src/HOL/Transcendental.thy
author huffman
Tue, 24 Feb 2009 11:12:58 -0800
changeset 30082 43c5b7bfc791
parent 29803 c56a5571f60a
child 30273 ecd6f0ca62ea
permissions -rw-r--r--
make more proofs work whether or not One_nat_def is a simp rule
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     1
(*  Title       : Transcendental.thy
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     2
    Author      : Jacques D. Fleuriot
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     3
    Copyright   : 1998,1999 University of Cambridge
13958
c1c67582c9b5 New material on integration, etc. Moving Hyperreal/ex
paulson
parents: 12196
diff changeset
     4
                  1999,2001 University of Edinburgh
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
     5
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     6
*)
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     7
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
     8
header{*Power Series, Transcendental Functions etc.*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
     9
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15086
diff changeset
    10
theory Transcendental
25600
73431bd8c4c4 joined EvenOdd theory with Parity
haftmann
parents: 25153
diff changeset
    11
imports Fact Series Deriv NthRoot
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15086
diff changeset
    12
begin
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    13
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
    14
subsection {* Properties of Power Series *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    15
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    16
lemma lemma_realpow_diff:
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    17
  fixes y :: "'a::recpower"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    18
  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    19
proof -
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    20
  assume "p \<le> n"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    21
  hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    22
  thus ?thesis by (simp add: power_Suc power_commutes)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    23
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    24
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    25
lemma lemma_realpow_diff_sumr:
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    26
  fixes y :: "'a::{recpower,comm_semiring_0}" shows
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    27
     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =  
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    28
      y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
    29
by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
    30
         del: setsum_op_ivl_Suc cong: strong_setsum_cong)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    31
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
    32
lemma lemma_realpow_diff_sumr2:
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    33
  fixes y :: "'a::{recpower,comm_ring}" shows
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
    34
     "x ^ (Suc n) - y ^ (Suc n) =  
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    35
      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
25153
af3c7e99fed0 tuned proof
haftmann
parents: 25062
diff changeset
    36
apply (induct n, simp add: power_Suc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    37
apply (simp add: power_Suc del: setsum_op_ivl_Suc)
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
    38
apply (subst setsum_op_ivl_Suc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    39
apply (subst lemma_realpow_diff_sumr)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    40
apply (simp add: right_distrib del: setsum_op_ivl_Suc)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    41
apply (subst mult_left_commute [where a="x - y"])
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    42
apply (erule subst)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29171
diff changeset
    43
apply (simp add: power_Suc algebra_simps)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    44
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    45
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
    46
lemma lemma_realpow_rev_sumr:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
    47
     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    48
      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    49
apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    50
apply (rule inj_onI, simp)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    51
apply auto
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    52
apply (rule_tac x="n - x" in image_eqI, simp, simp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    53
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    54
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    55
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    56
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    57
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
    58
lemma powser_insidea:
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    59
  fixes x z :: "'a::{real_normed_field,banach,recpower}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    60
  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    61
  assumes 2: "norm z < norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    62
  shows "summable (\<lambda>n. norm (f n * z ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    63
proof -
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    64
  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    65
  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    66
    by (rule summable_LIMSEQ_zero)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    67
  hence "convergent (\<lambda>n. f n * x ^ n)"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    68
    by (rule convergentI)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    69
  hence "Cauchy (\<lambda>n. f n * x ^ n)"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    70
    by (simp add: Cauchy_convergent_iff)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    71
  hence "Bseq (\<lambda>n. f n * x ^ n)"
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    72
    by (rule Cauchy_Bseq)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    73
  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
    74
    by (simp add: Bseq_def, safe)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    75
  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    76
                   K * norm (z ^ n) * inverse (norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    77
  proof (intro exI allI impI)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    78
    fix n::nat assume "0 \<le> n"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    79
    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    80
          norm (f n * x ^ n) * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    81
      by (simp add: norm_mult abs_mult)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    82
    also have "\<dots> \<le> K * norm (z ^ n)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    83
      by (simp only: mult_right_mono 4 norm_ge_zero)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    84
    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
    85
      by (simp add: x_neq_0)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    86
    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
    87
      by (simp only: mult_assoc)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    88
    finally show "norm (norm (f n * z ^ n)) \<le>
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    89
                  K * norm (z ^ n) * inverse (norm (x ^ n))"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    90
      by (simp add: mult_le_cancel_right x_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    91
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    92
  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
    93
  proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    94
    from 2 have "norm (norm (z * inverse x)) < 1"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    95
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    96
      by (simp add: nonzero_norm_divide divide_inverse [symmetric])
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    97
    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
    98
      by (rule summable_geometric)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
    99
    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
   100
      by (rule summable_mult)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   101
    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   102
      using x_neq_0
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   103
      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   104
                    power_inverse norm_power mult_assoc)
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   105
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   106
  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
   107
    by (rule summable_comparison_test)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   108
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   109
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   110
lemma powser_inside:
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   111
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach,recpower}" shows
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   112
     "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   113
      ==> summable (%n. f(n) * (z ^ n))"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   114
by (rule powser_insidea [THEN summable_norm_cancel])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   115
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   116
lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   117
  "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) = 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   118
   (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   119
proof (induct n)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   120
  case (Suc n)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   121
  have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) = 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   122
        (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   123
    using Suc.hyps 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
   124
  also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   125
  finally show ?case .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   126
qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   127
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   128
lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   129
  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
   130
  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
   131
proof (rule LIMSEQ_I)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   132
  fix r :: real assume "0 < r"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   133
  from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   134
  obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   135
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   136
  let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   137
  { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   138
    have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }" 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   139
      using sum_split_even_odd by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   140
    hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   141
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   142
    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
   143
    proof (cases "even m")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   144
      case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   145
    next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   146
      case False hence "even (Suc 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
   147
      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   148
      have eq: "Suc (2 * (m div 2)) = m" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   149
      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
   150
      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
   151
      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
   152
      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
   153
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   154
    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
   155
  }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   156
  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
   157
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   158
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   159
lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   160
  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
   161
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   162
  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   163
  { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   164
      by (cases B) auto } note if_sum = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   165
  have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   166
  { 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   167
    have "?s 0 = 0" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   168
    have Suc_m1: "\<And> n. Suc n - 1 = 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
   169
    { fix B T E have "(if \<not> B then T else E) = (if B then E else T)" by auto } note if_eq = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   170
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   171
    have "?s sums y" using sums_if'[OF `f sums y`] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   172
    from this[unfolded sums_def, THEN LIMSEQ_Suc] 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   173
    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   174
      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   175
                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   176
                even_nat_Suc Suc_m1 if_eq .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   177
  } from sums_add[OF g_sums this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   178
  show ?thesis unfolding if_sum .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   179
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   180
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   181
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
   182
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   183
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
   184
  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
   185
  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   186
  shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and> 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   187
             ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   188
  (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   189
proof -
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   190
  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
   191
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   192
  have "\<forall> n. ?f n \<le> ?f (Suc n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   193
  proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   194
  moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   195
  have "\<forall> n. ?g (Suc n) \<le> ?g n"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   196
  proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   197
    unfolding One_nat_def by auto qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   198
  moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   199
  have "\<forall> n. ?f n \<le> ?g n" 
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   200
  proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   201
    unfolding One_nat_def by auto qed
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   202
  moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   203
  have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   204
  proof (rule LIMSEQ_I)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   205
    fix r :: real assume "0 < r"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   206
    with `a ----> 0`[THEN LIMSEQ_D] 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   207
    obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < 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
   208
    hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < 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
   209
    thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < 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
   210
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   211
  ultimately
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   212
  show ?thesis by (rule lemma_nest_unique)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   213
qed 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   214
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   215
lemma summable_Leibniz': 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
   216
  assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   217
  and a_monotone: "\<And> n. a (Suc n) \<le> a n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   218
  shows summable: "summable (\<lambda> n. (-1)^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
   219
  and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   220
  and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   221
  and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   222
  and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   223
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   224
  let "?S n" = "(-1)^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
   225
  let "?P n" = "\<Sum>i=0..<n. ?S i"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   226
  let "?f n" = "?P (2 * n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   227
  let "?g n" = "?P (2 * n + 1)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   228
  obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   229
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   230
  
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   231
  let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   232
  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
   233
  proof (rule LIMSEQ_I)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   234
    fix r :: real assume "0 < r"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   235
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   236
    with `?f ----> l`[THEN LIMSEQ_D] 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   237
    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
   238
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   239
    from `0 < r` `?g ----> l`[THEN LIMSEQ_D] 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   240
    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
   241
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   242
    { fix n :: nat
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   243
      assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   244
      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
   245
      proof (cases "even n")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   246
	case True from even_nat_div_two_times_two[OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   247
	have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   248
	with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   249
	from f[OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   250
	show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   251
      next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   252
	case False hence "even (n - 1)" using even_num_iff odd_pos by auto 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   253
	from even_nat_div_two_times_two[OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   254
	have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   255
	hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   256
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   257
	from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   258
	from g[OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   259
	show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   260
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   261
    }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   262
    thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < 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
   263
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   264
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   265
  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
   266
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   267
  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
   268
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   269
  { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_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
   270
  { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_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
   271
  show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` 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
  show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   273
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   274
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   275
theorem summable_Leibniz: 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
   276
  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
   277
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   278
  and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   279
  and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   280
  and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   281
  and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   282
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   283
  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
   284
  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
   285
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   286
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a 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
   287
    { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=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
   288
    note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   289
    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
   290
    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
   291
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   292
    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
   293
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   294
    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
   295
    have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a 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
   296
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a 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
   297
    { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=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
   298
    note monotone = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   299
    note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF LIMSEQ_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   300
    have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   301
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" 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
   302
    from this[THEN sums_minus]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   303
    have "(\<lambda> n. (-1)^n * a n) sums -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
   304
    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
   305
    moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   306
    have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   307
    
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   308
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   309
    have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a 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
   310
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   311
    have ?pos 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
   312
    moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   313
    moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN LIMSEQ_minus_cancel] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   314
    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
   315
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   316
  from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   317
       this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   318
  show ?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
   319
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   320
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   321
subsection {* Term-by-Term Differentiability of Power Series *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   322
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   323
definition
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   324
  diffs :: "(nat => 'a::ring_1) => nat => 'a" where
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   325
  "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   326
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   327
text{*Lemma about distributing negation over it*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   328
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   329
by (simp add: diffs_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   330
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   331
lemma sums_Suc_imp:
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   332
  assumes f: "f 0 = 0"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   333
  shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   334
unfolding sums_def
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   335
apply (rule LIMSEQ_imp_Suc)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   336
apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   337
apply (simp only: setsum_shift_bounds_Suc_ivl)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   338
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   339
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   340
lemma diffs_equiv:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   341
     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   342
      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums  
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   343
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   344
unfolding diffs_def
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   345
apply (drule summable_sums)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   346
apply (rule sums_Suc_imp, simp_all)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   347
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   348
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   349
lemma lemma_termdiff1:
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   350
  fixes z :: "'a :: {recpower,comm_ring}" shows
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   351
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   352
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29171
diff changeset
   353
by(auto simp add: algebra_simps power_add [symmetric] cong: strong_setsum_cong)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   354
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   355
lemma sumr_diff_mult_const2:
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   356
  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   357
by (simp add: setsum_subtractf)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   358
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   359
lemma lemma_termdiff2:
23112
2bc882fbe51c remove division_by_zero requirement from termdiffs lemmas; cleaned up some proofs
huffman
parents: 23097
diff changeset
   360
  fixes h :: "'a :: {recpower,field}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   361
  assumes h: "h \<noteq> 0" shows
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   362
  "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   363
   h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   364
        (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   365
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   366
apply (simp add: right_diff_distrib diff_divide_distrib h)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   367
apply (simp add: mult_assoc [symmetric])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   368
apply (cases "n", simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   369
apply (simp add: lemma_realpow_diff_sumr2 h
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   370
                 right_diff_distrib [symmetric] mult_assoc
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   371
            del: realpow_Suc setsum_op_ivl_Suc of_nat_Suc)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   372
apply (subst lemma_realpow_rev_sumr)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   373
apply (subst sumr_diff_mult_const2)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   374
apply simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   375
apply (simp only: lemma_termdiff1 setsum_right_distrib)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   376
apply (rule setsum_cong [OF refl])
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   377
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   378
apply (clarify)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   379
apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   380
            del: setsum_op_ivl_Suc realpow_Suc)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   381
apply (subst mult_assoc [symmetric], subst power_add [symmetric])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   382
apply (simp add: mult_ac)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   383
done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   384
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   385
lemma real_setsum_nat_ivl_bounded2:
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   386
  fixes K :: "'a::ordered_semidom"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   387
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   388
  assumes K: "0 \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   389
  shows "setsum f {0..<n-k} \<le> of_nat n * K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   390
apply (rule order_trans [OF setsum_mono])
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   391
apply (rule f, simp)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   392
apply (simp add: mult_right_mono K)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   393
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   394
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   395
lemma lemma_termdiff3:
23112
2bc882fbe51c remove division_by_zero requirement from termdiffs lemmas; cleaned up some proofs
huffman
parents: 23097
diff changeset
   396
  fixes h z :: "'a::{real_normed_field,recpower}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   397
  assumes 1: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   398
  assumes 2: "norm z \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   399
  assumes 3: "norm (z + h) \<le> K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   400
  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
   401
          \<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
   402
proof -
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   403
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   404
        norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   405
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   406
    apply (subst lemma_termdiff2 [OF 1])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   407
    apply (subst norm_mult)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   408
    apply (rule mult_commute)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   409
    done
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   410
  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
   411
  proof (rule mult_right_mono [OF _ norm_ge_zero])
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   412
    from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   413
    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
   414
      apply (erule subst)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   415
      apply (simp only: norm_mult norm_power power_add)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   416
      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
   417
      done
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   418
    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   419
              (z + h) ^ q * z ^ (n - 2 - q))
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   420
          \<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
   421
      apply (intro
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   422
         order_trans [OF norm_setsum]
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   423
         real_setsum_nat_ivl_bounded2
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   424
         mult_nonneg_nonneg
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   425
         zero_le_imp_of_nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   426
         zero_le_power K)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   427
      apply (rule le_Kn, simp)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   428
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   429
  qed
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   430
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   431
    by (simp only: mult_assoc)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   432
  finally show ?thesis .
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   433
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   434
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   435
lemma lemma_termdiff4:
23112
2bc882fbe51c remove division_by_zero requirement from termdiffs lemmas; cleaned up some proofs
huffman
parents: 23097
diff changeset
   436
  fixes f :: "'a::{real_normed_field,recpower} \<Rightarrow>
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   437
              'b::real_normed_vector"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   438
  assumes k: "0 < (k::real)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   439
  assumes 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
   440
  shows "f -- 0 --> 0"
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   441
unfolding LIM_def diff_0_right
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   442
proof (safe)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   443
  let ?h = "of_real (k / 2)::'a"
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   444
  have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   445
  hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   446
  hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   447
  hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   448
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   449
  fix r::real assume r: "0 < r"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   450
  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   451
  proof (cases)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   452
    assume "K = 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   453
    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   454
      by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   455
    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   456
  next
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   457
    assume K_neq_zero: "K \<noteq> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   458
    with zero_le_K have K: "0 < K" by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   459
    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   460
    proof (rule exI, safe)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   461
      from k r K show "0 < min k (r * inverse K / 2)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   462
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   463
    next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   464
      fix x::'a
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   465
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   466
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   467
        by simp_all
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   468
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   469
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   470
        by (rule mult_strict_left_mono)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   471
      also have "\<dots> = r / 2"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   472
        using K_neq_zero by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   473
      also have "r / 2 < r"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   474
        using r by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   475
      finally show "norm (f x) < r" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   476
    qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   477
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   478
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   479
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   480
lemma lemma_termdiff5:
23112
2bc882fbe51c remove division_by_zero requirement from termdiffs lemmas; cleaned up some proofs
huffman
parents: 23097
diff changeset
   481
  fixes g :: "'a::{recpower,real_normed_field} \<Rightarrow>
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   482
              nat \<Rightarrow> 'b::banach"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   483
  assumes k: "0 < (k::real)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   484
  assumes f: "summable f"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   485
  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
   486
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   487
proof (rule lemma_termdiff4 [OF k])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   488
  fix h::'a assume "h \<noteq> 0" and "norm h < k"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   489
  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
   490
    by (simp add: le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   491
  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
   492
    by simp
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   493
  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
   494
    by (rule summable_mult2)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   495
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   496
    by (rule summable_comparison_test)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   497
  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
   498
    by (rule summable_norm)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   499
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   500
    by (rule summable_le)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   501
  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
   502
    by (rule suminf_mult2 [symmetric])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   503
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   504
qed
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   505
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   506
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   507
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   508
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   509
lemma termdiffs_aux:
23112
2bc882fbe51c remove division_by_zero requirement from termdiffs lemmas; cleaned up some proofs
huffman
parents: 23097
diff changeset
   510
  fixes x :: "'a::{recpower,real_normed_field,banach}"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   511
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   512
  assumes 2: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   513
  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
   514
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   515
proof -
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   516
  from dense [OF 2]
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   517
  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
   518
  from norm_ge_zero r1 have r: "0 < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   519
    by (rule order_le_less_trans)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   520
  hence r_neq_0: "r \<noteq> 0" by simp
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   521
  show ?thesis
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   522
  proof (rule lemma_termdiff5)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   523
    show "0 < r - norm x" using r1 by simp
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   524
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   525
    from r r2 have "norm (of_real r::'a) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   526
      by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   527
    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
   528
      by (rule powser_insidea)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   529
    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
   530
      using r
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   531
      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
   532
    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
   533
      by (rule diffs_equiv [THEN sums_summable])
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   534
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   535
      = (\<lambda>n. diffs (%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
   536
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   537
      apply (simp add: diffs_def)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   538
      apply (case_tac n, simp_all add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   539
      done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   540
    finally have "summable 
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   541
      (\<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
   542
      by (rule diffs_equiv [THEN sums_summable])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   543
    also have
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   544
      "(\<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
   545
           r ^ (n - Suc 0)) =
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   546
       (\<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
   547
      apply (rule ext)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   548
      apply (case_tac "n", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   549
      apply (case_tac "nat", simp)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   550
      apply (simp add: r_neq_0)
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   551
      done
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   552
    finally show
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   553
      "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
   554
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   555
    fix h::'a and n::nat
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   556
    assume h: "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   557
    assume "norm h < r - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   558
    hence "norm x + norm h < r" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   559
    with norm_triangle_ineq have xh: "norm (x + h) < r"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   560
      by (rule order_le_less_trans)
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   561
    show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   562
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   563
      apply (simp only: norm_mult mult_assoc)
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   564
      apply (rule mult_left_mono [OF _ norm_ge_zero])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   565
      apply (simp (no_asm) add: mult_assoc [symmetric])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   566
      apply (rule lemma_termdiff3)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   567
      apply (rule h)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   568
      apply (rule r1 [THEN order_less_imp_le])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   569
      apply (rule xh [THEN order_less_imp_le])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   570
      done
20849
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   571
  qed
389cd9c8cfe1 rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents: 20692
diff changeset
   572
qed
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   573
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   574
lemma termdiffs:
23112
2bc882fbe51c remove division_by_zero requirement from termdiffs lemmas; cleaned up some proofs
huffman
parents: 23097
diff changeset
   575
  fixes K x :: "'a::{recpower,real_normed_field,banach}"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   576
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   577
  assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   578
  assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   579
  assumes 4: "norm x < norm K"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   580
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   581
unfolding deriv_def
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   582
proof (rule LIM_zero_cancel)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   583
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   584
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   585
  proof (rule LIM_equal2)
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   586
    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
   587
  next
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   588
    fix h :: 'a
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   589
    assume "h \<noteq> 0"
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   590
    assume "norm (h - 0) < norm K - norm x"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   591
    hence "norm x + norm h < norm K" by simp
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   592
    hence 5: "norm (x + h) < norm K"
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   593
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   594
    have A: "summable (\<lambda>n. c n * x ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   595
      by (rule powser_inside [OF 1 4])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   596
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   597
      by (rule powser_inside [OF 1 5])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   598
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   599
      by (rule powser_inside [OF 2 4])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   600
    show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   601
             - (\<Sum>n. diffs c n * x ^ n) = 
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   602
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   603
      apply (subst sums_unique [OF diffs_equiv [OF C]])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   604
      apply (subst suminf_diff [OF B A])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   605
      apply (subst suminf_divide [symmetric])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   606
      apply (rule summable_diff [OF B A])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   607
      apply (subst suminf_diff)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   608
      apply (rule summable_divide)
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   609
      apply (rule summable_diff [OF B A])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   610
      apply (rule sums_summable [OF diffs_equiv [OF C]])
29163
e72d07a878f8 clean up some proofs; remove unused lemmas
huffman
parents: 28952
diff changeset
   611
      apply (rule arg_cong [where f="suminf"], rule ext)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29171
diff changeset
   612
      apply (simp add: algebra_simps)
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   613
      done
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   614
  next
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   615
    show "(\<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
   616
               of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
20860
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   617
        by (rule termdiffs_aux [OF 3 4])
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   618
  qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   619
qed
1a8efd618190 reorganize and speed up termdiffs proofs
huffman
parents: 20849
diff changeset
   620
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   621
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   622
subsection{* Some properties of factorials *}
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   623
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   624
lemma real_of_nat_fact_not_zero [simp]: "real (fact n) \<noteq> 0"
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   625
by auto
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   626
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   627
lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact n)"
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   628
by auto
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   629
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   630
lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact n)"
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   631
by simp
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   632
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   633
lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact n))"
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   634
by (auto simp add: positive_imp_inverse_positive)
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   635
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   636
lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact n))"
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   637
by (auto intro: order_less_imp_le)
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   638
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   639
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
   640
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   641
lemma DERIV_series': fixes f :: "real \<Rightarrow> 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
   642
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   643
  and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   644
  and "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
   645
  and "summable L" 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>"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   646
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   647
  unfolding deriv_def
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   648
proof (rule LIM_I)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   649
  fix r :: real assume "0 < r" hence "0 < r/3" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   650
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   651
  obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3" 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   652
    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
   653
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   654
  obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3" 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   655
    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
   656
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   657
  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
   658
  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
   659
    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
   660
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   661
  let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   662
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   663
  let ?r = "r / (3 * real ?N)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   664
  have "0 < 3 * real ?N" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   665
  from divide_pos_pos[OF `0 < r` this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   666
  have "0 < ?r" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   667
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   668
  let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   669
  def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   670
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   671
  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
   672
  proof (rule iffD2[OF Min_gr_iff])
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   673
    show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   674
    proof (rule ballI)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   675
      fix x assume "x \<in> ?s ` {0..<?N}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   676
      then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   677
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def] 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   678
      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)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   679
      have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   680
      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
   681
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   682
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   683
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   684
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   685
  hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   686
    by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   687
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   688
  { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   689
    hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   690
    
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   691
    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
   692
    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
   693
    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
   694
    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
   695
    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
   696
    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
   697
    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
   698
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   699
    { fix n
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   700
      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>" 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   701
	using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   702
      hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 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
   703
    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   704
    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   705
    have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   706
    hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   707
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   708
    have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   709
    also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   710
    proof (rule setsum_strict_mono)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   711
      fix n assume "n \<in> { 0 ..< ?N}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   712
      have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   713
      also have "S \<le> S'" using `S \<le> S'` .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   714
      also have "S' \<le> ?s n" 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
   715
      proof (rule Min_le_iff[THEN iffD2])
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   716
	have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?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
   717
	thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   718
      qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   719
      finally have "\<bar> x \<bar> < ?s n" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   720
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   721
      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
   722
      have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   723
      with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   724
      show "\<bar>?diff n x - f' x0 n\<bar> < ?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
   725
    qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   726
    also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   727
    also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   728
    also have "\<dots> = r/3" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   729
    finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   730
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   731
    from suminf_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
   732
    have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> = 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   733
                    \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   734
    also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   735
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   736
    also have "\<dots> < r /3 + r/3 + r/3" 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   737
      using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   738
    finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   739
      by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   740
  } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow> 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   741
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   742
    unfolding real_norm_def diff_0_right by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   743
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   744
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   745
lemma DERIV_power_series': fixes f :: "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
   746
  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   747
  and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   748
  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
   749
  (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
   750
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   751
  { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   752
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< 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
   753
    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
   754
    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
   755
      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
   756
      proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   757
	have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < 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
   758
	hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < 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
   759
	have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < 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
   760
	from powser_insidea[OF converges[OF in_Rball] this] 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
   761
      qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   762
      { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   763
	show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   764
	proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   765
	  have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>" 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   766
	    unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   767
	  also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>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
   768
	  proof (rule mult_left_mono)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   769
	    have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   770
	    also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   771
	    proof (rule setsum_mono)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   772
	      fix p assume "p \<in> {0..<Suc n}" hence "p \<le> 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
   773
	      { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   774
		hence "\<bar>x\<bar> \<le> 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
   775
		hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   776
	      } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   777
	      have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   778
	      thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> 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
   779
	    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   780
	    also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_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
   781
	    finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   782
	    show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   783
	  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   784
	  also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult real_mult_assoc[symmetric] by algebra
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   785
	  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
   786
	qed }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   787
      { fix n
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   788
	from DERIV_pow[of "Suc n" x0, THEN DERIV_cmult[where c="f n"]]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   789
	show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)" unfolding real_mult_assoc by auto }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   790
      { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < 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
   791
	have "summable (\<lambda> n. f n * x^n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   792
	proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   793
	  fix n
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   794
	  have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   795
	  show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   796
	    by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   797
	qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   798
	from this[THEN summable_mult2[where c=x], unfolded real_mult_assoc, unfolded real_mult_commute]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   799
	show "summable (?f x)" by auto }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   800
      show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   801
      show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   802
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   803
  } 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
   804
  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
   805
  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
   806
  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
   807
  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
   808
    case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   809
    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
   810
    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
   811
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   812
    case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   813
    have "- ?R < 0" 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
   814
    also have "\<dots> \<le> x0" using False by auto 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   815
    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
   816
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   817
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?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
   818
  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
   819
  show ?thesis .
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29695
diff changeset
   820
qed
29695
171146a93106 Added real related theorems from Fact.thy
chaieb
parents: 29667
diff changeset
   821
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
   822
subsection {* Exponential Function *}
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   823
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   824
definition
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   825
  exp :: "'a \<Rightarrow> 'a::{recpower,real_normed_field,banach}" where
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   826
  "exp x = (\<Sum>n. x ^ n /\<^sub>R real (fact n))"
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   827
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   828
lemma summable_exp_generic:
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   829
  fixes x :: "'a::{real_normed_algebra_1,recpower,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   830
  defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   831
  shows "summable S"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   832
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   833
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   834
    unfolding S_def by (simp add: power_Suc del: mult_Suc)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   835
  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
   836
    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
   837
  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
   838
    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
   839
  from r1 show ?thesis
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   840
  proof (rule ratio_test [rule_format])
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   841
    fix n :: nat
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   842
    assume n: "N \<le> n"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   843
    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
   844
      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
   845
    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
   846
      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
   847
    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
   848
      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
   849
    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
   850
      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
   851
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   852
      by (simp add: pos_divide_le_eq mult_ac)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   853
    thus "norm (S (Suc n)) \<le> r * norm (S n)"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   854
      by (simp add: S_Suc norm_scaleR inverse_eq_divide)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   855
  qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   856
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   857
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   858
lemma summable_norm_exp:
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   859
  fixes x :: "'a::{real_normed_algebra_1,recpower,banach}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   860
  shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   861
proof (rule summable_norm_comparison_test [OF exI, rule_format])
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   862
  show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   863
    by (rule summable_exp_generic)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   864
next
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   865
  fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   866
    by (simp add: norm_scaleR norm_power_ineq)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   867
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   868
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   869
lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   870
by (insert summable_exp_generic [where x=x], simp)
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   871
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   872
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   873
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   874
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
   875
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   876
lemma exp_fdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   877
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
23431
25ca91279a9b change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents: 23413
diff changeset
   878
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
23082
ffef77eed382 generalize powerseries and termdiffs lemmas using axclasses
huffman
parents: 23069
diff changeset
   879
         del: mult_Suc of_nat_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   880
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   881
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   882
by (simp add: diffs_def)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   883
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   884
lemma lemma_exp_ext: "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   885
by (auto intro!: ext simp add: exp_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   886
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   887
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   888
apply (simp add: exp_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   889
apply (subst lemma_exp_ext)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   890
apply (subgoal_tac "DERIV (\<lambda>u. \<Sum>n. of_real (inverse (real (fact n))) * u ^ n) x :> (\<Sum>n. diffs (\<lambda>n. of_real (inverse (real (fact n)))) n * x ^ n)")
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   891
apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   892
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   893
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   894
apply (simp del: of_real_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   895
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   896
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
   897
lemma isCont_exp [simp]: "isCont exp x"
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
   898
by (rule DERIV_exp [THEN DERIV_isCont])
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
   899
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
   900
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   901
subsubsection {* Properties of the Exponential Function *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   902
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   903
lemma powser_zero:
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   904
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1,recpower}"
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   905
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   906
proof -
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   907
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   908
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   909
  thus ?thesis unfolding One_nat_def by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   910
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   911
23278
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   912
lemma exp_zero [simp]: "exp 0 = 1"
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   913
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
375335bf619f clean up proofs of exp_zero, sin_zero, cos_zero
huffman
parents: 23255
diff changeset
   914
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   915
lemma setsum_cl_ivl_Suc2:
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   916
  "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
28069
ba4de3022862 moved lemma into SetInterval where it belongs
nipkow
parents: 27483
diff changeset
   917
by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   918
         del: setsum_cl_ivl_Suc)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   919
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   920
lemma exp_series_add:
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   921
  fixes x y :: "'a::{real_field,recpower}"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   922
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   923
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   924
proof (induct n)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   925
  case 0
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   926
  show ?case
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   927
    unfolding S_def by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   928
next
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   929
  case (Suc n)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   930
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   931
    unfolding S_def by (simp add: power_Suc del: mult_Suc)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   932
  hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   933
    by simp
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   934
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   935
  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   936
    by (simp only: times_S)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   937
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   938
    by (simp only: Suc)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   939
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   940
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   941
    by (rule left_distrib)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   942
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   943
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   944
    by (simp only: setsum_right_distrib mult_ac)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   945
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   946
                + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   947
    by (simp add: times_S Suc_diff_le)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   948
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   949
             (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   950
    by (subst setsum_cl_ivl_Suc2, simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   951
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   952
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   953
    by (subst setsum_cl_ivl_Suc, simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   954
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   955
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   956
             (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   957
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   958
              real_of_nat_add [symmetric], simp)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 23477
diff changeset
   959
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23115
diff changeset
   960
    by (simp only: scaleR_right.setsum)
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   961
  finally show
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   962
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   963
    by (simp add: scaleR_cancel_left del: setsum_cl_ivl_Suc)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   964
qed
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   965
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   966
lemma exp_add: "exp (x + y) = exp x * exp y"
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   967
unfolding exp_def
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   968
by (simp only: Cauchy_product summable_norm_exp exp_series_add)
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
   969
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   970
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   971
by (rule exp_add [symmetric])
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   972
23241
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   973
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   974
unfolding exp_def
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   975
apply (subst of_real.suminf)
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   976
apply (rule summable_exp_generic)
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   977
apply (simp add: scaleR_conv_of_real)
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   978
done
5f12b40a95bf add lemma exp_of_real
huffman
parents: 23177
diff changeset
   979
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   980
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   981
proof
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   982
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   983
  also assume "exp x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   984
  finally show "False" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   985
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   986
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   987
lemma exp_minus: "exp (- x) = inverse (exp x)"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   988
by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   989
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   990
lemma exp_diff: "exp (x - y) = exp x / exp y"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   991
  unfolding diff_minus divide_inverse
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   992
  by (simp add: exp_add exp_minus)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   993
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   994
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   995
subsubsection {* Properties of the Exponential Function on Reals *}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   996
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
   997
text {* Comparisons of @{term "exp x"} with zero. *}
29167
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   998
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
   999
text{*Proof: because every exponential can be seen as a square.*}
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1000
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1001
proof -
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1002
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1003
  thus ?thesis by (simp add: exp_add [symmetric])
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1004
qed
37a952bb9ebc rearranged subsections; cleaned up some proofs
huffman
parents: 29166
diff changeset
  1005
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1006
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1007
by (simp add: order_less_le)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1008
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1009
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1010
by (simp add: not_less)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1011
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1012
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1013
by (simp add: not_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1014
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1015
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1016
by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1017
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1018
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
  1019
apply (induct "n")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1020
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1021
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1022
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1023
text {* Strict monotonicity of exponential. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1024
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1025
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1026
apply (drule order_le_imp_less_or_eq, auto)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1027
apply (simp add: exp_def)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1028
apply (rule real_le_trans)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1029
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1030
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1031
done
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1032
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1033
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1034
proof -
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1035
  assume x: "0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1036
  hence "1 < 1 + x" by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1037
  also from x have "1 + x \<le> exp x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1038
    by (simp add: exp_ge_add_one_self_aux)
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1039
  finally show ?thesis .
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1040
qed
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1041
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1042
lemma exp_less_mono:
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1043
  fixes x y :: real
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1044
  assumes "x < y" shows "exp x < exp y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1045
proof -
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1046
  from `x < y` have "0 < y - x" by simp
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1047
  hence "1 < exp (y - x)" by (rule exp_gt_one)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1048
  hence "1 < exp y / exp x" by (simp only: exp_diff)
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1049
  thus "exp x < exp y" by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1050
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1051
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1052
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1053
apply (simp add: linorder_not_le [symmetric])
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1054
apply (auto simp add: order_le_less exp_less_mono)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1055
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1056
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1057
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1058
by (auto intro: exp_less_mono exp_less_cancel)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1059
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1060
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1061
by (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1062
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1063
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1064
by (simp add: order_eq_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1065
29170
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1066
text {* Comparisons of @{term "exp x"} with one. *}
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1067
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1068
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1069
  using exp_less_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1070
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1071
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1072
  using exp_less_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1073
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1074
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1075
  using exp_le_cancel_iff [where x=0 and y=x] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1076
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1077
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1078
  using exp_le_cancel_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1079
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1080
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1081
  using exp_inj_iff [where x=x and y=0] by simp
dad3933c88dd clean up lemmas about exp
huffman
parents: 29167
diff changeset
  1082
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1083
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1084
apply (rule IVT)
23045
95e04f335940 add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents: 23043
diff changeset
  1085
apply (auto intro: isCont_exp simp add: le_diff_eq)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1086
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
29165
562f95f06244 cleaned up some proofs; removed redundant simp rules
huffman
parents: 29164
diff changeset
  1087
apply simp
17014
ad5ceb90877d renamed exp_ge_add_one_self to exp_ge_add_one_self_aux
avigad
parents: 16924
diff changeset
  1088
apply (rule exp_ge_add_one_self_aux, simp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1089
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1090
23115
4615b2078592 generalized exp to work over any complete field; new proof of exp_add
huffman
parents: 23112
diff changeset
  1091
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1092
apply (rule_tac x = 1 and y = y in linorder_cases)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1093
apply (drule order_less_imp_le [THEN lemma_exp_total])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1094
apply (rule_tac [2] x = 0 in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1095
apply (frule_tac [3] real_inverse_gt_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1096
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1097
apply (rule_tac x = "-x" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1098
apply (simp add: exp_minus)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1099
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1100
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1101
29164
0d49c5b55046 move sin and cos to their own subsection
huffman
parents: 29163
diff changeset
  1102
subsection {* Natural Logarithm *}
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1103
23043
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1104
definition
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1105
  ln :: "real => real" where
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1106
  "ln x = (THE u. exp u = x)"
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1107
5dbfd67516a4 rearranged sections
huffman
parents: 23011
diff changeset
  1108
lemma ln_exp [simp]: "ln (exp x) = x"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1109
by (simp add: ln_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1110
22654
c2b6b5a9e136 new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents: 22653
diff changeset
  1111
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
c2b6b5a9e136 new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents: 22653
diff changeset
  1112
by (auto dest: exp_total)
c2b6b5a9e136 new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents: 22653
diff changeset
  1113
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1114
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1115
apply (rule iffI)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1116
apply (erule subst, rule exp_gt_zero)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1117
apply (erule exp_ln)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1118
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1119
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1120
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1121
by (erule subst, rule ln_exp)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1122
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1123
lemma ln_one [simp]: "ln 1 = 0"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1124
by (rule ln_unique, simp)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1125
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1126
lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1127
by (rule ln_unique, simp add: exp_add)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1128
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1129
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1130
by (rule ln_unique, simp add: exp_minus)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1131
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1132
lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1133
by (rule ln_unique, simp add: exp_diff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1134
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1135
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1136
by (rule ln_unique, simp add: exp_real_of_nat_mult)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1137
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1138
lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1139
by (subst exp_less_cancel_iff [symmetric], simp)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1140
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1141
lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1142
by (simp add: linorder_not_less [symmetric])
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1143
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1144
lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1145
by (simp add: order_eq_iff)
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1146
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1147
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1148
apply (rule exp_le_cancel_iff [THEN iffD1])
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1149
apply (simp add: exp_ge_add_one_self_aux)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1150
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1151
29171
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1152
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
5eff800a695f clean up lemmas about ln
huffman
parents: 29170
diff changeset
  1153
by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1154
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1155
lemma ln_ge_zero [simp]:
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1156
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1157
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1158
  have "0 < x" using x by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1159
  hence "exp 0 \<le> exp (ln x)"
22915
bb8a928a6bfa fix proofs
huffman
parents: 22722
diff changeset
  1160
    by (simp add: x)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1161
  thus ?thesis by (simp only: exp_le_cancel_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1162
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1163
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1164
lemma ln_ge_zero_imp_ge_one:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1165
  assumes ln: "0 \<le> ln x" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1166
      and x:  "0 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1167
  shows "1 \<le> x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1168
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1169
  from ln have "ln 1 \<le> ln x" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1170
  thus ?thesis by (simp add: x del: ln_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1171
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1172
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1173
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1174
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1175
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1176
lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1177
by (insert ln_ge_zero_iff [of x], arith)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1178
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1179
lemma ln_gt_zero:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1180
  assumes x: "1 < x" shows "0 < ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1181
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1182
  have "0 < x" using x by arith
22915
bb8a928a6bfa fix proofs
huffman
parents: 22722
diff changeset
  1183
  hence "exp 0 < exp (ln x)" by (simp add: x)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1184
  thus ?thesis  by (simp only: exp_less_cancel_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1185
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1186
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1187
lemma ln_gt_zero_imp_gt_one:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1188
  assumes ln: "0 < ln x" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1189
      and x:  "0 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1190
  shows "1 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1191
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1192
  from ln have "ln 1 < ln x" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1193
  thus ?thesis by (simp add: x del: ln_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1194
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1195
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1196
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1197
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1198
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1199
lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1200
by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1201
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1202
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1203
by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1204
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1205
lemma exp_ln_eq: "exp u = x ==> ln x = u"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff