exp_less_cancel)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1188
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1189
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
  1190
  by (auto simp add: linorder_not_less [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1191
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1192
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1193
  by (simp add: order_eq_iff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1194
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1195
text {* Comparisons of @{term "exp x"} with one. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1196
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1197
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1198
  using exp_less_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1199
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1200
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1201
  using exp_less_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1202
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1203
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1204
  using exp_le_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1205
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1206
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1207
  using exp_le_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1208
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1209
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1210
  using exp_inj_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1211
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1212
lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1213
proof (rule IVT)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1214
  assume "1 \<le> y"
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1215
  hence "0 \<le> y - 1" by simp
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1216
  hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1217
  thus "y \<le> exp (y - 1)" by simp
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1218
qed (simp_all add: le_diff_eq)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1219
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1220
lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff changeset
  1221
proof (rule linorder_le_cases [of 1 y])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1222
  assume "1 \<le> y"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1223
  thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
44755
257ac9da021f convert some proofs to Isar-style
huffman
parents: 44746
diff cha