src/HOL/Transcendental.thy
author paulson <lp15@cam.ac.uk>
Tue, 21 Apr 2015 17:19:00 +0100
changeset 60141 833adf7db7d8
parent 60036 218fcc645d22
child 60150 bd773c47ad0b
permissions -rw-r--r--
New material, mostly about limits. Consolidation.
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
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     1
(*  Title:      HOL/Transcendental.thy
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
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     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
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     3
    Author:     Lawrence C Paulson
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
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     4
    Author:     Jeremy Avigad
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
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parents:
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     5
*)
a3be6b3a9c0b new theories from Jacques Fleuriot
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parents:
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58889
5b7a9633cfa8 modernized header uniformly as section;
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section{*Power Series, Transcendental Functions etc.*}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
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     8
15131
c69542757a4d New theory header syntax.
nipkow
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theory Transcendental
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
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    10
imports Binomial Series Deriv NthRoot
15131
c69542757a4d New theory header syntax.
nipkow
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    11
begin
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    12
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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    13
lemma of_real_fact [simp]: "of_real (fact n) = fact n"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    14
  by (metis of_nat_fact of_real_of_nat_eq)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    15
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    16
lemma real_fact_nat [simp]: "real (fact n :: nat) = fact n"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    17
  by (simp add: real_of_nat_def)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    18
59731
paulson <lp15@cam.ac.uk>
parents: 59730 59688
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    19
lemma real_fact_int [simp]: "real (fact n :: int) = fact n"
paulson <lp15@cam.ac.uk>
parents: 59730 59688
diff changeset
    20
  by (metis of_int_of_nat_eq of_nat_fact real_of_int_def)
paulson <lp15@cam.ac.uk>
parents: 59730 59688
diff changeset
    21
57025
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    22
lemma root_test_convergence:
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    23
  fixes f :: "nat \<Rightarrow> 'a::banach"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    24
  assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    25
  assumes "x < 1"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    26
  shows "summable f"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    27
proof -
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    28
  have "0 \<le> x"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    29
    by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    30
  from `x < 1` obtain z where z: "x < z" "z < 1"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    31
    by (metis dense)
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    32
  from f `x < z`
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    33
  have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    34
    by (rule order_tendstoD)
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    35
  then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    36
    using eventually_ge_at_top
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    37
  proof eventually_elim
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    38
    fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    39
    from power_strict_mono[OF less, of n] n
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    40
    show "norm (f n) \<le> z ^ n"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    41
      by simp
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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  qed
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    43
  then show "summable f"
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    44
    unfolding eventually_sequentially
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    45
    using z `0 \<le> x` by (auto intro!: summable_comparison_test[OF _  summable_geometric])
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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qed
e7fd64f82876 add various lemmas
hoelzl
parents: 56952
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    47
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
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subsection {* Properties of Power Series *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
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    49
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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    50
lemma lemma_realpow_diff:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
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    51
  fixes y :: "'a::monoid_mult"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    52
  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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    53
proof -
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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    54
  assume "p \<le> n"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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    55
  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
    56
  thus ?thesis by (simp add: power_commutes)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
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    57
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    58
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
    59
lemma lemma_realpow_diff_sumr2:
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
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    60
  fixes y :: "'a::{comm_ring,monoid_mult}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
    61
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
    62
    "x ^ (Suc n) - y ^ (Suc n) =
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    63
      (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    64
proof (induct n)
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    65
  case (Suc n)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    66
  have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    67
    by simp
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    68
  also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
    69
    by (simp add: algebra_simps)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    70
  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
    71
    by (simp only: Suc)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    72
  also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
    73
    by (simp only: mult.left_commute)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    74
  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
    75
    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
    76
  finally show ?case .
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    77
qed simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    78
55832
8dd16f8dfe99 repaired document;
wenzelm
parents: 55734
diff changeset
    79
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
    80
  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
    81
  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
    82
using lemma_realpow_diff_sumr2[of x "n - 1" y]
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 55719
diff changeset
    83
by (cases "n = 0") (simp_all add: field_simps)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 55719
diff changeset
    84
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
    85
lemma lemma_realpow_rev_sumr:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
    86
   "(\<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
    87
    (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
    88
  by (subst nat_diff_setsum_reindex[symmetric]) simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    89
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
    90
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
    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> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
    93
using lemma_realpow_diff_sumr2 [of x _ 1]
55719
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^(n - Suc i))"
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
    99
using lemma_realpow_diff_sumr2 [of 1 _ x]
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
   100
  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
   101
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 55417
diff changeset
   102
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
   103
  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
   104
  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
   105
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
   106
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   107
lemma powser_zero:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   108
  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   109
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   110
proof -
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   111
  have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   112
    by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   113
  thus ?thesis unfolding One_nat_def by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   114
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   115
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   116
lemma powser_sums_zero:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   117
  fixes a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   118
  shows "(\<lambda>n. a n * 0^n) sums a 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   119
    using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   120
    by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   121
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   122
text{*Power series has a circle or radius of convergence: if it sums for @{term
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   123
  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
   124
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   125
lemma powser_insidea:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   126
  fixes x z :: "'a::real_normed_div_algebra"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   127
  assumes 1: "summable (\<lambda>n. f n * x^n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   128
    and 2: "norm z < norm x"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   129
  shows "summable (\<lambda>n. norm (f n * z ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   130
proof -
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   131
  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   132
  from 1 have "(\<lambda>n. f n * x^n) ----> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   133
    by (rule summable_LIMSEQ_zero)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   134
  hence "convergent (\<lambda>n. f n * x^n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   135
    by (rule convergentI)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   136
  hence "Cauchy (\<lambda>n. f n * x^n)"
44726
8478eab380e9 generalize some lemmas
huffman
parents: 44725
diff changeset
   137
    by (rule convergent_Cauchy)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   138
  hence "Bseq (\<lambda>n. f n * x^n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   139
    by (rule Cauchy_Bseq)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   140
  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
   141
    by (simp add: Bseq_def, safe)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   142
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   143
                   K * norm (z ^ n) * inverse (norm (x^n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   144
  proof (intro exI allI impI)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   145
    fix n::nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   146
    assume "0 \<le> n"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   147
    have "norm (norm (f n * z ^ n)) * norm (x^n) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   148
          norm (f n * x^n) * norm (z ^ n)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   149
      by (simp add: norm_mult abs_mult)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   150
    also have "\<dots> \<le> K * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   151
      by (simp only: mult_right_mono 4 norm_ge_zero)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   152
    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
   153
      by (simp add: x_neq_0)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   154
    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   155
      by (simp only: mult.assoc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   156
    finally show "norm (norm (f n * z ^ n)) \<le>
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   157
                  K * norm (z ^ n) * inverse (norm (x^n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   158
      by (simp add: mult_le_cancel_right x_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   159
  qed
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   160
  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
   161
  proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   162
    from 2 have "norm (norm (z * inverse x)) < 1"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   163
      using x_neq_0
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   164
      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
   165
    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   166
      by (rule summable_geometric)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   167
    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
   168
      by (rule summable_mult)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   169
    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   170
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   171
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   172
                    power_inverse norm_power mult.assoc)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   173
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   174
  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
   175
    by (rule summable_comparison_test)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   176
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   177
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   178
lemma powser_inside:
53599
78ea983f7987 generalize lemmas
huffman
parents: 53079
diff changeset
   179
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   180
  shows
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   181
    "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   182
      summable (\<lambda>n. f n * (z ^ n))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   183
  by (rule powser_insidea [THEN summable_norm_cancel])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   184
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   185
lemma powser_times_n_limit_0:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   186
  fixes x :: "'a::{real_normed_div_algebra,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   187
  assumes "norm x < 1"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   188
    shows "(\<lambda>n. of_nat n * x ^ n) ----> 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   189
proof -
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   190
  have "norm x / (1 - norm x) \<ge> 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   191
    using assms
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   192
    by (auto simp: divide_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   193
  moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   194
    using ex_le_of_int
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   195
    by (meson ex_less_of_int)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   196
  ultimately have N0: "N>0" 
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   197
    by auto
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   198
  then have *: "real (N + 1) * norm x / real N < 1"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   199
    using N assms
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   200
    by (auto simp: field_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   201
  { fix n::nat
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   202
    assume "N \<le> int n"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   203
    then have "real N * real_of_nat (Suc n) \<le> real_of_nat n * real (1 + N)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   204
      by (simp add: algebra_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   205
    then have "(real N * real_of_nat (Suc n)) * (norm x * norm (x ^ n))
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   206
               \<le> (real_of_nat n * real (1 + N)) * (norm x * norm (x ^ n))"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   207
      using N0 mult_mono by fastforce
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   208
    then have "real N * (norm x * (real_of_nat (Suc n) * norm (x ^ n)))
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   209
         \<le> real_of_nat n * (norm x * (real (1 + N) * norm (x ^ n)))"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   210
      by (simp add: algebra_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   211
  } note ** = this
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   212
  show ?thesis using *
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   213
    apply (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   214
    apply (simp add: N0 norm_mult nat_le_iff field_simps power_Suc ** 
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   215
                del: of_nat_Suc real_of_int_add)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   216
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   217
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   218
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   219
corollary lim_n_over_pown:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   220
  fixes x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   221
  shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) ---> 0) sequentially"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   222
using powser_times_n_limit_0 [of "inverse x"]
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   223
by (simp add: norm_divide divide_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   224
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   225
lemma sum_split_even_odd:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   226
  fixes f :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   227
  shows
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   228
    "(\<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
   229
     (\<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
   230
proof (induct n)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   231
  case 0
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   232
  then show ?case by simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   233
next
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   234
  case (Suc n)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   235
  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
   236
    (\<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
   237
    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
   238
  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
   239
    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
   240
  finally show ?case .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   241
qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   242
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   243
lemma sums_if':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   244
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   245
  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
   246
  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
   247
  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
   248
proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   249
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   250
  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
   251
  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
   252
  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
   253
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   254
  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
   255
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   256
    fix m
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   257
    assume "m \<ge> 2 * no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   258
    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
   259
    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
   260
      using sum_split_even_odd by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   261
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   262
      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
   263
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   264
    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
   265
    proof (cases "even m")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   266
      case True
58710
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58709
diff changeset
   267
      then show ?thesis by (auto simp add: even_two_times_div_two)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   268
    next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   269
      case False
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58740
diff changeset
   270
      then have eq: "Suc (2 * (m div 2)) = m" 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
   271
      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
   272
      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
   273
      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
   274
      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
   275
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   276
    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
   277
  }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   278
  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
   279
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   280
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   281
lemma sums_if:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   282
  fixes g :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   283
  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
   284
  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
   285
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   286
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   287
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   288
    fix B T E
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   289
    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
   290
      by (cases B) auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   291
  } note if_sum = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   292
  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
   293
    using sums_if'[OF `g sums x`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   294
  {
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 38642
diff changeset
   295
    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
   296
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   297
    have "?s sums y" using sums_if'[OF `f sums y`] .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   298
    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
   299
    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   300
      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
   301
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   302
  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
   303
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   304
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   305
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
   306
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   307
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
   308
  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
   309
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   310
  shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) ----> l) \<and>
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   311
             ((\<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
   312
  (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
   313
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
   314
  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
   315
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   316
  show "\<forall>n. ?f n \<le> ?f (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   317
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   318
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   319
    show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   320
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   321
  show "\<forall>n. ?g (Suc n) \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   322
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   323
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   324
    show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   325
      unfolding One_nat_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   326
  qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   327
  show "\<forall>n. ?f n \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   328
  proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   329
    fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   330
    show "?f n \<le> ?g n" using fg_diff a_pos
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   331
      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
   332
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   333
  show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   334
  proof (rule LIMSEQ_I)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   335
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   336
    assume "0 < r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   337
    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
   338
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   339
    hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   340
    thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   341
  qed
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   342
qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   343
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   344
lemma summable_Leibniz':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   345
  fixes a :: "nat \<Rightarrow> real"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   346
  assumes a_zero: "a ----> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   347
    and a_pos: "\<And> n. 0 \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   348
    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
   349
  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
   350
    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
   351
    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
   352
    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
   353
    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
   354
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   355
  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
   356
  let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   357
  let ?f = "\<lambda>n. ?P (2 * n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   358
  let ?g = "\<lambda>n. ?P (2 * n + 1)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   359
  obtain l :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   360
    where below_l: "\<forall> n. ?f n \<le> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   361
      and "?f ----> l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   362
      and above_l: "\<forall> n. l \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   363
      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
   364
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   365
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   366
  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
   367
  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
   368
  proof (rule LIMSEQ_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   369
    fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   370
    assume "0 < r"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   371
    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
   372
    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
   373
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   374
    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
   375
    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
   376
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   377
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   378
      fix n :: nat
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   379
      assume "n \<ge> (max (2 * f_no) (2 * g_no))"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   380
      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
   381
      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
   382
      proof (cases "even n")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   383
        case True
58710
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58709
diff changeset
   384
        then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   385
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   386
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   387
        from f[OF this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   388
          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
   389
      next
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   390
        case False
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   391
        hence "even (n - 1)" by simp
58710
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58709
diff changeset
   392
        then have n_eq: "2 * ((n - 1) div 2) = n - 1"
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58709
diff changeset
   393
          by (simp add: even_two_times_div_two)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   394
        hence range_eq: "n - 1 + 1 = n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   395
          using odd_pos[OF False] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   396
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   397
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   398
          by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   399
        from g[OF this] show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   400
          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
   401
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   402
    }
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   403
    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
   404
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   405
  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
   406
    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
   407
  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
   408
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   409
  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
   410
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   411
  fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   412
  show "suminf ?S \<le> ?g n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   413
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   414
  show "?f n \<le> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   415
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   416
  show "?g ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   417
    using `?g ----> l` `l = suminf ?S` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   418
  show "?f ----> suminf ?S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   419
    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
   420
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   421
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   422
theorem summable_Leibniz:
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   423
  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
   424
  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
   425
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   426
    and "0 < a 0 \<longrightarrow>
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   427
      (\<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
   428
    and "a 0 < 0 \<longrightarrow>
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   429
      (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg")
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   430
    and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?f")
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   431
    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
   432
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   433
  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
   434
  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
   435
    case True
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   436
    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
   437
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   438
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   439
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   440
      have "a (Suc n) \<le> a n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   441
        using ord[where n="Suc n" and m=n] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   442
    } note mono = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   443
    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
   444
    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
   445
    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
   446
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   447
    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
   448
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   449
    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
   450
    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
   451
    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
   452
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   453
    {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   454
      fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   455
      have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   456
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   457
    } note monotone = this
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   458
    note leibniz =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   459
      summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   460
        OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   461
    have "summable (\<lambda> n. (-1)^n * ?a n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   462
      using leibniz(1) by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   463
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   464
      unfolding summable_def by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   465
    from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   466
      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
   467
    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
   468
    moreover
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   469
    have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   470
      unfolding minus_diff_minus by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   471
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   472
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57514
diff changeset
   473
    have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   474
      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
   475
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   476
    have ?pos using `0 \<le> ?a 0` by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   477
    moreover have ?neg
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   478
      using leibniz(2,4)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   479
      unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   480
      by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   481
    moreover have ?f and ?g
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   482
      using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   483
      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
   484
    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
   485
  qed
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   486
  then show ?summable and ?pos and ?neg and ?f and ?g
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   487
    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
   488
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   489
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   490
subsection {* Term-by-Term Differentiability of Power Series *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   491
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   492
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
   493
  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
   494
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   495
text{*Lemma about distributing negation over it*}
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   496
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   497
  by (simp add: diffs_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   498
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   499
lemma sums_Suc_imp:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   500
  "(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
   501
  using sums_Suc_iff[of f] by simp
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 diffs_equiv:
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   504
  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
   505
  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
   506
      (\<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
   507
  unfolding diffs_def
54573
07864001495d cleaned up some messy proofs
paulson
parents: 54489
diff changeset
   508
  by (simp add: summable_sums sums_Suc_imp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   509
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   510
lemma lemma_termdiff1:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   511
  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
   512
  "(\<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
   513
   (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   514
  by (auto simp add: algebra_simps power_add [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   515
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   516
lemma sumr_diff_mult_const2:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   517
  "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
   518
  by (simp add: setsum_subtractf)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   519
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   520
lemma lemma_termdiff2:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   521
  fixes h :: "'a :: {field}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   522
  assumes h: "h \<noteq> 0"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   523
  shows
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   524
    "((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
   525
     h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   526
          (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   527
  apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   528
  apply (simp add: right_diff_distrib diff_divide_distrib h)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   529
  apply (simp add: mult.assoc [symmetric])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   530
  apply (cases "n", simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   531
  apply (simp add: lemma_realpow_diff_sumr2 h
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   532
                   right_diff_distrib [symmetric] mult.assoc
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   533
              del: power_Suc setsum_lessThan_Suc of_nat_Suc)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   534
  apply (subst lemma_realpow_rev_sumr)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   535
  apply (subst sumr_diff_mult_const2)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   536
  apply simp
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   537
  apply (simp only: lemma_termdiff1 setsum_right_distrib)
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   538
  apply (rule setsum.cong [OF refl])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   539
  apply (simp add: less_iff_Suc_add)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   540
  apply (clarify)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   541
  apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 ac_simps
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   542
              del: setsum_lessThan_Suc power_Suc)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   543
  apply (subst mult.assoc [symmetric], subst power_add [symmetric])
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   544
  apply (simp add: ac_simps)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   545
  done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   546
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   547
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
   548
  fixes K :: "'a::linordered_semidom"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   549
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   550
    and K: "0 \<le> K"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   551
  shows "setsum f {..<n-k} \<le> of_nat n * K"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   552
  apply (rule order_trans [OF setsum_mono])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   553
  apply (rule f, simp)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   554
  apply (simp add: mult_right_mono K)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   555
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   556
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   557
lemma lemma_termdiff3:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   558
  fixes h z :: "'a::{real_normed_field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   559
  assumes 1: "h \<noteq> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   560
    and 2: "norm z \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   561
    and 3: "norm (z + h) \<le> K"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   562
  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
   563
          \<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
   564
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   565
  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
   566
        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
   567
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   568
    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
   569
  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
   570
  proof (rule mult_right_mono [OF _ norm_ge_zero])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   571
    from norm_ge_zero 2 have K: "0 \<le> K"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   572
      by (rule order_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   573
    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
   574
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   575
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   576
      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
   577
      done
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   578
    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
   579
          \<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
   580
      apply (intro
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   581
         order_trans [OF norm_setsum]
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   582
         real_setsum_nat_ivl_bounded2
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   583
         mult_nonneg_nonneg
47489
04e7d09ade7a tuned some proofs;
huffman
parents: 47108
diff changeset
   584
         of_nat_0_le_iff
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   585
         zero_le_power K)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   586
      apply (rule le_Kn, simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   587
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   588
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   589
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   590
    by (simp only: mult.assoc)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   591
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   592
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   593
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   594
lemma lemma_termdiff4:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   595
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   596
  assumes k: "0 < (k::real)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   597
    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
   598
  shows "f -- 0 --> 0"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   599
proof (rule tendsto_norm_zero_cancel)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   600
  show "(\<lambda>h. norm (f h)) -- 0 --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   601
  proof (rule real_tendsto_sandwich)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   602
    show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   603
      by simp
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   604
    show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   605
      using k by (auto simp add: eventually_at dist_norm le)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   606
    show "(\<lambda>h. 0) -- (0::'a) --> (0::real)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   607
      by (rule tendsto_const)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   608
    have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   609
      by (intro tendsto_intros)
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   610
    then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0"
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   611
      by simp
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   612
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   613
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   614
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   615
lemma lemma_termdiff5:
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   616
  fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   617
  assumes k: "0 < (k::real)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   618
  assumes f: "summable f"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   619
  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
   620
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   621
proof (rule lemma_termdiff4 [OF k])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   622
  fix h::'a
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   623
  assume "h \<noteq> 0" and "norm h < k"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   624
  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
   625
    by (simp add: le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   626
  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
   627
    by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   628
  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
   629
    by (rule summable_mult2)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   630
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   631
    by (rule summable_comparison_test)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   632
  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
   633
    by (rule summable_norm)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   634
  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
   635
    by (rule suminf_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   636
  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
   637
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   638
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   639
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   640
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   641
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   642
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   643
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   644
lemma termdiffs_aux:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   645
  fixes x :: "'a::{real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   646
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   647
    and 2: "norm x < norm K"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   648
  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
   649
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   650
proof -
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   651
  from dense [OF 2]
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   652
  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
   653
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   654
    by (rule order_le_less_trans)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   655
  hence r_neq_0: "r \<noteq> 0" by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   656
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   657
  proof (rule lemma_termdiff5)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   658
    show "0 < r - norm x" using r1 by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   659
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   660
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   661
    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
   662
      by (rule powser_insidea)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   663
    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
   664
      using r
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   665
      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
   666
    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
   667
      by (rule diffs_equiv [THEN sums_summable])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   668
    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
   669
      (\<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
   670
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   671
      apply (simp add: diffs_def)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   672
      apply (case_tac n, simp_all add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   673
      done
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   674
    finally have "summable
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   675
      (\<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
   676
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   677
    also have
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   678
      "(\<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
   679
           r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   680
       (\<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
   681
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   682
      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
   683
      apply (rename_tac nat)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   684
      apply (case_tac "nat", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   685
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   686
      done
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   687
    finally
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   688
    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
   689
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   690
    fix h::'a and n::nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   691
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   692
    assume "norm h < r - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   693
    hence "norm x + norm h < r" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   694
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   695
      by (rule order_le_less_trans)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   696
    show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   697
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   698
      apply (simp only: norm_mult mult.assoc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   699
      apply (rule mult_left_mono [OF _ norm_ge_zero])
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   700
      apply (simp add: mult.assoc [symmetric])
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   701
      apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   702
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   703
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   704
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   705
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   706
lemma termdiffs:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   707
  fixes K x :: "'a::{real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   708
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
54575
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   709
      and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   710
      and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
0b9ca2c865cb cleaned up more messy proofs
paulson
parents: 54573
diff changeset
   711
      and 4: "norm x < norm K"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   712
  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
   713
  unfolding DERIV_def
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   714
proof (rule LIM_zero_cancel)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   715
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   716
            - suminf (\<lambda>n. diffs c n * x^n)) -- 0 --> 0"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   717
  proof (rule LIM_equal2)
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   718
    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
   719
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   720
    fix h :: 'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   721
    assume "norm (h - 0) < norm K - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   722
    hence "norm x + norm h < norm K" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   723
    hence 5: "norm (x + h) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   724
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   725
    have "summable (\<lambda>n. c n * x^n)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   726
      and "summable (\<lambda>n. c n * (x + h) ^ n)"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   727
      and "summable (\<lambda>n. diffs c n * x^n)"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   728
      using 1 2 4 5 by (auto elim: powser_inside)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   729
    then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   730
          (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
56167
ac8098b0e458 tuned proofs
huffman
parents: 55832
diff changeset
   731
      by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   732
    then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   733
          (\<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
   734
      by (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   735
  next
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   736
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   737
      by (rule termdiffs_aux [OF 3 4])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   738
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   739
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   740
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   741
subsection {* The Derivative of a Power Series Has the Same Radius of Convergence *}
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   742
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   743
lemma termdiff_converges:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   744
  fixes x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   745
  assumes K: "norm x < K"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   746
      and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   747
    shows "summable (\<lambda>n. diffs c n * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   748
proof (cases "x = 0")
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   749
  case True then show ?thesis
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   750
  using powser_sums_zero sums_summable by auto
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   751
next
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   752
  case False
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   753
  then have "K>0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   754
    using K less_trans zero_less_norm_iff by blast
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   755
  then obtain r::real where r: "norm x < norm r" "norm r < K" "r>0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   756
    using K False
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   757
    by (auto simp: abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   758
  have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) ----> 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   759
    using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   760
  then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   761
    using r unfolding LIMSEQ_iff
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   762
    apply (drule_tac x=1 in spec)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   763
    apply (auto simp: norm_divide norm_mult norm_power field_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   764
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   765
  have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   766
    apply (rule summable_comparison_test' [of "\<lambda>n. norm(c n * (of_real r) ^ n)" N])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   767
    apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   768
    using N r norm_of_real [of "r+K", where 'a = 'a]
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   769
    apply (auto simp add: norm_divide norm_mult norm_power )
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   770
    using less_eq_real_def by fastforce
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   771
  then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   772
    using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1]
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   773
    by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   774
  then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   775
    using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   776
    by (simp add: mult.assoc) (auto simp: power_Suc mult_ac)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   777
  then show ?thesis 
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   778
    by (simp add: diffs_def)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   779
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   780
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   781
lemma termdiff_converges_all:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   782
  fixes x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   783
  assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   784
    shows "summable (\<lambda>n. diffs c n * x^n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   785
  apply (rule termdiff_converges [where K = "1 + norm x"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   786
  using assms
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   787
  apply (auto simp: )
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   788
  done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   789
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   790
lemma termdiffs_strong:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   791
  fixes K x :: "'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   792
  assumes sm: "summable (\<lambda>n. c n * K ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   793
      and K: "norm x < norm K"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   794
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   795
proof -
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   796
  have [simp]: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   797
    using K
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   798
    apply (auto simp: norm_divide)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   799
    apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   800
    apply (auto simp: mult_2_right norm_triangle_mono)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   801
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   802
  have "summable (\<lambda>n. c n * (of_real (norm x + norm K) / 2) ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   803
    apply (rule summable_norm_cancel [OF powser_insidea [OF sm]])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   804
    using K
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   805
    apply (auto simp: algebra_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   806
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   807
  moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs c n * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   808
    by (blast intro: sm termdiff_converges powser_inside)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   809
  moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs(diffs c) n * x ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   810
    by (blast intro: sm termdiff_converges powser_inside)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   811
  ultimately show ?thesis
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   812
    apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   813
    apply (auto simp: algebra_simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   814
    using K
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   815
    apply (simp add: norm_divide)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   816
    apply (rule less_le_trans [of _ "of_real (norm K) + of_real (norm x)"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   817
    apply (simp_all add: of_real_add [symmetric] del: of_real_add)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   818
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   819
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   820
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   821
lemma powser_limit_0: 
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   822
  fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   823
  assumes s: "0 < s"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   824
      and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   825
    shows "(f ---> a 0) (at 0)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   826
proof -
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   827
  have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   828
    apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   829
    using s
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   830
    apply (auto simp: norm_divide)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   831
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   832
  then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   833
    apply (rule termdiffs_strong)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   834
    using s
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   835
    apply (auto simp: norm_divide)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   836
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   837
  then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   838
    by (blast intro: DERIV_continuous)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   839
  then have "((\<lambda>x. \<Sum>n. a n * x ^ n) ---> a 0) (at 0)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   840
    by (simp add: continuous_within powser_zero)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   841
  then show ?thesis 
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   842
    apply (rule Lim_transform)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   843
    apply (auto simp add: LIM_eq)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   844
    apply (rule_tac x="s" in exI)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   845
    using s 
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   846
    apply (auto simp: sm [THEN sums_unique])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   847
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   848
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   849
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   850
lemma powser_limit_0_strong: 
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   851
  fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   852
  assumes s: "0 < s"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   853
      and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   854
    shows "(f ---> a 0) (at 0)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   855
proof -
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   856
  have *: "((\<lambda>x. if x = 0 then a 0 else f x) ---> a 0) (at 0)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   857
    apply (rule powser_limit_0 [OF s])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   858
    apply (case_tac "x=0")
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   859
    apply (auto simp add: powser_sums_zero sm)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   860
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   861
  show ?thesis
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   862
    apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"])
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   863
    apply (simp_all add: *)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   864
    done
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   865
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60036
diff changeset
   866
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   867
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   868
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
   869
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   870
lemma DERIV_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   871
  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
   872
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   873
    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
   874
    and "summable (f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   875
    and "summable L"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   876
    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
   877
  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
   878
  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
   879
proof (rule LIM_I)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   880
  fix r :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   881
  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
   882
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   883
  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
   884
    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
   885
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   886
  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
   887
    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
   888
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   889
  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
   890
  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
   891
    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
   892
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   893
  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
   894
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   895
  let ?r = "r / (3 * real ?N)"
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   896
  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
   897
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   898
  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
   899
  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
   900
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   901
  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
   902
  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
   903
    show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   904
    proof
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   905
      fix x
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   906
      assume "x \<in> ?s ` {..<?N}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   907
      then obtain n where "x = ?s n" and "n \<in> {..<?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   908
        using image_iff[THEN iffD1] by blast
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   909
      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
   910
      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
   911
        by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   912
      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
   913
      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
   914
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   915
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   916
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   917
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   918
  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
   919
    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
   920
    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
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   922
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   923
    fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   924
    assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   925
    hence x_in_I: "x0 + x \<in> { a <..< b }"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   926
      using S_a S_b by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   927
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   928
    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
   929
    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
   930
    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
   931
    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
   932
    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
   933
    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
   934
    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
   935
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   936
    { fix n
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   937
      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
   938
        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
   939
        unfolding abs_divide .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   940
      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
   941
        using `x \<noteq> 0` by auto }
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   942
    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
   943
    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
   944
      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
   945
    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
   946
      using L_estimate by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   947
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   948
    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
   949
    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
   950
    proof (rule setsum_strict_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   951
      fix n
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   952
      assume "n \<in> {..< ?N}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   953
      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
   954
      also have "S \<le> S'" using `S \<le> S'` .
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   955
      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
   956
      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
   957
        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
   958
          using `n \<in> {..< ?N}` by auto
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   959
        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
   960
      qed auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   961
      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
   962
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   963
      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
   964
      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
   965
      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
   966
        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
   967
    qed auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
   968
    also have "\<dots> = of_nat (card {..<?N}) * ?r"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   969
      by (rule setsum_constant)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   970
    also have "\<dots> = real ?N * ?r"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   971
      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
   972
    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
   973
    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
   974
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   975
    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
   976
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   977
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   978
      unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   979
      using suminf_divide[OF diff_smbl, symmetric] by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   980
    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
   981
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   982
      unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   983
      apply (subst (5) add.commute)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   984
      by (rule abs_triangle_ineq)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   985
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   986
      using abs_triangle_ineq4 by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
   987
    also have "\<dots> < r /3 + r/3 + r/3"
36842
99745a4b9cc9 fix some linarith_split_limit warnings
huffman
parents: 36824
diff changeset
   988
      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
   989
      by (rule add_strict_mono [OF add_less_le_mono])
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   990
    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
   991
      by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   992
  }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   993
  thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   994
      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
   995
    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
   996
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   997
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   998
lemma DERIV_power_series':
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
   999
  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
  1000
  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
  1001
    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
  1002
  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
  1003
  (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
  1004
proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1005
  {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1006
    fix R'
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1007
    assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1008
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1009
      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
  1010
    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
  1011
    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
  1012
      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
  1013
      proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1014
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1015
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1016
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1017
          using `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1018
        have "norm R' < norm ((R' + R) / 2)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1019
          using `0 < R'` `0 < R` `R' < R` by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1020
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1021
          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
  1022
      qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1023
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1024
        fix n x y
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1025
        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
  1026
        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
  1027
        proof -
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1028
          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
  1029
            (\<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
  1030
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1031
            by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1032
          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
  1033
          proof (rule mult_left_mono)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1034
            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
  1035
              by (rule setsum_abs)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1036
            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
  1037
            proof (rule setsum_mono)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1038
              fix p
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1039
              assume "p \<in> {..<Suc n}"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1040
              hence "p \<le> n" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1041
              {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1042
                fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1043
                fix x :: real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1044
                assume "x \<in> {-R'<..<R'}"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  1045
                hence "\<bar>x\<bar> \<le> R'"  by auto
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1046
                hence "\<bar>x^n\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1047
                  unfolding power_abs by (rule power_mono, auto)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1048
              }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1049
              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
  1050
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1051
                unfolding abs_mult by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1052
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1053
                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
  1054
            qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1055
            also have "\<dots> = real (Suc n) * R' ^ n"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1056
              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
  1057
            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
  1058
              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
  1059
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1060
              unfolding abs_mult[symmetric] by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  1061
          qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1062
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1063
            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
  1064
          finally show ?thesis .
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1065
        qed
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1066
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1067
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1068
        fix n
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1069
        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
  1070
          by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1071
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1072
      {
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1073
        fix x
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1074
        assume "x \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1075
        hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1076
          using assms `R' < R` by auto
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32047
diff changeset
  1077
        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
  1078
        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
  1079
          fix n
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1080
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1081
            by (rule mult_left_mono) auto
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1082
          show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1083
            unfolding real_norm_def abs_mult
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1084
            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
  1085
        qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`])
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1086
        from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute]
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1087
        show "summable (?f x)" by auto
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1088
      }
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1089
      show "summable (?f' x0)"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1090
        using converges[OF `x0 \<in> {-R <..< R}`] .
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1091
      show "x0 \<in> {-R' <..< R'}"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1092
        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
  1093
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1094
  } 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
  1095
  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
  1096
  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
  1097
  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
  1098
  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
  1099
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1100
    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
  1101
    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
  1102
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1103
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1104
    have "- ?R < 0" using assms by auto
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1105
    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
  1106
    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
  1107
  qed
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1108
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1109
    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
  1110
  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
  1111
  show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
  1112
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
  1113
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1114
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  1115
subsection {* Exponential Function *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1116
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1117
definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1118
  where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1119
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1120
lemma summable_exp_generic:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1121
  fixes x :: "'a::{real_normed_algebra_1,banach}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1122
  defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1123
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1124
proof -
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1125
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30082
diff changeset
  1126
    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
  1127
  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
  1128
    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
  1129
  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
  1130
    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
  1131
  from r1 show ?thesis
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56181
diff changeset
  1132
  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
  1133
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1134
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1135
    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
  1136
      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
  1137
    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
  1138
      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
  1139
    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
  1140
      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
  1141
    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
  1142
      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
  1143
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
  1144
      by (simp add: pos_divide_le_eq ac_simps)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1145
    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
  1146
      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
  1147
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1148
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1149
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1150
lemma summable_norm_exp:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
  1151
  fixes x :: "'a::{real_normed_algebra_1,banach}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1152
  shows "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1153
proof (rule summable_norm_comparison_test [OF exI, rule_format])
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1154
  show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1155
    by (rule summable_exp_generic)
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1156
  fix n
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1157
  show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35213
diff changeset
  1158
    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
  1159
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1160
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1161
lemma summable_exp: 
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1162
  fixes x :: "'a::{real_normed_field,banach}"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1163
  shows "summable (\<lambda>n. inverse (fact n) * x^n)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1164
  using summable_exp_generic [where x=x]
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1165
  by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1166
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1167
lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1168
  unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1169
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 41550
diff changeset
  1170
lemma exp_fdiffs:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1171
  fixes XXX :: "'a::{real_normed_field,banach}"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1172
  shows "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a))"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1173
  by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1174
           del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1175
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1176
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
  1177
  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
  1178
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1179
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
53079
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1180
  unfolding exp_def scaleR_conv_of_real
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1181
  apply (rule DERIV_cong)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1182
  apply (rule termdiffs [where K="of_real (1 + norm x)"])
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1183
  apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1184
  apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1185
  apply (simp del: of_real_add)
ade63ccd6f4e tuned proofs;
wenzelm
parents: 53076
diff changeset
  1186
  done
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1187
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  1188
declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
51527
bd62e7ff103b move Ln.thy and Log.thy to Transcendental.thy
hoelzl
parents: 51482
diff changeset
  1189
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1190
lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1191
proof -
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1192
  from summable_norm[OF summable_norm_exp, of x]
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
  1193
  have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))"
58656
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1194
    by (simp add: exp_def)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1195
  also have "\<dots> \<le> exp (norm x)"
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1196
    using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset
  1197
    by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
7f14d5d9b933 relaxed class constraints for exp
immler
parents: 58410
diff changeset