src/HOL/Limits.thy
author desharna
Fri, 27 May 2022 13:46:40 +0200
changeset 75465 d9b23902692d
parent 74513 67d87d224e00
child 76724 7ff71bdcf731
permissions -rw-r--r--
excluded dummy ATPs from Sledgehammer's default provers
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
52265
bb907eba5902 tuned headers;
wenzelm
parents: 51642
diff changeset
     1
(*  Title:      HOL/Limits.thy
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
     2
    Author:     Brian Huffman
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
     3
    Author:     Jacques D. Fleuriot, University of Cambridge
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
     4
    Author:     Lawrence C Paulson
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
     5
    Author:     Jeremy Avigad
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     6
*)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     7
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
     8
section \<open>Limits on Real Vector Spaces\<close>
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     9
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    10
theory Limits
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    11
  imports Real_Vector_Spaces
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    12
begin
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    13
73795
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    14
text \<open>Lemmas related to shifting/scaling\<close>
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    15
lemma range_add [simp]:
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    16
  fixes a::"'a::group_add" shows "range ((+) a) = UNIV"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    17
  by (metis add_minus_cancel surjI)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    18
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    19
lemma range_diff [simp]:
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    20
  fixes a::"'a::group_add" shows "range ((-) a) = UNIV"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    21
  by (metis (full_types) add_minus_cancel diff_minus_eq_add surj_def)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    22
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    23
lemma range_mult [simp]:
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    24
  fixes a::"real" shows "range ((*) a) = (if a=0 then {0} else UNIV)"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    25
  by (simp add: surj_def) (meson dvdE dvd_field_iff)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    26
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
    27
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
    28
subsection \<open>Filter going to infinity norm\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
    29
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    30
definition at_infinity :: "'a::real_normed_vector filter"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    31
  where "at_infinity = (INF r. principal {x. r \<le> norm x})"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
    32
57276
49c51eeaa623 filters are easier to define with INF on filters.
hoelzl
parents: 57275
diff changeset
    33
lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
49c51eeaa623 filters are easier to define with INF on filters.
hoelzl
parents: 57275
diff changeset
    34
  unfolding at_infinity_def
49c51eeaa623 filters are easier to define with INF on filters.
hoelzl
parents: 57275
diff changeset
    35
  by (subst eventually_INF_base)
49c51eeaa623 filters are easier to define with INF on filters.
hoelzl
parents: 57275
diff changeset
    36
     (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    37
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62369
diff changeset
    38
corollary eventually_at_infinity_pos:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    39
  "eventually p at_infinity \<longleftrightarrow> (\<exists>b. 0 < b \<and> (\<forall>x. norm x \<ge> b \<longrightarrow> p x))"
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    40
  unfolding eventually_at_infinity
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    41
  by (meson le_less_trans norm_ge_zero not_le zero_less_one)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    42
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    43
lemma at_infinity_eq_at_top_bot: "(at_infinity :: real filter) = sup at_top at_bot"
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    44
proof -
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    45
  have 1: "\<lbrakk>\<forall>n\<ge>u. A n; \<forall>n\<le>v. A n\<rbrakk>
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    46
       \<Longrightarrow> \<exists>b. \<forall>x. b \<le> \<bar>x\<bar> \<longrightarrow> A x" for A and u v::real
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    47
    by (rule_tac x="max (- v) u" in exI) (auto simp: abs_real_def)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    48
  have 2: "\<forall>x. u \<le> \<bar>x\<bar> \<longrightarrow> A x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. A n" for A and u::real
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    49
    by (meson abs_less_iff le_cases less_le_not_le)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    50
  have 3: "\<forall>x. u \<le> \<bar>x\<bar> \<longrightarrow> A x \<Longrightarrow> \<exists>N. \<forall>n\<le>N. A n" for A and u::real
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    51
    by (metis (full_types) abs_ge_self abs_minus_cancel le_minus_iff order_trans)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    52
  show ?thesis
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
    53
    by (auto simp: filter_eq_iff eventually_sup eventually_at_infinity
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    54
      eventually_at_top_linorder eventually_at_bot_linorder intro: 1 2 3)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
    55
qed
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
    56
57276
49c51eeaa623 filters are easier to define with INF on filters.
hoelzl
parents: 57275
diff changeset
    57
lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
    58
  unfolding at_infinity_eq_at_top_bot by simp
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
    59
57276
49c51eeaa623 filters are easier to define with INF on filters.
hoelzl
parents: 57275
diff changeset
    60
lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
    61
  unfolding at_infinity_eq_at_top_bot by simp
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
    62
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    63
lemma filterlim_at_top_imp_at_infinity: "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    64
  for f :: "_ \<Rightarrow> real"
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56541
diff changeset
    65
  by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56541
diff changeset
    66
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
    67
lemma filterlim_real_at_infinity_sequentially: "filterlim real at_infinity sequentially"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
    68
  by (simp add: filterlim_at_top_imp_at_infinity filterlim_real_sequentially)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
    69
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    70
lemma lim_infinity_imp_sequentially: "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    71
  by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
    72
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
    73
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
    74
subsubsection \<open>Boundedness\<close>
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    75
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    76
definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    77
  where Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    78
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    79
abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    80
  where "Bseq X \<equiv> Bfun X sequentially"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    81
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    82
lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    83
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    84
lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    85
  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    86
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    87
lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
    88
  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
    89
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    90
lemma Bfun_def: "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
51474
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
    91
  unfolding Bfun_metric_def norm_conv_dist
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
    92
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    93
  fix y K
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
    94
  assume K: "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
51474
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
    95
  moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
    96
    by (intro always_eventually) (metis dist_commute dist_triangle)
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
    97
  with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
    98
    by eventually_elim auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
    99
  with \<open>0 < K\<close> show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
51474
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
   100
    by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62369
diff changeset
   101
qed (force simp del: norm_conv_dist [symmetric])
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   102
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   103
lemma BfunI:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   104
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   105
  shows "Bfun f F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   106
  unfolding Bfun_def
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   107
proof (intro exI conjI allI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   108
  show "0 < max K 1" by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   109
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   110
    using K by (rule eventually_mono) simp
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   111
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   112
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   113
lemma BfunE:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   114
  assumes "Bfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   115
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   116
  using assms unfolding Bfun_def by blast
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   117
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   118
lemma Cauchy_Bseq:
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   119
  assumes "Cauchy X" shows "Bseq X"
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   120
proof -
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   121
  have "\<exists>y K. 0 < K \<and> (\<exists>N. \<forall>n\<ge>N. dist (X n) y \<le> K)"
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   122
    if "\<And>m n. \<lbrakk>m \<ge> M; n \<ge> M\<rbrakk> \<Longrightarrow> dist (X m) (X n) < 1" for M
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   123
    by (meson order.order_iff_strict that zero_less_one)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   124
  with assms show ?thesis
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   125
    by (force simp: Cauchy_def Bfun_metric_def eventually_sequentially)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   126
qed
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   127
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   128
subsubsection \<open>Bounded Sequences\<close>
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   129
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   130
lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   131
  by (intro BfunI) (auto simp: eventually_sequentially)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   132
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   133
lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   134
  unfolding Bfun_def eventually_sequentially
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   135
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   136
  fix N K
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   137
  assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   138
  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54263
diff changeset
   139
    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   140
       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   141
qed auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   142
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   143
lemma BseqE: "Bseq X \<Longrightarrow> (\<And>K. 0 < K \<Longrightarrow> \<forall>n. norm (X n) \<le> K \<Longrightarrow> Q) \<Longrightarrow> Q"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   144
  unfolding Bseq_def by auto
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   145
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   146
lemma BseqD: "Bseq X \<Longrightarrow> \<exists>K. 0 < K \<and> (\<forall>n. norm (X n) \<le> K)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   147
  by (simp add: Bseq_def)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   148
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   149
lemma BseqI: "0 < K \<Longrightarrow> \<forall>n. norm (X n) \<le> K \<Longrightarrow> Bseq X"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
   150
  by (auto simp: Bseq_def)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   151
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   152
lemma Bseq_bdd_above: "Bseq X \<Longrightarrow> bdd_above (range X)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   153
  for X :: "nat \<Rightarrow> real"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   154
proof (elim BseqE, intro bdd_aboveI2)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   155
  fix K n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   156
  assume "0 < K" "\<forall>n. norm (X n) \<le> K"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   157
  then show "X n \<le> K"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   158
    by (auto elim!: allE[of _ n])
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   159
qed
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   160
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   161
lemma Bseq_bdd_above': "Bseq X \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   162
  for X :: "nat \<Rightarrow> 'a :: real_normed_vector"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   163
proof (elim BseqE, intro bdd_aboveI2)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   164
  fix K n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   165
  assume "0 < K" "\<forall>n. norm (X n) \<le> K"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   166
  then show "norm (X n) \<le> K"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   167
    by (auto elim!: allE[of _ n])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   168
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   169
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   170
lemma Bseq_bdd_below: "Bseq X \<Longrightarrow> bdd_below (range X)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   171
  for X :: "nat \<Rightarrow> real"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   172
proof (elim BseqE, intro bdd_belowI2)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   173
  fix K n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   174
  assume "0 < K" "\<forall>n. norm (X n) \<le> K"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   175
  then show "- K \<le> X n"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   176
    by (auto elim!: allE[of _ n])
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   177
qed
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
   178
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   179
lemma Bseq_eventually_mono:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   180
  assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   181
  shows "Bseq f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   182
proof -
67958
732c0b059463 tuned proofs and generalized some lemmas about limits
huffman
parents: 67950
diff changeset
   183
  from assms(2) obtain K where "0 < K" and "eventually (\<lambda>n. norm (g n) \<le> K) sequentially"
732c0b059463 tuned proofs and generalized some lemmas about limits
huffman
parents: 67950
diff changeset
   184
    unfolding Bfun_def by fast
732c0b059463 tuned proofs and generalized some lemmas about limits
huffman
parents: 67950
diff changeset
   185
  with assms(1) have "eventually (\<lambda>n. norm (f n) \<le> K) sequentially"
732c0b059463 tuned proofs and generalized some lemmas about limits
huffman
parents: 67950
diff changeset
   186
    by (fast elim: eventually_elim2 order_trans)
69272
15e9ed5b28fb isabelle update_cartouches -t;
wenzelm
parents: 69064
diff changeset
   187
  with \<open>0 < K\<close> show "Bseq f"
67958
732c0b059463 tuned proofs and generalized some lemmas about limits
huffman
parents: 67950
diff changeset
   188
    unfolding Bfun_def by fast
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   189
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   190
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   191
lemma lemma_NBseq_def: "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   192
proof safe
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   193
  fix K :: real
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   194
  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   195
  then have "K \<le> real (Suc n)" by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   196
  moreover assume "\<forall>m. norm (X m) \<le> K"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   197
  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   198
    by (blast intro: order_trans)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   199
  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   200
next
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   201
  show "\<And>N. \<forall>n. norm (X n) \<le> real (Suc N) \<Longrightarrow> \<exists>K>0. \<forall>n. norm (X n) \<le> K"
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   202
    using of_nat_0_less_iff by blast
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   203
qed
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   204
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   205
text \<open>Alternative definition for \<open>Bseq\<close>.\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   206
lemma Bseq_iff: "Bseq X \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   207
  by (simp add: Bseq_def) (simp add: lemma_NBseq_def)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   208
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   209
lemma lemma_NBseq_def2: "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   210
proof -
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   211
  have *: "\<And>N. \<forall>n. norm (X n) \<le> 1 + real N \<Longrightarrow>
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   212
         \<exists>N. \<forall>n. norm (X n) < 1 + real N"
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   213
    by (metis add.commute le_less_trans less_add_one of_nat_Suc)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   214
  then show ?thesis
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   215
    unfolding lemma_NBseq_def
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   216
    by (metis less_le_not_le not_less_iff_gr_or_eq of_nat_Suc)
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   217
qed
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   218
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   219
text \<open>Yet another definition for Bseq.\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   220
lemma Bseq_iff1a: "Bseq X \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) < real (Suc N))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   221
  by (simp add: Bseq_def lemma_NBseq_def2)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   222
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   223
subsubsection \<open>A Few More Equivalence Theorems for Boundedness\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   224
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   225
text \<open>Alternative formulation for boundedness.\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   226
lemma Bseq_iff2: "Bseq X \<longleftrightarrow> (\<exists>k > 0. \<exists>x. \<forall>n. norm (X n + - x) \<le> k)"
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   227
  by (metis BseqE BseqI' add.commute add_cancel_right_left add_uminus_conv_diff norm_add_leD
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   228
            norm_minus_cancel norm_minus_commute)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   229
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   230
text \<open>Alternative formulation for boundedness.\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   231
lemma Bseq_iff3: "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   232
  (is "?P \<longleftrightarrow> ?Q")
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53381
diff changeset
   233
proof
0ae3db699a3e tuned proofs
haftmann
parents: 53381
diff changeset
   234
  assume ?P
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   235
  then obtain K where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
   236
    by (auto simp: Bseq_def)
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53381
diff changeset
   237
  from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   238
  from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   239
    by (auto intro: order_trans norm_triangle_ineq4)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   240
  then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   241
    by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   242
  with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53381
diff changeset
   243
next
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   244
  assume ?Q
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
   245
  then show ?P by (auto simp: Bseq_iff2)
53602
0ae3db699a3e tuned proofs
haftmann
parents: 53381
diff changeset
   246
qed
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   247
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   248
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   249
subsubsection \<open>Upper Bounds and Lubs of Bounded Sequences\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   250
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   251
lemma Bseq_minus_iff: "Bseq (\<lambda>n. - (X n) :: 'a::real_normed_vector) \<longleftrightarrow> Bseq X"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   252
  by (simp add: Bseq_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   253
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
   254
lemma Bseq_add:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   255
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   256
  assumes "Bseq f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   257
  shows "Bseq (\<lambda>x. f x + c)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   258
proof -
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   259
  from assms obtain K where K: "\<And>x. norm (f x) \<le> K"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   260
    unfolding Bseq_def by blast
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   261
  {
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   262
    fix x :: nat
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   263
    have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   264
    also have "norm (f x) \<le> K" by (rule K)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   265
    finally have "norm (f x + c) \<le> K + norm c" by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   266
  }
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   267
  then show ?thesis by (rule BseqI')
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   268
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   269
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   270
lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   271
  for f :: "nat \<Rightarrow> 'a::real_normed_vector"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   272
  using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   273
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
   274
lemma Bseq_mult:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   275
  fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   276
  assumes "Bseq f" and "Bseq g"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   277
  shows "Bseq (\<lambda>x. f x * g x)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   278
proof -
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   279
  from assms obtain K1 K2 where K: "norm (f x) \<le> K1" "K1 > 0" "norm (g x) \<le> K2" "K2 > 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   280
    for x
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   281
    unfolding Bseq_def by blast
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   282
  then have "norm (f x * g x) \<le> K1 * K2" for x
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   283
    by (auto simp: norm_mult intro!: mult_mono)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   284
  then show ?thesis by (rule BseqI')
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   285
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   286
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   287
lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   288
  unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   289
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   290
lemma Bseq_cmult_iff:
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   291
  fixes c :: "'a::real_normed_field"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   292
  assumes "c \<noteq> 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   293
  shows "Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   294
proof
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   295
  assume "Bseq (\<lambda>x. c * f x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   296
  with Bfun_const have "Bseq (\<lambda>x. inverse c * (c * f x))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   297
    by (rule Bseq_mult)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   298
  with \<open>c \<noteq> 0\<close> show "Bseq f"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70804
diff changeset
   299
    by (simp add: field_split_simps)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   300
qed (intro Bseq_mult Bfun_const)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   301
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   302
lemma Bseq_subseq: "Bseq f \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   303
  for f :: "nat \<Rightarrow> 'a::real_normed_vector"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   304
  unfolding Bseq_def by auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   305
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   306
lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   307
  for f :: "nat \<Rightarrow> 'a::real_normed_vector"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   308
  using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   309
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   310
lemma increasing_Bseq_subseq_iff:
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
   311
  assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a::real_normed_vector) \<le> norm (f y)" "strict_mono g"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   312
  shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   313
proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   314
  assume "Bseq (\<lambda>x. f (g x))"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   315
  then obtain K where K: "\<And>x. norm (f (g x)) \<le> K"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   316
    unfolding Bseq_def by auto
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   317
  {
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   318
    fix x :: nat
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   319
    from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   320
      by (auto simp: filterlim_at_top eventually_at_top_linorder)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   321
    then have "norm (f x) \<le> norm (f (g y))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   322
      using assms(1) by blast
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   323
    also have "norm (f (g y)) \<le> K" by (rule K)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   324
    finally have "norm (f x) \<le> K" .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   325
  }
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   326
  then show "Bseq f" by (rule BseqI')
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   327
qed (use Bseq_subseq[of f g] in simp_all)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   328
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   329
lemma nonneg_incseq_Bseq_subseq_iff:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   330
  fixes f :: "nat \<Rightarrow> real"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   331
    and g :: "nat \<Rightarrow> nat"
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
   332
  assumes "\<And>x. f x \<ge> 0" "incseq f" "strict_mono g"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   333
  shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   334
  using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   335
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   336
lemma Bseq_eq_bounded: "range f \<subseteq> {a..b} \<Longrightarrow> Bseq f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   337
  for a b :: real
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   338
proof (rule BseqI'[where K="max (norm a) (norm b)"])
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   339
  fix n assume "range f \<subseteq> {a..b}"
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   340
  then have "f n \<in> {a..b}"
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   341
    by blast
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   342
  then show "norm (f n) \<le> max (norm a) (norm b)"
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   343
    by auto
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
   344
qed
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   345
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   346
lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> B \<Longrightarrow> Bseq X"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   347
  for B :: real
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   348
  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   349
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   350
lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. B \<le> X i \<Longrightarrow> Bseq X"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   351
  for B :: real
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   352
  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   353
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   354
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   355
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Polynomal function extremal theorem, from HOL Light\<close>
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   356
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
   357
lemma polyfun_extremal_lemma: 
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   358
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   359
  assumes "0 < e"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   360
    shows "\<exists>M. \<forall>z. M \<le> norm(z) \<longrightarrow> norm (\<Sum>i\<le>n. c(i) * z^i) \<le> e * norm(z) ^ (Suc n)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   361
proof (induct n)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   362
  case 0 with assms
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   363
  show ?case
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   364
    apply (rule_tac x="norm (c 0) / e" in exI)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   365
    apply (auto simp: field_simps)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   366
    done
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   367
next
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   368
  case (Suc n)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   369
  obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   370
    using Suc assms by blast
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   371
  show ?case
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   372
  proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   373
    fix z::'a
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   374
    assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   375
    then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   376
      using assms by (simp add: field_simps)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   377
    have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   378
      using M [OF z1] by simp
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   379
    then have "norm (\<Sum>i\<le>n. c i * z^i) + norm (c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   380
      by simp
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   381
    then have "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   382
      by (blast intro: norm_triangle_le elim: )
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   383
    also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   384
      by (simp add: norm_power norm_mult algebra_simps)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   385
    also have "... \<le> (e * norm z) * norm z ^ Suc n"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   386
      by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   387
    finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   388
      by simp
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   389
  qed
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   390
qed
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   391
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   392
lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   393
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   394
  assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   395
    shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   396
using kn
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   397
proof (induction n)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   398
  case 0
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   399
  then show ?case
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
   400
    using k by simp
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   401
next
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   402
  case (Suc m)
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
   403
  show ?case
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   404
  proof (cases "c (Suc m) = 0")
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   405
    case True
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
   406
    then show ?thesis using Suc k
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   407
      by auto (metis antisym_conv less_eq_Suc_le not_le)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   408
  next
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   409
    case False
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   410
    then obtain M where M:
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   411
          "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>m. c i * z^i) \<le> norm (c (Suc m)) / 2 * norm z ^ Suc m"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   412
      using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   413
      by auto
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   414
    have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   415
    proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   416
      fix z::'a
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   417
      assume z1: "M \<le> norm z" "1 \<le> norm z"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   418
         and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   419
      then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   420
        using False by (simp add: field_simps)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   421
      have nz: "norm z \<le> norm z ^ Suc m"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   422
        by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   423
      have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   424
        by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   425
      have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   426
            \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   427
        using M [of z] Suc z1  by auto
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   428
      also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   429
        using nz by (simp add: mult_mono del: power_Suc)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   430
      finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   431
        using Suc.IH
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   432
        apply (auto simp: eventually_at_infinity)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   433
        apply (rule *)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   434
        apply (simp add: field_simps norm_mult norm_power)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   435
        done
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   436
    qed
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
   437
    then show ?thesis
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   438
      by (simp add: eventually_at_infinity)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   439
  qed
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   440
qed
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   441
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   442
subsection \<open>Convergence to Zero\<close>
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   443
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   444
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   445
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   446
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   447
lemma ZfunI: "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   448
  by (simp add: Zfun_def)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   449
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   450
lemma ZfunD: "Zfun f F \<Longrightarrow> 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   451
  by (simp add: Zfun_def)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   452
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   453
lemma Zfun_ssubst: "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   454
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   455
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   456
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   457
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   458
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   459
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   460
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   461
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   462
lemma Zfun_imp_Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   463
  assumes f: "Zfun f F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   464
    and g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   465
  shows "Zfun (\<lambda>x. g x) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   466
proof (cases "0 < K")
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   467
  case K: True
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   468
  show ?thesis
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   469
  proof (rule ZfunI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   470
    fix r :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   471
    assume "0 < r"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   472
    then have "0 < r / K" using K by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   473
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   474
      using ZfunD [OF f] by blast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   475
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   476
    proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   477
      case (elim x)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   478
      then have "norm (f x) * K < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   479
        by (simp add: pos_less_divide_eq K)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   480
      then show ?case
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   481
        by (simp add: order_le_less_trans [OF elim(1)])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   482
    qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   483
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   484
next
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   485
  case False
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   486
  then have K: "K \<le> 0" by (simp only: not_less)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   487
  show ?thesis
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   488
  proof (rule ZfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   489
    fix r :: real
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   490
    assume "0 < r"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   491
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   492
    proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   493
      case (elim x)
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   494
      also have "norm (f x) * K \<le> norm (f x) * 0"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   495
        using K norm_ge_zero by (rule mult_left_mono)
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   496
      finally show ?case
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   497
        using \<open>0 < r\<close> by simp
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   498
    qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   499
  qed
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   500
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   501
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   502
lemma Zfun_le: "Zfun g F \<Longrightarrow> \<forall>x. norm (f x) \<le> norm (g x) \<Longrightarrow> Zfun f F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   503
  by (erule Zfun_imp_Zfun [where K = 1]) simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   504
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   505
lemma Zfun_add:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   506
  assumes f: "Zfun f F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   507
    and g: "Zfun g F"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   508
  shows "Zfun (\<lambda>x. f x + g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   509
proof (rule ZfunI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   510
  fix r :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   511
  assume "0 < r"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   512
  then have r: "0 < r / 2" by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   513
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   514
    using f r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   515
  moreover
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   516
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   517
    using g r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   518
  ultimately
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   519
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   520
  proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   521
    case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   522
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   523
      by (rule norm_triangle_ineq)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   524
    also have "\<dots> < r/2 + r/2"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   525
      using elim by (rule add_strict_mono)
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   526
    finally show ?case
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   527
      by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   528
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   529
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   530
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   531
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   532
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   533
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   534
lemma Zfun_diff: "Zfun f F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53602
diff changeset
   535
  using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   536
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   537
lemma (in bounded_linear) Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   538
  assumes g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   539
  shows "Zfun (\<lambda>x. f (g x)) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   540
proof -
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   541
  obtain K where "norm (f x) \<le> norm x * K" for x
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   542
    using bounded by blast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   543
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   544
    by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   545
  with g show ?thesis
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   546
    by (rule Zfun_imp_Zfun)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   547
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   548
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   549
lemma (in bounded_bilinear) Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   550
  assumes f: "Zfun f F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   551
    and g: "Zfun g F"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   552
  shows "Zfun (\<lambda>x. f x ** g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   553
proof (rule ZfunI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   554
  fix r :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   555
  assume r: "0 < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   556
  obtain K where K: "0 < K"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   557
    and norm_le: "norm (x ** y) \<le> norm x * norm y * K" for x y
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   558
    using pos_bounded by blast
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   559
  from K have K': "0 < inverse K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   560
    by (rule positive_imp_inverse_positive)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   561
  have "eventually (\<lambda>x. norm (f x) < r) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   562
    using f r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   563
  moreover
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   564
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   565
    using g K' by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   566
  ultimately
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   567
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   568
  proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   569
    case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   570
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   571
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   572
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   573
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   574
    also from K have "r * inverse K * K = r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   575
      by simp
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   576
    finally show ?case .
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   577
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   578
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   579
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   580
lemma (in bounded_bilinear) Zfun_left: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   581
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   582
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   583
lemma (in bounded_bilinear) Zfun_right: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   584
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   585
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   586
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   587
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   588
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   589
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   590
lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   591
  by (simp only: tendsto_iff Zfun_def dist_norm)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   592
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   593
lemma tendsto_0_le:
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   594
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F \<Longrightarrow> (g \<longlongrightarrow> 0) F"
56366
0362c3bb4d02 new theorem about zero limits
paulson <lp15@cam.ac.uk>
parents: 56330
diff changeset
   595
  by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
0362c3bb4d02 new theorem about zero limits
paulson <lp15@cam.ac.uk>
parents: 56330
diff changeset
   596
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   597
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   598
subsubsection \<open>Distance and norms\<close>
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   599
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   600
lemma tendsto_dist [tendsto_intros]:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   601
  fixes l m :: "'a::metric_space"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   602
  assumes f: "(f \<longlongrightarrow> l) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   603
    and g: "(g \<longlongrightarrow> m) F"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   604
  shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   605
proof (rule tendstoI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   606
  fix e :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   607
  assume "0 < e"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   608
  then have e2: "0 < e/2" by simp
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   609
  from tendstoD [OF f e2] tendstoD [OF g e2]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   610
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   611
  proof (eventually_elim)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   612
    case (elim x)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   613
    then show "dist (dist (f x) (g x)) (dist l m) < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   614
      unfolding dist_real_def
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   615
      using dist_triangle2 [of "f x" "g x" "l"]
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   616
        and dist_triangle2 [of "g x" "l" "m"]
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   617
        and dist_triangle3 [of "l" "m" "f x"]
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   618
        and dist_triangle [of "f x" "m" "g x"]
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   619
      by arith
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   620
  qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   621
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   622
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   623
lemma continuous_dist[continuous_intros]:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   624
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   625
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   626
  unfolding continuous_def by (rule tendsto_dist)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   627
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   628
lemma continuous_on_dist[continuous_intros]:
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   629
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   630
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   631
  unfolding continuous_on_def by (auto intro: tendsto_dist)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
   632
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   633
lemma continuous_at_dist: "isCont (dist a) b"
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   634
  using continuous_on_dist [OF continuous_on_const continuous_on_id] continuous_on_eq_continuous_within by blast
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   635
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   636
lemma tendsto_norm [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   637
  unfolding norm_conv_dist by (intro tendsto_intros)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   638
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   639
lemma continuous_norm [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   640
  unfolding continuous_def by (rule tendsto_norm)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   641
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   642
lemma continuous_on_norm [continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   643
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   644
  unfolding continuous_on_def by (auto intro: tendsto_norm)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   645
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   646
lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   647
  by (intro continuous_on_id continuous_on_norm)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 70999
diff changeset
   648
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   649
lemma tendsto_norm_zero: "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   650
  by (drule tendsto_norm) simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   651
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   652
lemma tendsto_norm_zero_cancel: "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   653
  unfolding tendsto_iff dist_norm by simp
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   654
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   655
lemma tendsto_norm_zero_iff: "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   656
  unfolding tendsto_iff dist_norm by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   657
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   658
lemma tendsto_rabs [tendsto_intros]: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   659
  for l :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   660
  by (fold real_norm_def) (rule tendsto_norm)
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   661
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   662
lemma continuous_rabs [continuous_intros]:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   663
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   664
  unfolding real_norm_def[symmetric] by (rule continuous_norm)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   665
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   666
lemma continuous_on_rabs [continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   667
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   668
  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   669
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   670
lemma tendsto_rabs_zero: "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   671
  by (fold real_norm_def) (rule tendsto_norm_zero)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   672
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   673
lemma tendsto_rabs_zero_cancel: "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   674
  by (fold real_norm_def) (rule tendsto_norm_zero_cancel)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   675
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   676
lemma tendsto_rabs_zero_iff: "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   677
  by (fold real_norm_def) (rule tendsto_norm_zero_iff)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   678
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   679
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   680
subsection \<open>Topological Monoid\<close>
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   681
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   682
class topological_monoid_add = topological_space + monoid_add +
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   683
  assumes tendsto_add_Pair: "LIM x (nhds a \<times>\<^sub>F nhds b). fst x + snd x :> nhds (a + b)"
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   684
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   685
class topological_comm_monoid_add = topological_monoid_add + comm_monoid_add
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   686
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   687
lemma tendsto_add [tendsto_intros]:
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   688
  fixes a b :: "'a::topological_monoid_add"
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   689
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   690
  using filterlim_compose[OF tendsto_add_Pair, of "\<lambda>x. (f x, g x)" a b F]
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   691
  by (simp add: nhds_prod[symmetric] tendsto_Pair)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   692
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   693
lemma continuous_add [continuous_intros]:
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   694
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   695
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   696
  unfolding continuous_def by (rule tendsto_add)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   697
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   698
lemma continuous_on_add [continuous_intros]:
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   699
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   700
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   701
  unfolding continuous_on_def by (auto intro: tendsto_add)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   702
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   703
lemma tendsto_add_zero:
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   704
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   705
  shows "(f \<longlongrightarrow> 0) F \<Longrightarrow> (g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   706
  by (drule (1) tendsto_add) simp
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   707
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
   708
lemma tendsto_sum [tendsto_intros]:
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   709
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_add"
63915
bab633745c7f tuned proofs;
wenzelm
parents: 63721
diff changeset
   710
  shows "(\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F) \<Longrightarrow> ((\<lambda>x. \<Sum>i\<in>I. f i x) \<longlongrightarrow> (\<Sum>i\<in>I. a i)) F"
bab633745c7f tuned proofs;
wenzelm
parents: 63721
diff changeset
   711
  by (induct I rule: infinite_finite_induct) (simp_all add: tendsto_add)
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   712
67673
c8caefb20564 lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   713
lemma tendsto_null_sum:
c8caefb20564 lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   714
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_add"
c8caefb20564 lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   715
  assumes "\<And>i. i \<in> I \<Longrightarrow> ((\<lambda>x. f x i) \<longlongrightarrow> 0) F"
c8caefb20564 lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   716
  shows "((\<lambda>i. sum (f i) I) \<longlongrightarrow> 0) F"
c8caefb20564 lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   717
  using tendsto_sum [of I "\<lambda>x y. f y x" "\<lambda>x. 0"] assms by simp
c8caefb20564 lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   718
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
   719
lemma continuous_sum [continuous_intros]:
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   720
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
63301
d3c87eb0bad2 new results about topology
paulson <lp15@cam.ac.uk>
parents: 63263
diff changeset
   721
  shows "(\<And>i. i \<in> I \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>I. f i x)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
   722
  unfolding continuous_def by (rule tendsto_sum)
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
   723
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
   724
lemma continuous_on_sum [continuous_intros]:
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   725
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::topological_comm_monoid_add"
63301
d3c87eb0bad2 new results about topology
paulson <lp15@cam.ac.uk>
parents: 63263
diff changeset
   726
  shows "(\<And>i. i \<in> I \<Longrightarrow> continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<Sum>i\<in>I. f i x)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
   727
  unfolding continuous_on_def by (auto intro: tendsto_sum)
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   728
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62368
diff changeset
   729
instance nat :: topological_comm_monoid_add
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   730
  by standard
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   731
    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62368
diff changeset
   732
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62368
diff changeset
   733
instance int :: topological_comm_monoid_add
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   734
  by standard
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   735
    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   736
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62368
diff changeset
   737
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   738
subsubsection \<open>Topological group\<close>
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   739
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   740
class topological_group_add = topological_monoid_add + group_add +
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   741
  assumes tendsto_uminus_nhds: "(uminus \<longlongrightarrow> - a) (nhds a)"
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   742
begin
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   743
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   744
lemma tendsto_minus [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> - a) F"
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   745
  by (rule filterlim_compose[OF tendsto_uminus_nhds])
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   746
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   747
end
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   748
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   749
class topological_ab_group_add = topological_group_add + ab_group_add
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   750
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   751
instance topological_ab_group_add < topological_comm_monoid_add ..
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   752
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   753
lemma continuous_minus [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   754
  for f :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   755
  unfolding continuous_def by (rule tendsto_minus)
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   756
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   757
lemma continuous_on_minus [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   758
  for f :: "_ \<Rightarrow> 'b::topological_group_add"
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   759
  unfolding continuous_on_def by (auto intro: tendsto_minus)
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   760
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   761
lemma tendsto_minus_cancel: "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   762
  for a :: "'a::topological_group_add"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   763
  by (drule tendsto_minus) simp
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   764
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   765
lemma tendsto_minus_cancel_left:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   766
  "(f \<longlongrightarrow> - (y::_::topological_group_add)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   767
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   768
  by auto
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   769
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   770
lemma tendsto_diff [tendsto_intros]:
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   771
  fixes a b :: "'a::topological_group_add"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   772
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   773
  using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   774
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   775
lemma continuous_diff [continuous_intros]:
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   776
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   777
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   778
  unfolding continuous_def by (rule tendsto_diff)
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   779
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   780
lemma continuous_on_diff [continuous_intros]:
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   781
  fixes f g :: "_ \<Rightarrow> 'b::topological_group_add"
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   782
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   783
  unfolding continuous_on_def by (auto intro: tendsto_diff)
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   784
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67371
diff changeset
   785
lemma continuous_on_op_minus: "continuous_on (s::'a::topological_group_add set) ((-) x)"
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   786
  by (rule continuous_intros | simp)+
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   787
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   788
instance real_normed_vector < topological_ab_group_add
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   789
proof
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   790
  fix a b :: 'a
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   791
  show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   792
    unfolding tendsto_Zfun_iff add_diff_add
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   793
    using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   794
    by (intro Zfun_add)
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
   795
       (auto simp: tendsto_Zfun_iff[symmetric] nhds_prod[symmetric] intro!: tendsto_fst)
63081
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   796
  show "(uminus \<longlongrightarrow> - a) (nhds a)"
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   797
    unfolding tendsto_Zfun_iff minus_diff_minus
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   798
    using filterlim_ident[of "nhds a"]
5a5beb3dbe7e introduced class topological_group between topological_monoid and real_normed_vector
immler
parents: 63040
diff changeset
   799
    by (intro Zfun_minus) (simp add: tendsto_Zfun_iff)
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   800
qed
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
   801
65204
d23eded35a33 modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
immler
parents: 65036
diff changeset
   802
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'a=real]
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50880
diff changeset
   803
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   804
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
   805
subsubsection \<open>Linear operators and multiplication\<close>
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   806
70999
5b753486c075 Inverse function theorem + lemmas
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
   807
lemma linear_times [simp]: "linear (\<lambda>x. c * x)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   808
  for c :: "'a::real_algebra"
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
   809
  by (auto simp: linearI distrib_left)
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
   810
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   811
lemma (in bounded_linear) tendsto: "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   812
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   813
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   814
lemma (in bounded_linear) continuous: "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   815
  using tendsto[of g _ F] by (auto simp: continuous_def)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   816
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   817
lemma (in bounded_linear) continuous_on: "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   818
  using tendsto[of g] by (auto simp: continuous_on_def)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   819
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   820
lemma (in bounded_linear) tendsto_zero: "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   821
  by (drule tendsto) (simp only: zero)
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   822
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   823
lemma (in bounded_bilinear) tendsto:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   824
  "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   825
  by (simp only: tendsto_Zfun_iff prod_diff_prod Zfun_add Zfun Zfun_left Zfun_right)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   826
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   827
lemma (in bounded_bilinear) continuous:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   828
  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   829
  using tendsto[of f _ F g] by (auto simp: continuous_def)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   830
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   831
lemma (in bounded_bilinear) continuous_on:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   832
  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   833
  using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   834
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   835
lemma (in bounded_bilinear) tendsto_zero:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   836
  assumes f: "(f \<longlongrightarrow> 0) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   837
    and g: "(g \<longlongrightarrow> 0) F"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   838
  shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   839
  using tendsto [OF f g] by (simp add: zero_left)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   840
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   841
lemma (in bounded_bilinear) tendsto_left_zero:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   842
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) \<longlongrightarrow> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   843
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   844
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   845
lemma (in bounded_bilinear) tendsto_right_zero:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   846
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) \<longlongrightarrow> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   847
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   848
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   849
lemmas tendsto_of_real [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   850
  bounded_linear.tendsto [OF bounded_linear_of_real]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   851
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   852
lemmas tendsto_scaleR [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   853
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   854
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   855
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   856
text\<open>Analogous type class for multiplication\<close>
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   857
class topological_semigroup_mult = topological_space + semigroup_mult +
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   858
  assumes tendsto_mult_Pair: "LIM x (nhds a \<times>\<^sub>F nhds b). fst x * snd x :> nhds (a * b)"
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   859
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   860
instance real_normed_algebra < topological_semigroup_mult
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   861
proof
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   862
  fix a b :: 'a
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   863
  show "((\<lambda>x. fst x * snd x) \<longlongrightarrow> a * b) (nhds a \<times>\<^sub>F nhds b)"
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   864
    unfolding nhds_prod[symmetric]
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   865
    using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   866
    by (simp add: bounded_bilinear.tendsto [OF bounded_bilinear_mult])
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   867
qed
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   868
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   869
lemma tendsto_mult [tendsto_intros]:
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   870
  fixes a b :: "'a::topological_semigroup_mult"
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   871
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x * g x) \<longlongrightarrow> a * b) F"
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   872
  using filterlim_compose[OF tendsto_mult_Pair, of "\<lambda>x. (f x, g x)" a b F]
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   873
  by (simp add: nhds_prod[symmetric] tendsto_Pair)
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   874
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   875
lemma tendsto_mult_left: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   876
  for c :: "'a::topological_semigroup_mult"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   877
  by (rule tendsto_mult [OF tendsto_const])
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   878
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   879
lemma tendsto_mult_right: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
   880
  for c :: "'a::topological_semigroup_mult"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   881
  by (rule tendsto_mult [OF _ tendsto_const])
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
   882
70804
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   883
lemma tendsto_mult_left_iff [simp]:
70803
2d658afa1fc0 Generalised two results concerning limits from the real numbers to type classes
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   884
   "c \<noteq> 0 \<Longrightarrow> tendsto(\<lambda>x. c * f x) (c * l) F \<longleftrightarrow> tendsto f l F" for c :: "'a::{topological_semigroup_mult,field}"
70688
3d894e1cfc75 new material on Analysis, plus some rearrangements
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
   885
  by (auto simp: tendsto_mult_left dest: tendsto_mult_left [where c = "1/c"])
3d894e1cfc75 new material on Analysis, plus some rearrangements
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
   886
70804
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   887
lemma tendsto_mult_right_iff [simp]:
70803
2d658afa1fc0 Generalised two results concerning limits from the real numbers to type classes
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   888
   "c \<noteq> 0 \<Longrightarrow> tendsto(\<lambda>x. f x * c) (l * c) F \<longleftrightarrow> tendsto f l F" for c :: "'a::{topological_semigroup_mult,field}"
2d658afa1fc0 Generalised two results concerning limits from the real numbers to type classes
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   889
  by (auto simp: tendsto_mult_right dest: tendsto_mult_left [where c = "1/c"])
70688
3d894e1cfc75 new material on Analysis, plus some rearrangements
paulson <lp15@cam.ac.uk>
parents: 70532
diff changeset
   890
70804
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   891
lemma tendsto_zero_mult_left_iff [simp]:
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   892
  fixes c::"'a::{topological_semigroup_mult,field}" assumes "c \<noteq> 0" shows "(\<lambda>n. c * a n)\<longlonglongrightarrow> 0 \<longleftrightarrow> a \<longlonglongrightarrow> 0"
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   893
  using assms tendsto_mult_left tendsto_mult_left_iff by fastforce
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   894
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   895
lemma tendsto_zero_mult_right_iff [simp]:
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   896
  fixes c::"'a::{topological_semigroup_mult,field}" assumes "c \<noteq> 0" shows "(\<lambda>n. a n * c)\<longlonglongrightarrow> 0 \<longleftrightarrow> a \<longlonglongrightarrow> 0"
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   897
  using assms tendsto_mult_right tendsto_mult_right_iff by fastforce
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   898
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   899
lemma tendsto_zero_divide_iff [simp]:
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   900
  fixes c::"'a::{topological_semigroup_mult,field}" assumes "c \<noteq> 0" shows "(\<lambda>n. a n / c)\<longlonglongrightarrow> 0 \<longleftrightarrow> a \<longlonglongrightarrow> 0"
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   901
  using tendsto_zero_mult_right_iff [of "1/c" a] assms by (simp add: field_simps)
4eef7c6ef7bf More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents: 70803
diff changeset
   902
70365
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   903
lemma lim_const_over_n [tendsto_intros]:
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   904
  fixes a :: "'a::real_normed_field"
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   905
  shows "(\<lambda>n. a / of_nat n) \<longlonglongrightarrow> 0"
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   906
  using tendsto_mult [OF tendsto_const [of a] lim_1_over_n] by simp
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   907
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   908
lemmas continuous_of_real [continuous_intros] =
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   909
  bounded_linear.continuous [OF bounded_linear_of_real]
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   910
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   911
lemmas continuous_scaleR [continuous_intros] =
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   912
  bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   913
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   914
lemmas continuous_mult [continuous_intros] =
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   915
  bounded_bilinear.continuous [OF bounded_bilinear_mult]
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   916
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   917
lemmas continuous_on_of_real [continuous_intros] =
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   918
  bounded_linear.continuous_on [OF bounded_linear_of_real]
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   919
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   920
lemmas continuous_on_scaleR [continuous_intros] =
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   921
  bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   922
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   923
lemmas continuous_on_mult [continuous_intros] =
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   924
  bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   925
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   926
lemmas tendsto_mult_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   927
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   928
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   929
lemmas tendsto_mult_left_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   930
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   931
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   932
lemmas tendsto_mult_right_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   933
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   934
68296
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   935
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   936
lemma continuous_mult_left:
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   937
  fixes c::"'a::real_normed_algebra"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   938
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   939
by (rule continuous_mult [OF continuous_const])
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   940
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   941
lemma continuous_mult_right:
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   942
  fixes c::"'a::real_normed_algebra"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   943
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   944
by (rule continuous_mult [OF _ continuous_const])
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   945
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   946
lemma continuous_on_mult_left:
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   947
  fixes c::"'a::real_normed_algebra"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   948
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   949
by (rule continuous_on_mult [OF continuous_on_const])
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   950
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   951
lemma continuous_on_mult_right:
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   952
  fixes c::"'a::real_normed_algebra"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   953
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   954
by (rule continuous_on_mult [OF _ continuous_on_const])
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   955
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   956
lemma continuous_on_mult_const [simp]:
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   957
  fixes c::"'a::real_normed_algebra"
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68860
diff changeset
   958
  shows "continuous_on s ((*) c)"
68296
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   959
  by (intro continuous_on_mult_left continuous_on_id)
69d680e94961 tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   960
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   961
lemma tendsto_divide_zero:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   962
  fixes c :: "'a::real_normed_field"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   963
  shows "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x / c) \<longlongrightarrow> 0) F"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   964
  by (cases "c=0") (simp_all add: divide_inverse tendsto_mult_left_zero)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   965
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   966
lemma tendsto_power [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   967
  for f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
58729
e8ecc79aee43 add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents: 57512
diff changeset
   968
  by (induct n) (simp_all add: tendsto_mult)
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   969
65680
378a2f11bec9 Simplification of some proofs. Also key lemmas using !! rather than ! in premises
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
   970
lemma tendsto_null_power: "\<lbrakk>(f \<longlongrightarrow> 0) F; 0 < n\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> 0) F"
378a2f11bec9 Simplification of some proofs. Also key lemmas using !! rather than ! in premises
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
   971
    for f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra_1}"
378a2f11bec9 Simplification of some proofs. Also key lemmas using !! rather than ! in premises
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
   972
  using tendsto_power [of f 0 F n] by (simp add: power_0_left)
378a2f11bec9 Simplification of some proofs. Also key lemmas using !! rather than ! in premises
paulson <lp15@cam.ac.uk>
parents: 65578
diff changeset
   973
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   974
lemma continuous_power [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   975
  for f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   976
  unfolding continuous_def by (rule tendsto_power)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   977
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
   978
lemma continuous_on_power [continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   979
  fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   980
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   981
  unfolding continuous_on_def by (auto intro: tendsto_power)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   982
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
   983
lemma tendsto_prod [tendsto_intros]:
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   984
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
63915
bab633745c7f tuned proofs;
wenzelm
parents: 63721
diff changeset
   985
  shows "(\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F) \<Longrightarrow> ((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
bab633745c7f tuned proofs;
wenzelm
parents: 63721
diff changeset
   986
  by (induct S rule: infinite_finite_induct) (simp_all add: tendsto_mult)
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   987
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
   988
lemma continuous_prod [continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   989
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   990
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
   991
  unfolding continuous_def by (rule tendsto_prod)
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
   992
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
   993
lemma continuous_on_prod [continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   994
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   995
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
   996
  unfolding continuous_on_def by (auto intro: tendsto_prod)
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   997
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   998
lemma tendsto_of_real_iff:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
   999
  "((\<lambda>x. of_real (f x) :: 'a::real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1000
  unfolding tendsto_iff by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1001
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1002
lemma tendsto_add_const_iff:
74475
409ca22dee4c new notion of infinite sums in HOL-Analysis, ordering on complex numbers
eberlm <eberlm@in.tum.de>
parents: 73885
diff changeset
  1003
  "((\<lambda>x. c + f x :: 'a::topological_group_add) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  1004
  using tendsto_add[OF tendsto_const[of c], of f d]
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1005
    and tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1006
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1007
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1008
class topological_monoid_mult = topological_semigroup_mult + monoid_mult
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1009
class topological_comm_monoid_mult = topological_monoid_mult + comm_monoid_mult
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1010
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1011
lemma tendsto_power_strong [tendsto_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1012
  fixes f :: "_ \<Rightarrow> 'b :: topological_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1013
  assumes "(f \<longlongrightarrow> a) F" "(g \<longlongrightarrow> b) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1014
  shows   "((\<lambda>x. f x ^ g x) \<longlongrightarrow> a ^ b) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1015
proof -
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1016
  have "((\<lambda>x. f x ^ b) \<longlongrightarrow> a ^ b) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1017
    by (induction b) (auto intro: tendsto_intros assms)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1018
  also from assms(2) have "eventually (\<lambda>x. g x = b) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1019
    by (simp add: nhds_discrete filterlim_principal)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1020
  hence "eventually (\<lambda>x. f x ^ b = f x ^ g x) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1021
    by eventually_elim simp
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1022
  hence "((\<lambda>x. f x ^ b) \<longlongrightarrow> a ^ b) F \<longleftrightarrow> ((\<lambda>x. f x ^ g x) \<longlongrightarrow> a ^ b) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1023
    by (intro filterlim_cong refl)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1024
  finally show ?thesis .
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1025
qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1026
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1027
lemma continuous_mult' [continuous_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1028
  fixes f g :: "_ \<Rightarrow> 'b::topological_semigroup_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1029
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1030
  unfolding continuous_def by (rule tendsto_mult)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1031
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1032
lemma continuous_power' [continuous_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1033
  fixes f :: "_ \<Rightarrow> 'b::topological_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1034
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ^ g x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1035
  unfolding continuous_def by (rule tendsto_power_strong) auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1036
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1037
lemma continuous_on_mult' [continuous_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1038
  fixes f g :: "_ \<Rightarrow> 'b::topological_semigroup_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1039
  shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. f x * g x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1040
  unfolding continuous_on_def by (auto intro: tendsto_mult)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1041
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1042
lemma continuous_on_power' [continuous_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1043
  fixes f :: "_ \<Rightarrow> 'b::topological_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1044
  shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. f x ^ g x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1045
  unfolding continuous_on_def by (auto intro: tendsto_power_strong)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1046
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1047
lemma tendsto_mult_one:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1048
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1049
  shows "(f \<longlongrightarrow> 1) F \<Longrightarrow> (g \<longlongrightarrow> 1) F \<Longrightarrow> ((\<lambda>x. f x * g x) \<longlongrightarrow> 1) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1050
  by (drule (1) tendsto_mult) simp
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1051
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1052
lemma tendsto_prod' [tendsto_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1053
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1054
  shows "(\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F) \<Longrightarrow> ((\<lambda>x. \<Prod>i\<in>I. f i x) \<longlongrightarrow> (\<Prod>i\<in>I. a i)) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1055
  by (induct I rule: infinite_finite_induct) (simp_all add: tendsto_mult)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1056
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1057
lemma tendsto_one_prod':
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1058
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1059
  assumes "\<And>i. i \<in> I \<Longrightarrow> ((\<lambda>x. f x i) \<longlongrightarrow> 1) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1060
  shows "((\<lambda>i. prod (f i) I) \<longlongrightarrow> 1) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1061
  using tendsto_prod' [of I "\<lambda>x y. f y x" "\<lambda>x. 1"] assms by simp
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1062
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1063
lemma continuous_prod' [continuous_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1064
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1065
  shows "(\<And>i. i \<in> I \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>I. f i x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1066
  unfolding continuous_def by (rule tendsto_prod')
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1067
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1068
lemma continuous_on_prod' [continuous_intros]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1069
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::topological_comm_monoid_mult"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1070
  shows "(\<And>i. i \<in> I \<Longrightarrow> continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<Prod>i\<in>I. f i x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1071
  unfolding continuous_on_def by (auto intro: tendsto_prod')
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1072
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1073
instance nat :: topological_comm_monoid_mult
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1074
  by standard
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1075
    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1076
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1077
instance int :: topological_comm_monoid_mult
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1078
  by standard
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1079
    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1080
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1081
class comm_real_normed_algebra_1 = real_normed_algebra_1 + comm_monoid_mult
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1082
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1083
context real_normed_field
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1084
begin
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1085
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1086
subclass comm_real_normed_algebra_1
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1087
proof
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1088
  from norm_mult[of "1 :: 'a" 1] show "norm 1 = 1" by simp 
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1089
qed (simp_all add: norm_mult)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1090
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1091
end
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68721
diff changeset
  1092
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1093
subsubsection \<open>Inverse and division\<close>
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1094
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1095
lemma (in bounded_bilinear) Zfun_prod_Bfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1096
  assumes f: "Zfun f F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1097
    and g: "Bfun g F"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1098
  shows "Zfun (\<lambda>x. f x ** g x) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1099
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1100
  obtain K where K: "0 \<le> K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1101
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  1102
    using nonneg_bounded by blast
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1103
  obtain B where B: "0 < B"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1104
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1105
    using g by (rule BfunE)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1106
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1107
  using norm_g proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1108
    case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1109
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1110
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1111
    also have "\<dots> \<le> norm (f x) * B * K"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1112
      by (intro mult_mono' order_refl norm_g norm_ge_zero mult_nonneg_nonneg K elim)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1113
    also have "\<dots> = norm (f x) * (B * K)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57447
diff changeset
  1114
      by (rule mult.assoc)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1115
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1116
  qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1117
  with f show ?thesis
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1118
    by (rule Zfun_imp_Zfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1119
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1120
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1121
lemma (in bounded_bilinear) Bfun_prod_Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1122
  assumes f: "Bfun f F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1123
    and g: "Zfun g F"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1124
  shows "Zfun (\<lambda>x. f x ** g x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1125
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1126
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1127
lemma Bfun_inverse:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1128
  fixes a :: "'a::real_normed_div_algebra"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1129
  assumes f: "(f \<longlongrightarrow> a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1130
  assumes a: "a \<noteq> 0"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1131
  shows "Bfun (\<lambda>x. inverse (f x)) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1132
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1133
  from a have "0 < norm a" by simp
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1134
  then have "\<exists>r>0. r < norm a" by (rule dense)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1135
  then obtain r where r1: "0 < r" and r2: "r < norm a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1136
    by blast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1137
  have "eventually (\<lambda>x. dist (f x) a < r) F"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  1138
    using tendstoD [OF f r1] by blast
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1139
  then have "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1140
  proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1141
    case (elim x)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1142
    then have 1: "norm (f x - a) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1143
      by (simp add: dist_norm)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1144
    then have 2: "f x \<noteq> 0" using r2 by auto
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1145
    then have "norm (inverse (f x)) = inverse (norm (f x))"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1146
      by (rule nonzero_norm_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1147
    also have "\<dots> \<le> inverse (norm a - r)"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1148
    proof (rule le_imp_inverse_le)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1149
      show "0 < norm a - r"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1150
        using r2 by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1151
      have "norm a - norm (f x) \<le> norm (a - f x)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1152
        by (rule norm_triangle_ineq2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1153
      also have "\<dots> = norm (f x - a)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1154
        by (rule norm_minus_commute)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1155
      also have "\<dots> < r" using 1 .
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1156
      finally show "norm a - r \<le> norm (f x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1157
        by simp
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1158
    qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1159
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1160
  qed
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1161
  then show ?thesis by (rule BfunI)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1162
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1163
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
  1164
lemma tendsto_inverse [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1165
  fixes a :: "'a::real_normed_div_algebra"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1166
  assumes f: "(f \<longlongrightarrow> a) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1167
    and a: "a \<noteq> 0"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1168
  shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1169
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1170
  from a have "0 < norm a" by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1171
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1172
    by (rule tendstoD)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1173
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1174
    unfolding dist_norm by (auto elim!: eventually_mono)
44627
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1175
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1176
    - (inverse (f x) * (f x - a) * inverse a)) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1177
    by (auto elim!: eventually_mono simp: inverse_diff_inverse)
44627
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1178
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1179
    by (intro Zfun_minus Zfun_mult_left
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1180
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1181
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1182
  ultimately show ?thesis
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1183
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1184
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1185
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1186
lemma continuous_inverse:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1187
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1188
  assumes "continuous F f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1189
    and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1190
  shows "continuous F (\<lambda>x. inverse (f x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1191
  using assms unfolding continuous_def by (rule tendsto_inverse)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1192
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1193
lemma continuous_at_within_inverse[continuous_intros]:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1194
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1195
  assumes "continuous (at a within s) f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1196
    and "f a \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1197
  shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1198
  using assms unfolding continuous_within by (rule tendsto_inverse)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1199
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
  1200
lemma continuous_on_inverse[continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1201
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1202
  assumes "continuous_on s f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1203
    and "\<forall>x\<in>s. f x \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1204
  shows "continuous_on s (\<lambda>x. inverse (f x))"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  1205
  using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1206
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
  1207
lemma tendsto_divide [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1208
  fixes a b :: "'a::real_normed_field"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1209
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1210
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1211
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1212
lemma continuous_divide:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1213
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1214
  assumes "continuous F f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1215
    and "continuous F g"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1216
    and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1217
  shows "continuous F (\<lambda>x. (f x) / (g x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1218
  using assms unfolding continuous_def by (rule tendsto_divide)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1219
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1220
lemma continuous_at_within_divide[continuous_intros]:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1221
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1222
  assumes "continuous (at a within s) f" "continuous (at a within s) g"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1223
    and "g a \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1224
  shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1225
  using assms unfolding continuous_within by (rule tendsto_divide)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1226
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1227
lemma isCont_divide[continuous_intros, simp]:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1228
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1229
  assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1230
  shows "isCont (\<lambda>x. (f x) / g x) a"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1231
  using assms unfolding continuous_at by (rule tendsto_divide)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1232
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
  1233
lemma continuous_on_divide[continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1234
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1235
  assumes "continuous_on s f" "continuous_on s g"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1236
    and "\<forall>x\<in>s. g x \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1237
  shows "continuous_on s (\<lambda>x. (f x) / (g x))"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  1238
  using assms unfolding continuous_on_def by (blast intro: tendsto_divide)
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1239
71837
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1240
lemma tendsto_power_int [tendsto_intros]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1241
  fixes a :: "'a::real_normed_div_algebra"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1242
  assumes f: "(f \<longlongrightarrow> a) F"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1243
    and a: "a \<noteq> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1244
  shows "((\<lambda>x. power_int (f x) n) \<longlongrightarrow> power_int a n) F"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1245
  using assms by (cases n rule: int_cases4) (auto intro!: tendsto_intros simp: power_int_minus)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1246
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1247
lemma continuous_power_int:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1248
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1249
  assumes "continuous F f"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1250
    and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1251
  shows "continuous F (\<lambda>x. power_int (f x) n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1252
  using assms unfolding continuous_def by (rule tendsto_power_int)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1253
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1254
lemma continuous_at_within_power_int[continuous_intros]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1255
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1256
  assumes "continuous (at a within s) f"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1257
    and "f a \<noteq> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1258
  shows "continuous (at a within s) (\<lambda>x. power_int (f x) n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1259
  using assms unfolding continuous_within by (rule tendsto_power_int)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1260
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1261
lemma continuous_on_power_int [continuous_intros]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1262
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1263
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1264
  shows "continuous_on s (\<lambda>x. power_int (f x) n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1265
  using assms unfolding continuous_on_def by (blast intro: tendsto_power_int)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71827
diff changeset
  1266
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1267
lemma tendsto_sgn [tendsto_intros]: "(f \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1268
  for l :: "'a::real_normed_vector"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1269
  unfolding sgn_div_norm by (simp add: tendsto_intros)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1270
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1271
lemma continuous_sgn:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1272
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1273
  assumes "continuous F f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1274
    and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1275
  shows "continuous F (\<lambda>x. sgn (f x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1276
  using assms unfolding continuous_def by (rule tendsto_sgn)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1277
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1278
lemma continuous_at_within_sgn[continuous_intros]:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1279
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1280
  assumes "continuous (at a within s) f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1281
    and "f a \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1282
  shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1283
  using assms unfolding continuous_within by (rule tendsto_sgn)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1284
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1285
lemma isCont_sgn[continuous_intros]:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1286
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1287
  assumes "isCont f a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1288
    and "f a \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1289
  shows "isCont (\<lambda>x. sgn (f x)) a"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1290
  using assms unfolding continuous_at by (rule tendsto_sgn)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1291
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56366
diff changeset
  1292
lemma continuous_on_sgn[continuous_intros]:
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1293
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1294
  assumes "continuous_on s f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1295
    and "\<forall>x\<in>s. f x \<noteq> 0"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1296
  shows "continuous_on s (\<lambda>x. sgn (f x))"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  1297
  using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1298
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1299
lemma filterlim_at_infinity:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60974
diff changeset
  1300
  fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1301
  assumes "0 \<le> c"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1302
  shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1303
  unfolding filterlim_iff eventually_at_infinity
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1304
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1305
  fix P :: "'a \<Rightarrow> bool"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1306
  fix b
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1307
  assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1308
  assume P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1309
  have "max b (c + 1) > c" by auto
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1310
  with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1311
    by auto
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1312
  then show "eventually (\<lambda>x. P (f x)) F"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1313
  proof eventually_elim
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1314
    case (elim x)
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1315
    with P show "P (f x)" by auto
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1316
  qed
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1317
qed force
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1318
67371
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1319
lemma filterlim_at_infinity_imp_norm_at_top:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1320
  fixes F
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1321
  assumes "filterlim f at_infinity F"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1322
  shows   "filterlim (\<lambda>x. norm (f x)) at_top F"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1323
proof -
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1324
  {
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1325
    fix r :: real
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1326
    have "\<forall>\<^sub>F x in F. r \<le> norm (f x)" using filterlim_at_infinity[of 0 f F] assms
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1327
      by (cases "r > 0")
67371
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1328
         (auto simp: not_less intro: always_eventually order.trans[OF _ norm_ge_zero])
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1329
  }
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1330
  thus ?thesis by (auto simp: filterlim_at_top)
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1331
qed
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1332
67371
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1333
lemma filterlim_norm_at_top_imp_at_infinity:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1334
  fixes F
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1335
  assumes "filterlim (\<lambda>x. norm (f x)) at_top F"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1336
  shows   "filterlim f at_infinity F"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1337
  using filterlim_at_infinity[of 0 f F] assms by (auto simp: filterlim_at_top)
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1338
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1339
lemma filterlim_norm_at_top: "filterlim norm at_top at_infinity"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1340
  by (rule filterlim_at_infinity_imp_norm_at_top) (rule filterlim_ident)
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1341
67950
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1342
lemma filterlim_at_infinity_conv_norm_at_top:
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1343
  "filterlim f at_infinity G \<longleftrightarrow> filterlim (\<lambda>x. norm (f x)) at_top G"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1344
  by (auto simp: filterlim_at_infinity[OF order.refl] filterlim_at_top_gt[of _ _ 0])
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1345
67371
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1346
lemma eventually_not_equal_at_infinity:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1347
  "eventually (\<lambda>x. x \<noteq> (a :: 'a :: {real_normed_vector})) at_infinity"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1348
proof -
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1349
  from filterlim_norm_at_top[where 'a = 'a]
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1350
    have "\<forall>\<^sub>F x in at_infinity. norm a < norm (x::'a)" by (auto simp: filterlim_at_top_dense)
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1351
  thus ?thesis by eventually_elim auto
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1352
qed
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1353
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1354
lemma filterlim_int_of_nat_at_topD:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1355
  fixes F
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1356
  assumes "filterlim (\<lambda>x. f (int x)) F at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1357
  shows   "filterlim f F at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1358
proof -
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1359
  have "filterlim (\<lambda>x. f (int (nat x))) F at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1360
    by (rule filterlim_compose[OF assms filterlim_nat_sequentially])
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1361
  also have "?this \<longleftrightarrow> filterlim f F at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1362
    by (intro filterlim_cong refl eventually_mono [OF eventually_ge_at_top[of "0::int"]]) auto
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1363
  finally show ?thesis .
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1364
qed
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1365
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1366
lemma filterlim_int_sequentially [tendsto_intros]:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1367
  "filterlim int at_top sequentially"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1368
  unfolding filterlim_at_top
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1369
proof
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1370
  fix C :: int
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1371
  show "eventually (\<lambda>n. int n \<ge> C) at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1372
    using eventually_ge_at_top[of "nat \<lceil>C\<rceil>"] by eventually_elim linarith
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1373
qed
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1374
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1375
lemma filterlim_real_of_int_at_top [tendsto_intros]:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1376
  "filterlim real_of_int at_top at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1377
  unfolding filterlim_at_top
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1378
proof
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1379
  fix C :: real
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1380
  show "eventually (\<lambda>n. real_of_int n \<ge> C) at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1381
    using eventually_ge_at_top[of "\<lceil>C\<rceil>"] by eventually_elim linarith
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1382
qed
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1383
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1384
lemma filterlim_abs_real: "filterlim (abs::real \<Rightarrow> real) at_top at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1385
proof (subst filterlim_cong[OF refl refl])
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1386
  from eventually_ge_at_top[of "0::real"] show "eventually (\<lambda>x::real. \<bar>x\<bar> = x) at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1387
    by eventually_elim simp
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1388
qed (simp_all add: filterlim_ident)
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1389
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1390
lemma filterlim_of_real_at_infinity [tendsto_intros]:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1391
  "filterlim (of_real :: real \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity at_top"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1392
  by (intro filterlim_norm_at_top_imp_at_infinity) (auto simp: filterlim_abs_real)
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1393
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1394
lemma not_tendsto_and_filterlim_at_infinity:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1395
  fixes c :: "'a::real_normed_vector"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1396
  assumes "F \<noteq> bot"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1397
    and "(f \<longlongrightarrow> c) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1398
    and "filterlim f at_infinity F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1399
  shows False
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1400
proof -
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  1401
  from tendstoD[OF assms(2), of "1/2"]
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1402
  have "eventually (\<lambda>x. dist (f x) c < 1/2) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1403
    by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1404
  moreover
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1405
  from filterlim_at_infinity[of "norm c" f F] assms(3)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1406
  have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1407
  ultimately have "eventually (\<lambda>x. False) F"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1408
  proof eventually_elim
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1409
    fix x
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1410
    assume A: "dist (f x) c < 1/2"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1411
    assume "norm (f x) \<ge> norm c + 1"
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62369
diff changeset
  1412
    also have "norm (f x) = dist (f x) 0" by simp
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1413
    also have "\<dots> \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62369
diff changeset
  1414
    finally show False using A by simp
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1415
  qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1416
  with assms show False by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1417
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1418
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1419
lemma filterlim_at_infinity_imp_not_convergent:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1420
  assumes "filterlim f at_infinity sequentially"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1421
  shows "\<not> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1422
  by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1423
     (simp_all add: convergent_LIMSEQ_iff)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1424
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1425
lemma filterlim_at_infinity_imp_eventually_ne:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1426
  assumes "filterlim f at_infinity F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1427
  shows "eventually (\<lambda>z. f z \<noteq> c) F"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1428
proof -
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1429
  have "norm c + 1 > 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1430
    by (intro add_nonneg_pos) simp_all
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1431
  with filterlim_at_infinity[OF order.refl, of f F] assms
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1432
  have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1433
    by blast
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1434
  then show ?thesis
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1435
    by eventually_elim auto
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1436
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1437
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  1438
lemma tendsto_of_nat [tendsto_intros]:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1439
  "filterlim (of_nat :: nat \<Rightarrow> 'a::real_normed_algebra_1) at_infinity sequentially"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1440
proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62393
diff changeset
  1441
  fix r :: real
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62393
diff changeset
  1442
  assume r: "r > 0"
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62393
diff changeset
  1443
  define n where "n = nat \<lceil>r\<rceil>"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1444
  from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1445
    unfolding n_def by linarith
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1446
  from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1447
    by eventually_elim (use n in simp_all)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1448
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1449
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1450
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  1451
subsection \<open>Relate \<^const>\<open>at\<close>, \<^const>\<open>at_left\<close> and \<^const>\<open>at_right\<close>\<close>
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1452
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1453
text \<open>
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  1454
  This lemmas are useful for conversion between \<^term>\<open>at x\<close> to \<^term>\<open>at_left x\<close> and
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  1455
  \<^term>\<open>at_right x\<close> and also \<^term>\<open>at_right 0\<close>.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1456
\<close>
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1457
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents: 51360
diff changeset
  1458
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1459
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1460
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1461
  for a d :: "'a::real_normed_vector"
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1462
  by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1463
    (auto intro!: tendsto_eq_intros filterlim_ident)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1464
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1465
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1466
  for a :: "'a::real_normed_vector"
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1467
  by (rule filtermap_fun_inverse[where g=uminus])
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1468
    (auto intro!: tendsto_eq_intros filterlim_ident)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1469
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1470
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1471
  for a d :: "'a::real_normed_vector"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1472
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1473
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1474
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1475
  for a d :: "real"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1476
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1477
73795
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1478
lemma filterlim_shift:
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1479
  fixes d :: "'a::real_normed_vector"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1480
  assumes "filterlim f F (at a)"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1481
  shows "filterlim (f \<circ> (+) d) F (at (a - d))"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1482
  unfolding filterlim_iff
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1483
proof (intro strip)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1484
  fix P
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1485
  assume "eventually P F"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1486
  then have "\<forall>\<^sub>F x in filtermap (\<lambda>y. y - d) (at a). P (f (d + x))"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1487
    using assms by (force simp add: filterlim_iff eventually_filtermap)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1488
  then show "(\<forall>\<^sub>F x in at (a - d). P ((f \<circ> (+) d) x))"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1489
    by (force simp add: filtermap_at_shift)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1490
qed
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1491
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1492
lemma filterlim_shift_iff:
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1493
  fixes d :: "'a::real_normed_vector"
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1494
  shows "filterlim (f \<circ> (+) d) F (at (a - d)) = filterlim f F (at a)"   (is "?lhs = ?rhs")
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1495
proof
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1496
  assume L: ?lhs show ?rhs
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1497
    using filterlim_shift [OF L, of "-d"] by (simp add: filterlim_iff)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1498
qed (metis filterlim_shift)
8893e0ed263a new lemmas mostly about paths
paulson <lp15@cam.ac.uk>
parents: 72245
diff changeset
  1499
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1500
lemma at_right_to_0: "at_right a = filtermap (\<lambda>x. x + a) (at_right 0)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1501
  for a :: real
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1502
  using filtermap_at_right_shift[of "-a" 0] by simp
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1503
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1504
lemma filterlim_at_right_to_0:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1505
  "filterlim f F (at_right a) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1506
  for a :: real
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1507
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1508
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1509
lemma eventually_at_right_to_0:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1510
  "eventually P (at_right a) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1511
  for a :: real
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1512
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1513
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1514
lemma at_to_0: "at a = filtermap (\<lambda>x. x + a) (at 0)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1515
  for a :: "'a::real_normed_vector"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1516
  using filtermap_at_shift[of "-a" 0] by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1517
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1518
lemma filterlim_at_to_0:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1519
  "filterlim f F (at a) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at 0)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1520
  for a :: "'a::real_normed_vector"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1521
  unfolding filterlim_def filtermap_filtermap at_to_0[of a] ..
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1522
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1523
lemma eventually_at_to_0:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1524
  "eventually P (at a) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at 0)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1525
  for a ::  "'a::real_normed_vector"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1526
  unfolding at_to_0[of a] by (simp add: eventually_filtermap)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1527
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1528
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1529
  for a :: "'a::real_normed_vector"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1530
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1531
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1532
lemma at_left_minus: "at_left a = filtermap (\<lambda>x. - x) (at_right (- a))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1533
  for a :: real
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1534
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1535
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1536
lemma at_right_minus: "at_right a = filtermap (\<lambda>x. - x) (at_left (- a))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1537
  for a :: real
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1538
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1539
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1540
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1541
lemma filterlim_at_left_to_right:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1542
  "filterlim f F (at_left a) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1543
  for a :: real
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1544
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1545
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1546
lemma eventually_at_left_to_right:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1547
  "eventually P (at_left a) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1548
  for a :: real
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1549
  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1550
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1551
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1552
  unfolding filterlim_at_top eventually_at_bot_dense
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1553
  by (metis leI minus_less_iff order_less_asym)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1554
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1555
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1556
  unfolding filterlim_at_bot eventually_at_top_dense
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1557
  by (metis leI less_minus_iff order_less_asym)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1558
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1559
lemma at_bot_mirror :
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1560
  shows "(at_bot::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_top"
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  1561
proof (rule filtermap_fun_inverse[symmetric])
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  1562
  show "filterlim uminus at_top (at_bot::'a filter)"
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  1563
    using eventually_at_bot_linorder filterlim_at_top le_minus_iff by force
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  1564
  show "filterlim uminus (at_bot::'a filter) at_top"
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  1565
    by (simp add: filterlim_at_bot minus_le_iff)
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  1566
qed auto
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68296
diff changeset
  1567
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1568
lemma at_top_mirror :
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1569
  shows "(at_top::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_bot"
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68296
diff changeset
  1570
  apply (subst at_bot_mirror)
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  1571
  by (auto simp: filtermap_filtermap)
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1572
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1573
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1574
  unfolding filterlim_def at_top_mirror filtermap_filtermap ..
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1575
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1576
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1577
  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1578
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1579
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1580
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1581
    and filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1582
  by auto
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1583
68721
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1584
lemma tendsto_at_botI_sequentially:
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1585
  fixes f :: "real \<Rightarrow> 'b::first_countable_topology"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1586
  assumes *: "\<And>X. filterlim X at_bot sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1587
  shows "(f \<longlongrightarrow> y) at_bot"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1588
  unfolding filterlim_at_bot_mirror
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1589
proof (rule tendsto_at_topI_sequentially)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1590
  fix X :: "nat \<Rightarrow> real" assume "filterlim X at_top sequentially"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1591
  thus "(\<lambda>n. f (-X n)) \<longlonglongrightarrow> y" by (intro *) (auto simp: filterlim_uminus_at_top)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1592
qed
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68615
diff changeset
  1593
67950
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1594
lemma filterlim_at_infinity_imp_filterlim_at_top:
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1595
  assumes "filterlim (f :: 'a \<Rightarrow> real) at_infinity F"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1596
  assumes "eventually (\<lambda>x. f x > 0) F"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1597
  shows   "filterlim f at_top F"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1598
proof -
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1599
  from assms(2) have *: "eventually (\<lambda>x. norm (f x) = f x) F" by eventually_elim simp
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1600
  from assms(1) show ?thesis unfolding filterlim_at_infinity_conv_norm_at_top
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1601
    by (subst (asm) filterlim_cong[OF refl refl *])
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1602
qed
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1603
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1604
lemma filterlim_at_infinity_imp_filterlim_at_bot:
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1605
  assumes "filterlim (f :: 'a \<Rightarrow> real) at_infinity F"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1606
  assumes "eventually (\<lambda>x. f x < 0) F"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1607
  shows   "filterlim f at_bot F"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1608
proof -
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1609
  from assms(2) have *: "eventually (\<lambda>x. norm (f x) = -f x) F" by eventually_elim simp
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1610
  from assms(1) have "filterlim (\<lambda>x. - f x) at_top F"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1611
    unfolding filterlim_at_infinity_conv_norm_at_top
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1612
    by (subst (asm) filterlim_cong[OF refl refl *])
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1613
  thus ?thesis by (simp add: filterlim_uminus_at_top)
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1614
qed
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67707
diff changeset
  1615
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1616
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1617
  unfolding filterlim_uminus_at_top by simp
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1618
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1619
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1620
  unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1621
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1622
  fix Z :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1623
  assume [arith]: "0 < Z"
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1624
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  1625
    by (auto simp: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1626
  then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1627
    by (auto elim!: eventually_mono simp: inverse_eq_divide field_simps)
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1628
qed
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1629
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1630
lemma tendsto_inverse_0:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60974
diff changeset
  1631
  fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1632
  shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1633
  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1634
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1635
  fix r :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1636
  assume "0 < r"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1637
  show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1638
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1639
    fix x :: 'a
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1640
    from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1641
    also assume *: "inverse (r / 2) \<le> norm x"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1642
    finally show "norm (inverse x) < r"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1643
      using * \<open>0 < r\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1644
      by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1645
  qed
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1646
qed
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1647
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1648
lemma tendsto_add_filterlim_at_infinity:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1649
  fixes c :: "'b::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1650
    and F :: "'a filter"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1651
  assumes "(f \<longlongrightarrow> c) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1652
    and "filterlim g at_infinity F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1653
  shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1654
proof (subst filterlim_at_infinity[OF order_refl], safe)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1655
  fix r :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1656
  assume r: "r > 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1657
  from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1658
    by (rule tendsto_norm)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1659
  then have "eventually (\<lambda>x. norm (f x) < norm c + 1) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1660
    by (rule order_tendstoD) simp_all
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1661
  moreover from r have "r + norm c + 1 > 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1662
    by (intro add_pos_nonneg) simp_all
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1663
  with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1664
    unfolding filterlim_at_infinity[OF order_refl]
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1665
    by (elim allE[of _ "r + norm c + 1"]) simp_all
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1666
  ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1667
  proof eventually_elim
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1668
    fix x :: 'a
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1669
    assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1670
    from A B have "r \<le> norm (g x) - norm (f x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1671
      by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1672
    also have "norm (g x) - norm (f x) \<le> norm (g x + f x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1673
      by (rule norm_diff_ineq)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1674
    finally show "r \<le> norm (f x + g x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1675
      by (simp add: add_ac)
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1676
  qed
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1677
qed
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1678
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1679
lemma tendsto_add_filterlim_at_infinity':
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1680
  fixes c :: "'b::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1681
    and F :: "'a filter"
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1682
  assumes "filterlim f at_infinity F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1683
    and "(g \<longlongrightarrow> c) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1684
  shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1685
  by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
  1686
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1687
lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1688
  unfolding filterlim_at
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1689
  by (auto simp: eventually_at_top_dense)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1690
     (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1691
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1692
lemma filterlim_inverse_at_top:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1693
  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1694
  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1695
     (simp add: filterlim_def eventually_filtermap eventually_mono at_within_def le_principal)
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1696
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1697
lemma filterlim_inverse_at_bot_neg:
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1698
  "LIM x (at_left (0::real)). inverse x :> at_bot"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1699
  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1700
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1701
lemma filterlim_inverse_at_bot:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1702
  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1703
  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1704
  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1705
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1706
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1707
  by (intro filtermap_fun_inverse[symmetric, where g=inverse])
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60182
diff changeset
  1708
     (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1709
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1710
lemma eventually_at_right_to_top:
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1711
  "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1712
  unfolding at_right_to_top eventually_filtermap ..
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1713
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1714
lemma filterlim_at_right_to_top:
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1715
  "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1716
  unfolding filterlim_def at_right_to_top filtermap_filtermap ..
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1717
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1718
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1719
  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1720
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1721
lemma eventually_at_top_to_right:
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1722
  "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1723
  unfolding at_top_to_right eventually_filtermap ..
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1724
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1725
lemma filterlim_at_top_to_right:
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1726
  "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1727
  unfolding filterlim_def at_top_to_right filtermap_filtermap ..
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1728
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1729
lemma filterlim_inverse_at_infinity:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60974
diff changeset
  1730
  fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1731
  shows "filterlim inverse at_infinity (at (0::'a))"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1732
  unfolding filterlim_at_infinity[OF order_refl]
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1733
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1734
  fix r :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1735
  assume "0 < r"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1736
  then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1737
    unfolding eventually_at norm_inverse
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1738
    by (intro exI[of _ "inverse r"])
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1739
       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1740
qed
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1741
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1742
lemma filterlim_inverse_at_iff:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60974
diff changeset
  1743
  fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1744
  shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1745
  unfolding filterlim_def filtermap_filtermap[symmetric]
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1746
proof
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1747
  assume "filtermap g F \<le> at_infinity"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1748
  then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1749
    by (rule filtermap_mono)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1750
  also have "\<dots> \<le> at 0"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1751
    using tendsto_inverse_0[where 'a='b]
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1752
    by (auto intro!: exI[of _ 1]
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1753
        simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1754
  finally show "filtermap inverse (filtermap g F) \<le> at 0" .
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1755
next
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1756
  assume "filtermap inverse (filtermap g F) \<le> at 0"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1757
  then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1758
    by (rule filtermap_mono)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1759
  with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1760
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1761
qed
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
  1762
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1763
lemma tendsto_mult_filterlim_at_infinity:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1764
  fixes c :: "'a::real_normed_field"
64394
141e1ed8d5a0 more new material
paulson <lp15@cam.ac.uk>
parents: 64287
diff changeset
  1765
  assumes  "(f \<longlongrightarrow> c) F" "c \<noteq> 0"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1766
  assumes "filterlim g at_infinity F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1767
  shows "filterlim (\<lambda>x. f x * g x) at_infinity F"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1768
proof -
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1769
  have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1770
    by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1771
  then have "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1772
    unfolding filterlim_at
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1773
    using assms
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1774
    by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1775
  then show ?thesis
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1776
    by (subst filterlim_inverse_at_iff[symmetric]) simp_all
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1777
qed
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  1778
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1779
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
73885
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1780
  by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1781
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1782
lemma filterlim_inverse_at_top_iff:
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1783
  "eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> (LIM x F. inverse (f x) :> at_top) \<longleftrightarrow> (f \<longlongrightarrow> (0 :: real)) F"
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1784
  by (auto dest: tendsto_inverse_0_at_top filterlim_inverse_at_top)
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1785
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1786
lemma filterlim_at_top_iff_inverse_0:
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1787
  "eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> (LIM x F. f x :> at_top) \<longleftrightarrow> ((inverse \<circ> f) \<longlongrightarrow> (0 :: real)) F"
26171a89466a A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents: 73795
diff changeset
  1788
  using filterlim_inverse_at_top_iff [of "inverse \<circ> f"] by auto
50419
3177d0374701 add exponential and uniform distributions
hoelzl
parents: 50347
diff changeset
  1789
63556
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1790
lemma real_tendsto_divide_at_top:
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1791
  fixes c::"real"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1792
  assumes "(f \<longlongrightarrow> c) F"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1793
  assumes "filterlim g at_top F"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1794
  shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1795
  by (auto simp: divide_inverse_commute
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1796
      intro!: tendsto_mult[THEN tendsto_eq_rhs] tendsto_inverse_0_at_top assms)
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1797
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1798
lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x) at_top sequentially"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1799
  for c :: nat
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  1800
  by (rule filterlim_subseq) (auto simp: strict_mono_def)
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1801
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1802
lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c) at_top sequentially"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1803
  for c :: nat
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 65680
diff changeset
  1804
  by (rule filterlim_subseq) (auto simp: strict_mono_def)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1805
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1806
lemma filterlim_times_pos:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1807
  "LIM x F1. c * f x :> at_right l"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1808
  if "filterlim f (at_right p) F1" "0 < c" "l = c * p"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1809
  for c::"'a::{linordered_field, linorder_topology}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1810
  unfolding filterlim_iff
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1811
proof safe
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1812
  fix P
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1813
  assume "\<forall>\<^sub>F x in at_right l. P x"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1814
  then obtain d where "c * p < d" "\<And>y. y > c * p \<Longrightarrow> y < d \<Longrightarrow> P y"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1815
    unfolding \<open>l = _ \<close> eventually_at_right_field
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1816
    by auto
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1817
  then have "\<forall>\<^sub>F a in at_right p. P (c * a)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1818
    by (auto simp: eventually_at_right_field \<open>0 < c\<close> field_simps intro!: exI[where x="d/c"])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1819
  from that(1)[unfolded filterlim_iff, rule_format, OF this]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1820
  show "\<forall>\<^sub>F x in F1. P (c * f x)" .
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1821
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1822
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1823
lemma filtermap_nhds_times: "c \<noteq> 0 \<Longrightarrow> filtermap (times c) (nhds a) = nhds (c * a)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1824
  for a c :: "'a::real_normed_field"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1825
  by (rule filtermap_fun_inverse[where g="\<lambda>x. inverse c * x"])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1826
    (auto intro!: tendsto_eq_intros filterlim_ident)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1827
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1828
lemma filtermap_times_pos_at_right:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1829
  fixes c::"'a::{linordered_field, linorder_topology}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1830
  assumes "c > 0"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1831
  shows "filtermap (times c) (at_right p) = at_right (c * p)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1832
  using assms
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1833
  by (intro filtermap_fun_inverse[where g="\<lambda>x. inverse c * x"])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1834
    (auto intro!: filterlim_ident filterlim_times_pos)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  1835
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1836
lemma at_to_infinity: "(at (0::'a::{real_normed_field,field})) = filtermap inverse at_infinity"
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1837
proof (rule antisym)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1838
  have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1839
    by (fact tendsto_inverse_0)
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1840
  then show "filtermap inverse at_infinity \<le> at (0::'a)"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  1841
    using filterlim_def filterlim_ident filterlim_inverse_at_iff by fastforce
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1842
next
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1843
  have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1844
    using filterlim_inverse_at_infinity unfolding filterlim_def
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1845
    by (rule filtermap_mono)
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1846
  then show "at (0::'a) \<le> filtermap inverse at_infinity"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1847
    by (simp add: filtermap_ident filtermap_filtermap)
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1848
qed
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1849
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1850
lemma lim_at_infinity_0:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1851
  fixes l :: "'a::{real_normed_field,field}"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1852
  shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f \<circ> inverse) \<longlongrightarrow> l) (at (0::'a))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1853
  by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1854
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1855
lemma lim_zero_infinity:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1856
  fixes l :: "'a::{real_normed_field,field}"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1857
  shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1858
  by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1859
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
  1860
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1861
text \<open>
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1862
  We only show rules for multiplication and addition when the functions are either against a real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1863
  value or against infinity. Further rules are easy to derive by using @{thm
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1864
  filterlim_uminus_at_top}.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1865
\<close>
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1866
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1867
lemma filterlim_tendsto_pos_mult_at_top:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1868
  assumes f: "(f \<longlongrightarrow> c) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1869
    and c: "0 < c"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1870
    and g: "LIM x F. g x :> at_top"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1871
  shows "LIM x F. (f x * g x :: real) :> at_top"
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1872
  unfolding filterlim_at_top_gt[where c=0]
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1873
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1874
  fix Z :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1875
  assume "0 < Z"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1876
  from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1877
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1878
        simp: dist_real_def abs_real_def split: if_split_asm)
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1879
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1880
    unfolding filterlim_at_top by auto
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1881
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1882
  proof eventually_elim
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1883
    case (elim x)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1884
    with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1885
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1886
    with \<open>0 < c\<close> show "Z \<le> f x * g x"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1887
       by simp
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1888
  qed
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1889
qed
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1890
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1891
lemma filterlim_at_top_mult_at_top:
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1892
  assumes f: "LIM x F. f x :> at_top"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1893
    and g: "LIM x F. g x :> at_top"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1894
  shows "LIM x F. (f x * g x :: real) :> at_top"
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1895
  unfolding filterlim_at_top_gt[where c=0]
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1896
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1897
  fix Z :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1898
  assume "0 < Z"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1899
  from f have "eventually (\<lambda>x. 1 \<le> f x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1900
    unfolding filterlim_at_top by auto
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1901
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1902
    unfolding filterlim_at_top by auto
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1903
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1904
  proof eventually_elim
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1905
    case (elim x)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  1906
    with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1907
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1908
    then show "Z \<le> f x * g x"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1909
       by simp
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1910
  qed
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1911
qed
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1912
63556
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1913
lemma filterlim_at_top_mult_tendsto_pos:
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1914
  assumes f: "(f \<longlongrightarrow> c) F"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1915
    and c: "0 < c"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1916
    and g: "LIM x F. g x :> at_top"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1917
  shows "LIM x F. (g x * f x:: real) :> at_top"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1918
  by (auto simp: mult.commute intro!: filterlim_tendsto_pos_mult_at_top f c g)
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  1919
50419
3177d0374701 add exponential and uniform distributions
hoelzl
parents: 50347
diff changeset
  1920
lemma filterlim_tendsto_pos_mult_at_bot:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1921
  fixes c :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1922
  assumes "(f \<longlongrightarrow> c) F" "0 < c" "filterlim g at_bot F"
50419
3177d0374701 add exponential and uniform distributions
hoelzl
parents: 50347
diff changeset
  1923
  shows "LIM x F. f x * g x :> at_bot"
3177d0374701 add exponential and uniform distributions
hoelzl
parents: 50347
diff changeset
  1924
  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
3177d0374701 add exponential and uniform distributions
hoelzl
parents: 50347
diff changeset
  1925
  unfolding filterlim_uminus_at_bot by simp
3177d0374701 add exponential and uniform distributions
hoelzl
parents: 50347
diff changeset
  1926
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60141
diff changeset
  1927
lemma filterlim_tendsto_neg_mult_at_bot:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1928
  fixes c :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1929
  assumes c: "(f \<longlongrightarrow> c) F" "c < 0" and g: "filterlim g at_top F"
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60141
diff changeset
  1930
  shows "LIM x F. f x * g x :> at_bot"
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60141
diff changeset
  1931
  using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60141
diff changeset
  1932
  unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60141
diff changeset
  1933
56330
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1934
lemma filterlim_pow_at_top:
63721
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63556
diff changeset
  1935
  fixes f :: "'a \<Rightarrow> real"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1936
  assumes "0 < n"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1937
    and f: "LIM x F. f x :> at_top"
56330
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1938
  shows "LIM x F. (f x)^n :: real :> at_top"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1939
  using \<open>0 < n\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1940
proof (induct n)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1941
  case 0
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1942
  then show ?case by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1943
next
56330
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1944
  case (Suc n) with f show ?case
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1945
    by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1946
qed
56330
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1947
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1948
lemma filterlim_pow_at_bot_even:
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1949
  fixes f :: "real \<Rightarrow> real"
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1950
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1951
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1952
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1953
lemma filterlim_pow_at_bot_odd:
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1954
  fixes f :: "real \<Rightarrow> real"
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1955
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1956
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
5c4d3be7a6b0 add limits of power at top and bot
hoelzl
parents: 55415
diff changeset
  1957
67371
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1958
lemma filterlim_power_at_infinity [tendsto_intros]:
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1959
  fixes F and f :: "'a \<Rightarrow> 'b :: real_normed_div_algebra"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1960
  assumes "filterlim f at_infinity F" "n > 0"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1961
  shows   "filterlim (\<lambda>x. f x ^ n) at_infinity F"
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1962
  by (rule filterlim_norm_at_top_imp_at_infinity)
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  1963
     (auto simp: norm_power intro!: filterlim_pow_at_top assms
67371
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1964
           intro: filterlim_at_infinity_imp_norm_at_top)
2d9cf74943e1 moved in some material from Euler-MacLaurin
paulson <lp15@cam.ac.uk>
parents: 67091
diff changeset
  1965
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1966
lemma filterlim_tendsto_add_at_top:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1967
  assumes f: "(f \<longlongrightarrow> c) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1968
    and g: "LIM x F. g x :> at_top"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1969
  shows "LIM x F. (f x + g x :: real) :> at_top"
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1970
  unfolding filterlim_at_top_gt[where c=0]
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1971
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1972
  fix Z :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1973
  assume "0 < Z"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1974
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1975
    by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1976
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1977
    unfolding filterlim_at_top by auto
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1978
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1979
    by eventually_elim simp
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1980
qed
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1981
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1982
lemma LIM_at_top_divide:
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1983
  fixes f g :: "'a \<Rightarrow> real"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  1984
  assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1985
    and g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1986
  shows "LIM x F. f x / g x :> at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1987
  unfolding divide_inverse
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1988
  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1989
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1990
lemma filterlim_at_top_add_at_top:
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1991
  assumes f: "LIM x F. f x :> at_top"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1992
    and g: "LIM x F. g x :> at_top"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1993
  shows "LIM x F. (f x + g x :: real) :> at_top"
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1994
  unfolding filterlim_at_top_gt[where c=0]
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1995
proof safe
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1996
  fix Z :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  1997
  assume "0 < Z"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1998
  from f have "eventually (\<lambda>x. 0 \<le> f x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  1999
    unfolding filterlim_at_top by auto
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  2000
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  2001
    unfolding filterlim_at_top by auto
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  2002
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  2003
    by eventually_elim simp
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  2004
qed
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
  2005
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2006
lemma tendsto_divide_0:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60974
diff changeset
  2007
  fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2008
  assumes f: "(f \<longlongrightarrow> c) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2009
    and g: "LIM x F. g x :> at_infinity"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2010
  shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2011
  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]]
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2012
  by (simp add: divide_inverse)
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2013
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2014
lemma linear_plus_1_le_power:
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2015
  fixes x :: real
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2016
  assumes x: "0 \<le> x"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2017
  shows "real n * x + 1 \<le> (x + 1) ^ n"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2018
proof (induct n)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2019
  case 0
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2020
  then show ?case by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2021
next
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2022
  case (Suc n)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2023
  from x have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2024
    by (simp add: field_simps)
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2025
  also have "\<dots> \<le> (x + 1)^Suc n"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2026
    using Suc x by (simp add: mult_left_mono)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2027
  finally show ?case .
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2028
qed
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2029
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2030
lemma filterlim_realpow_sequentially_gt1:
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2031
  fixes x :: "'a :: real_normed_div_algebra"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2032
  assumes x[arith]: "1 < norm x"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2033
  shows "LIM n sequentially. x ^ n :> at_infinity"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2034
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2035
  fix y :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2036
  assume "0 < y"
72219
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  2037
  obtain N :: nat where "y < real N * (norm x - 1)"
0f38c96a0a74 tidying up some theorem statements
paulson <lp15@cam.ac.uk>
parents: 71837
diff changeset
  2038
    by (meson diff_gt_0_iff_gt reals_Archimedean3 x)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2039
  also have "\<dots> \<le> real N * (norm x - 1) + 1"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2040
    by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2041
  also have "\<dots> \<le> (norm x - 1 + 1) ^ N"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2042
    by (rule linear_plus_1_le_power) simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2043
  also have "\<dots> = norm x ^ N"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2044
    by simp
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2045
  finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2046
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2047
  then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2048
    unfolding eventually_sequentially
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2049
    by (auto simp: norm_power)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2050
qed simp
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  2051
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents: 51360
diff changeset
  2052
66456
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2053
lemma filterlim_divide_at_infinity:
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2054
  fixes f g :: "'a \<Rightarrow> 'a :: real_normed_field"
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2055
  assumes "filterlim f (nhds c) F" "filterlim g (at 0) F" "c \<noteq> 0"
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2056
  shows   "filterlim (\<lambda>x. f x / g x) at_infinity F"
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2057
proof -
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2058
  have "filterlim (\<lambda>x. f x * inverse (g x)) at_infinity F"
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2059
    by (intro tendsto_mult_filterlim_at_infinity[OF assms(1,3)]
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2060
          filterlim_compose [OF filterlim_inverse_at_infinity assms(2)])
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2061
  thus ?thesis by (simp add: field_simps)
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2062
qed
621897f47fab Various lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents: 66447
diff changeset
  2063
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2064
subsection \<open>Floor and Ceiling\<close>
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2065
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2066
lemma eventually_floor_less:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2067
  fixes f :: "'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2068
  assumes f: "(f \<longlongrightarrow> l) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2069
    and l: "l \<notin> \<int>"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2070
  shows "\<forall>\<^sub>F x in F. of_int (floor l) < f x"
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2071
  by (intro order_tendstoD[OF f]) (metis Ints_of_int antisym_conv2 floor_correct l)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2072
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2073
lemma eventually_less_ceiling:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2074
  fixes f :: "'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2075
  assumes f: "(f \<longlongrightarrow> l) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2076
    and l: "l \<notin> \<int>"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2077
  shows "\<forall>\<^sub>F x in F. f x < of_int (ceiling l)"
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2078
  by (intro order_tendstoD[OF f]) (metis Ints_of_int l le_of_int_ceiling less_le)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2079
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2080
lemma eventually_floor_eq:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2081
  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2082
  assumes f: "(f \<longlongrightarrow> l) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2083
    and l: "l \<notin> \<int>"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2084
  shows "\<forall>\<^sub>F x in F. floor (f x) = floor l"
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2085
  using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2086
  by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2087
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2088
lemma eventually_ceiling_eq:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2089
  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2090
  assumes f: "(f \<longlongrightarrow> l) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2091
    and l: "l \<notin> \<int>"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2092
  shows "\<forall>\<^sub>F x in F. ceiling (f x) = ceiling l"
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2093
  using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2094
  by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2095
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2096
lemma tendsto_of_int_floor:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2097
  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2098
  assumes "(f \<longlongrightarrow> l) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2099
    and "l \<notin> \<int>"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2100
  shows "((\<lambda>x. of_int (floor (f x)) :: 'c::{ring_1,topological_space}) \<longlongrightarrow> of_int (floor l)) F"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2101
  using eventually_floor_eq[OF assms]
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2102
  by (simp add: eventually_mono topological_tendstoI)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2103
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2104
lemma tendsto_of_int_ceiling:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2105
  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2106
  assumes "(f \<longlongrightarrow> l) F"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2107
    and "l \<notin> \<int>"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2108
  shows "((\<lambda>x. of_int (ceiling (f x)):: 'c::{ring_1,topological_space}) \<longlongrightarrow> of_int (ceiling l)) F"
63263
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2109
  using eventually_ceiling_eq[OF assms]
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2110
  by (simp add: eventually_mono topological_tendstoI)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2111
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2112
lemma continuous_on_of_int_floor:
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2113
  "continuous_on (UNIV - \<int>::'a::{order_topology, floor_ceiling} set)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2114
    (\<lambda>x. of_int (floor x)::'b::{ring_1, topological_space})"
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2115
  unfolding continuous_on_def
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2116
  by (auto intro!: tendsto_of_int_floor)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2117
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2118
lemma continuous_on_of_int_ceiling:
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2119
  "continuous_on (UNIV - \<int>::'a::{order_topology, floor_ceiling} set)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2120
    (\<lambda>x. of_int (ceiling x)::'b::{ring_1, topological_space})"
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2121
  unfolding continuous_on_def
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2122
  by (auto intro!: tendsto_of_int_ceiling)
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2123
c6c95d64607a approximation, derivative, and continuity of floor and ceiling
immler
parents: 63104
diff changeset
  2124
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2125
subsection \<open>Limits of Sequences\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2126
62368
106569399cd6 add type class for topological monoids
hoelzl
parents: 62101
diff changeset
  2127
lemma [trans]: "X = Y \<Longrightarrow> Y \<longlonglongrightarrow> z \<Longrightarrow> X \<longlonglongrightarrow> z"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2128
  by simp
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2129
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2130
lemma LIMSEQ_iff:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2131
  fixes L :: "'a::real_normed_vector"
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2132
  shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
60017
b785d6d06430 Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
  2133
unfolding lim_sequentially dist_norm ..
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2134
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2135
lemma LIMSEQ_I: "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2136
  for L :: "'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2137
  by (simp add: LIMSEQ_iff)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2138
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2139
lemma LIMSEQ_D: "X \<longlonglongrightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2140
  for L :: "'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2141
  by (simp add: LIMSEQ_iff)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2142
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2143
lemma LIMSEQ_linear: "X \<longlonglongrightarrow> x \<Longrightarrow> l > 0 \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2144
  unfolding tendsto_def eventually_sequentially
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57447
diff changeset
  2145
  by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2146
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2147
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2148
text \<open>Transformation of limit.\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2149
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2150
lemma Lim_transform: "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2151
  for a b :: "'a::real_normed_vector"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2152
  using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2153
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2154
lemma Lim_transform2: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (g \<longlongrightarrow> a) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2155
  for a b :: "'a::real_normed_vector"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2156
  by (erule Lim_transform) (simp add: tendsto_minus_cancel)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2157
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2158
proposition Lim_transform_eq: "((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F \<longleftrightarrow> (g \<longlongrightarrow> a) F"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2159
  for a :: "'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2160
  using Lim_transform Lim_transform2 by blast
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62369
diff changeset
  2161
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2162
lemma Lim_transform_eventually:
70532
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  2163
  "\<lbrakk>(f \<longlongrightarrow> l) F; eventually (\<lambda>x. f x = g x) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> l) F"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2164
  using eventually_elim2 by (fastforce simp add: tendsto_def)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2165
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2166
lemma Lim_transform_within:
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2167
  assumes "(f \<longlongrightarrow> l) (at x within S)"
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2168
    and "0 < d"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2169
    and "\<And>x'. x'\<in>S \<Longrightarrow> 0 < dist x' x \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x'"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2170
  shows "(g \<longlongrightarrow> l) (at x within S)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2171
proof (rule Lim_transform_eventually)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2172
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2173
    using assms by (auto simp: eventually_at)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2174
  show "(f \<longlongrightarrow> l) (at x within S)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2175
    by fact
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2176
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2177
67706
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2178
lemma filterlim_transform_within:
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2179
  assumes "filterlim g G (at x within S)"
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2180
  assumes "G \<le> F" "0<d" "(\<And>x'. x' \<in> S \<Longrightarrow> 0 < dist x' x \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') "
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2181
  shows "filterlim f F (at x within S)"
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2182
  using assms
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2183
  apply (elim filterlim_mono_eventually)
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2184
  unfolding eventually_at by auto
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67673
diff changeset
  2185
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2186
text \<open>Common case assuming being away from some crucial point like 0.\<close>
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2187
lemma Lim_transform_away_within:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2188
  fixes a b :: "'a::t1_space"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2189
  assumes "a \<noteq> b"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2190
    and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2191
    and "(f \<longlongrightarrow> l) (at a within S)"
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2192
  shows "(g \<longlongrightarrow> l) (at a within S)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2193
proof (rule Lim_transform_eventually)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2194
  show "(f \<longlongrightarrow> l) (at a within S)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2195
    by fact
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2196
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2197
    unfolding eventually_at_topological
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2198
    by (rule exI [where x="- {b}"]) (simp add: open_Compl assms)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2199
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2200
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2201
lemma Lim_transform_away_at:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2202
  fixes a b :: "'a::t1_space"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2203
  assumes ab: "a \<noteq> b"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2204
    and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2205
    and fl: "(f \<longlongrightarrow> l) (at a)"
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2206
  shows "(g \<longlongrightarrow> l) (at a)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2207
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2208
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2209
text \<open>Alternatively, within an open set.\<close>
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2210
lemma Lim_transform_within_open:
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2211
  assumes "(f \<longlongrightarrow> l) (at a within T)"
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2212
    and "open s" and "a \<in> s"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2213
    and "\<And>x. x\<in>s \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x = g x"
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2214
  shows "(g \<longlongrightarrow> l) (at a within T)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2215
proof (rule Lim_transform_eventually)
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2216
  show "eventually (\<lambda>x. f x = g x) (at a within T)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2217
    unfolding eventually_at_topological
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2218
    using assms by auto
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2219
  show "(f \<longlongrightarrow> l) (at a within T)" by fact
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2220
qed
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2221
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2222
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2223
text \<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2224
72220
bb29e4eb938d but not the [cong] rule
paulson <lp15@cam.ac.uk>
parents: 72219
diff changeset
  2225
lemma Lim_cong_within:
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2226
  assumes "a = b"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2227
    and "x = y"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2228
    and "S = T"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2229
    and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2230
  shows "(f \<longlongrightarrow> x) (at a within S) \<longleftrightarrow> (g \<longlongrightarrow> y) (at b within T)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2231
  unfolding tendsto_def eventually_at_topological
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2232
  using assms by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2233
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2234
text \<open>An unbounded sequence's inverse tends to 0.\<close>
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65204
diff changeset
  2235
lemma LIMSEQ_inverse_zero:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65204
diff changeset
  2236
  assumes "\<And>r::real. \<exists>N. \<forall>n\<ge>N. r < X n"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65204
diff changeset
  2237
  shows "(\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2238
  apply (rule filterlim_compose[OF tendsto_inverse_0])
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2239
  by (metis assms eventually_at_top_linorderI filterlim_at_top_dense filterlim_at_top_imp_at_infinity)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2240
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  2241
text \<open>The sequence \<^term>\<open>1/n\<close> tends to 0 as \<^term>\<open>n\<close> tends to infinity.\<close>
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2242
lemma LIMSEQ_inverse_real_of_nat: "(\<lambda>n. inverse (real (Suc n))) \<longlonglongrightarrow> 0"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2243
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2244
      filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2245
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2246
text \<open>
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  2247
  The sequence \<^term>\<open>r + 1/n\<close> tends to \<^term>\<open>r\<close> as \<^term>\<open>n\<close> tends to
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2248
  infinity is now easily proved.
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2249
\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2250
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2251
lemma LIMSEQ_inverse_real_of_nat_add: "(\<lambda>n. r + inverse (real (Suc n))) \<longlonglongrightarrow> r"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2252
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2253
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2254
lemma LIMSEQ_inverse_real_of_nat_add_minus: "(\<lambda>n. r + -inverse (real (Suc n))) \<longlonglongrightarrow> r"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2255
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2256
  by auto
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2257
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2258
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult: "(\<lambda>n. r * (1 + - inverse (real (Suc n)))) \<longlonglongrightarrow> r"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2259
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2260
  by auto
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2261
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2262
lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2263
  using lim_1_over_n by (simp add: inverse_eq_divide)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2264
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  2265
lemma lim_inverse_n': "((\<lambda>n. 1 / n) \<longlongrightarrow> 0) sequentially"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  2266
  using lim_inverse_n
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  2267
  by (simp add: inverse_eq_divide)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67673
diff changeset
  2268
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2269
lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2270
proof (rule Lim_transform_eventually)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2271
  show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2272
    using eventually_gt_at_top[of "0::nat"]
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2273
    by eventually_elim (simp add: field_simps)
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2274
  have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2275
    by (intro tendsto_add tendsto_const lim_inverse_n)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2276
  then show "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2277
    by simp
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2278
qed
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2279
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2280
lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2281
proof (rule Lim_transform_eventually)
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2282
  show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) =
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2283
      of_nat n / of_nat (Suc n)) sequentially"
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2284
    using eventually_gt_at_top[of "0::nat"]
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2285
    by eventually_elim (simp add: field_simps del: of_nat_Suc)
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2286
  have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2287
    by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2288
  then show "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2289
    by simp
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2290
qed
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61169
diff changeset
  2291
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2292
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2293
subsection \<open>Convergence on sequences\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2294
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2295
lemma convergent_cong:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2296
  assumes "eventually (\<lambda>x. f x = g x) sequentially"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2297
  shows "convergent f \<longleftrightarrow> convergent g"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2298
  unfolding convergent_def
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2299
  by (subst filterlim_cong[OF refl refl assms]) (rule refl)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2300
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2301
lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
71827
5e315defb038 the Uniq quantifier
paulson <lp15@cam.ac.uk>
parents: 71167
diff changeset
  2302
  by (auto simp: convergent_def filterlim_sequentially_Suc)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2303
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2304
lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2305
proof (induct m arbitrary: f)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2306
  case 0
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2307
  then show ?case by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2308
next
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2309
  case (Suc m)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2310
  have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2311
    by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2312
  also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2313
    by (rule convergent_Suc_iff)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2314
  also have "\<dots> \<longleftrightarrow> convergent f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2315
    by (rule Suc)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2316
  finally show ?case .
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2317
qed
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2318
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2319
lemma convergent_add:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2320
  fixes X Y :: "nat \<Rightarrow> 'a::topological_monoid_add"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2321
  assumes "convergent (\<lambda>n. X n)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2322
    and "convergent (\<lambda>n. Y n)"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2323
  shows "convergent (\<lambda>n. X n + Y n)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  2324
  using assms unfolding convergent_def by (blast intro: tendsto_add)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2325
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
  2326
lemma convergent_sum:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2327
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::topological_comm_monoid_add"
63915
bab633745c7f tuned proofs;
wenzelm
parents: 63721
diff changeset
  2328
  shows "(\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)) \<Longrightarrow> convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
bab633745c7f tuned proofs;
wenzelm
parents: 63721
diff changeset
  2329
  by (induct A rule: infinite_finite_induct) (simp_all add: convergent_const convergent_add)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2330
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2331
lemma (in bounded_linear) convergent:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2332
  assumes "convergent (\<lambda>n. X n)"
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2333
  shows "convergent (\<lambda>n. f (X n))"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  2334
  using assms unfolding convergent_def by (blast intro: tendsto)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2335
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2336
lemma (in bounded_bilinear) convergent:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2337
  assumes "convergent (\<lambda>n. X n)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2338
    and "convergent (\<lambda>n. Y n)"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2339
  shows "convergent (\<lambda>n. X n ** Y n)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  2340
  using assms unfolding convergent_def by (blast intro: tendsto)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2341
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2342
lemma convergent_minus_iff:
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2343
  fixes X :: "nat \<Rightarrow> 'a::topological_group_add"
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2344
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2345
  unfolding convergent_def by (force dest: tendsto_minus)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2346
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2347
lemma convergent_diff:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2348
  fixes X Y :: "nat \<Rightarrow> 'a::topological_group_add"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2349
  assumes "convergent (\<lambda>n. X n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2350
  assumes "convergent (\<lambda>n. Y n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2351
  shows "convergent (\<lambda>n. X n - Y n)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  2352
  using assms unfolding convergent_def by (blast intro: tendsto_diff)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2353
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2354
lemma convergent_norm:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2355
  assumes "convergent f"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2356
  shows "convergent (\<lambda>n. norm (f n))"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2357
proof -
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2358
  from assms have "f \<longlonglongrightarrow> lim f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2359
    by (simp add: convergent_LIMSEQ_iff)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2360
  then have "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2361
    by (rule tendsto_norm)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2362
  then show ?thesis
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2363
    by (auto simp: convergent_def)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2364
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2365
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2366
lemma convergent_of_real:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2367
  "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a::real_normed_algebra_1)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2368
  unfolding convergent_def by (blast intro!: tendsto_of_real)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2369
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2370
lemma convergent_add_const_iff:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2371
  "convergent (\<lambda>n. c + f n :: 'a::topological_ab_group_add) \<longleftrightarrow> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2372
proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2373
  assume "convergent (\<lambda>n. c + f n)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2374
  from convergent_diff[OF this convergent_const[of c]] show "convergent f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2375
    by simp
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2376
next
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2377
  assume "convergent f"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2378
  from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2379
    by simp
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2380
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2381
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2382
lemma convergent_add_const_right_iff:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2383
  "convergent (\<lambda>n. f n + c :: 'a::topological_ab_group_add) \<longleftrightarrow> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2384
  using convergent_add_const_iff[of c f] by (simp add: add_ac)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2385
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2386
lemma convergent_diff_const_right_iff:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2387
  "convergent (\<lambda>n. f n - c :: 'a::topological_ab_group_add) \<longleftrightarrow> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2388
  using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2389
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2390
lemma convergent_mult:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2391
  fixes X Y :: "nat \<Rightarrow> 'a::topological_semigroup_mult"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2392
  assumes "convergent (\<lambda>n. X n)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2393
    and "convergent (\<lambda>n. Y n)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2394
  shows "convergent (\<lambda>n. X n * Y n)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  2395
  using assms unfolding convergent_def by (blast intro: tendsto_mult)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2396
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2397
lemma convergent_mult_const_iff:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2398
  assumes "c \<noteq> 0"
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2399
  shows "convergent (\<lambda>n. c * f n :: 'a::{field,topological_semigroup_mult}) \<longleftrightarrow> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2400
proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2401
  assume "convergent (\<lambda>n. c * f n)"
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61976
diff changeset
  2402
  from assms convergent_mult[OF this convergent_const[of "inverse c"]]
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2403
    show "convergent f" by (simp add: field_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2404
next
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2405
  assume "convergent f"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2406
  from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2407
    by simp
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2408
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2409
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2410
lemma convergent_mult_const_right_iff:
68064
b249fab48c76 type class generalisations; some work on infinite products
paulson <lp15@cam.ac.uk>
parents: 67958
diff changeset
  2411
  fixes c :: "'a::{field,topological_semigroup_mult}"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2412
  assumes "c \<noteq> 0"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2413
  shows "convergent (\<lambda>n. f n * c) \<longleftrightarrow> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2414
  using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2415
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2416
lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2417
  by (simp add: Cauchy_Bseq convergent_Cauchy)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2418
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2419
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2420
text \<open>A monotone sequence converges to its least upper bound.\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2421
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2422
lemma LIMSEQ_incseq_SUP:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2423
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder,linorder_topology}"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2424
  assumes u: "bdd_above (range X)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2425
    and X: "incseq X"
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2426
  shows "X \<longlonglongrightarrow> (SUP i. X i)"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2427
  by (rule order_tendstoI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2428
    (auto simp: eventually_sequentially u less_cSUP_iff
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2429
      intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2430
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2431
lemma LIMSEQ_decseq_INF:
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2432
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2433
  assumes u: "bdd_below (range X)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2434
    and X: "decseq X"
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2435
  shows "X \<longlonglongrightarrow> (INF i. X i)"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2436
  by (rule order_tendstoI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2437
     (auto simp: eventually_sequentially u cINF_less_iff
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2438
       intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2439
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2440
text \<open>Main monotonicity theorem.\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2441
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2442
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent X"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2443
  for X :: "nat \<Rightarrow> real"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2444
  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2445
      dest: Bseq_bdd_above Bseq_bdd_below)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2446
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2447
lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent X"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2448
  for X :: "nat \<Rightarrow> real"
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  2449
  by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2450
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2451
lemma monoseq_imp_convergent_iff_Bseq: "monoseq f \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2452
  for f :: "nat \<Rightarrow> real"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2453
  using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2454
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2455
lemma Bseq_monoseq_convergent'_inc:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2456
  fixes f :: "nat \<Rightarrow> real"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2457
  shows "Bseq (\<lambda>n. f (n + M)) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2458
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2459
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2460
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2461
lemma Bseq_monoseq_convergent'_dec:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2462
  fixes f :: "nat \<Rightarrow> real"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2463
  shows "Bseq (\<lambda>n. f (n + M)) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
  2464
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2465
    (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2466
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2467
lemma Cauchy_iff: "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2468
  for X :: "nat \<Rightarrow> 'a::real_normed_vector"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2469
  unfolding Cauchy_def dist_norm ..
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2470
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2471
lemma CauchyI: "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2472
  for X :: "nat \<Rightarrow> 'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2473
  by (simp add: Cauchy_iff)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2474
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2475
lemma CauchyD: "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2476
  for X :: "nat \<Rightarrow> 'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2477
  by (simp add: Cauchy_iff)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2478
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2479
lemma incseq_convergent:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2480
  fixes X :: "nat \<Rightarrow> real"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2481
  assumes "incseq X"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2482
    and "\<forall>i. X i \<le> B"
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2483
  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2484
proof atomize_elim
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2485
  from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2486
  obtain L where "X \<longlonglongrightarrow> L"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2487
    by (auto simp: convergent_def monoseq_def incseq_def)
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2488
  with \<open>incseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. X i \<le> L)"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2489
    by (auto intro!: exI[of _ L] incseq_le)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2490
qed
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2491
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2492
lemma decseq_convergent:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2493
  fixes X :: "nat \<Rightarrow> real"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2494
  assumes "decseq X"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2495
    and "\<forall>i. B \<le> X i"
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2496
  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2497
proof atomize_elim
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2498
  from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2499
  obtain L where "X \<longlonglongrightarrow> L"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2500
    by (auto simp: convergent_def monoseq_def decseq_def)
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2501
  with \<open>decseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. L \<le> X i)"
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68296
diff changeset
  2502
    by (auto intro!: exI[of _ L] decseq_ge)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2503
qed
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2504
70694
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2505
lemma monoseq_convergent:
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2506
  fixes X :: "nat \<Rightarrow> real"
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2507
  assumes X: "monoseq X" and B: "\<And>i. \<bar>X i\<bar> \<le> B"
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2508
  obtains L where "X \<longlonglongrightarrow> L"
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2509
  using X unfolding monoseq_iff
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2510
proof
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2511
  assume "incseq X"
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2512
  show thesis
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2513
    using abs_le_D1 [OF B] incseq_convergent [OF \<open>incseq X\<close>] that by meson
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2514
next
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2515
  assume "decseq X"
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2516
  show thesis
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2517
    using decseq_convergent [OF \<open>decseq X\<close>] that
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2518
    by (metis B abs_le_iff add.inverse_inverse neg_le_iff_le)
ae37b8fbf023 New theory Equivalence_Measurable_On_Borel, with the HOL Light notion of measurable_on and its equivalence to ours
paulson <lp15@cam.ac.uk>
parents: 70688
diff changeset
  2519
qed
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2520
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2521
subsection \<open>Power Sequences\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2522
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2523
lemma Bseq_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> Bseq (\<lambda>n. x ^ n)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2524
  for x :: real
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2525
  by (metis decseq_bounded decseq_def power_decreasing zero_le_power)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2526
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2527
lemma monoseq_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> monoseq (\<lambda>n. x ^ n)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2528
  for x :: real
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2529
  using monoseq_def power_decreasing by blast
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2530
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2531
lemma convergent_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> convergent (\<lambda>n. x ^ n)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2532
  for x :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2533
  by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2534
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2535
lemma LIMSEQ_inverse_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2536
  for x :: real
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2537
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2538
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2539
lemma LIMSEQ_realpow_zero:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2540
  fixes x :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2541
  assumes "0 \<le> x" "x < 1"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2542
  shows "(\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2543
proof (cases "x = 0")
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2544
  case False
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2545
  with \<open>0 \<le> x\<close> have x0: "0 < x" by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2546
  then have "1 < inverse x"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2547
    using \<open>x < 1\<close> by (rule one_less_inverse)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2548
  then have "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2549
    by (rule LIMSEQ_inverse_realpow_zero)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2550
  then show ?thesis by (simp add: power_inverse)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2551
next
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2552
  case True
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2553
  show ?thesis
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2554
    by (rule LIMSEQ_imp_Suc) (simp add: True)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2555
qed
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2556
70723
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70694
diff changeset
  2557
lemma LIMSEQ_power_zero [tendsto_intros]: "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2558
  for x :: "'a::real_normed_algebra_1"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2559
  apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2560
  by (simp add: Zfun_le norm_power_ineq tendsto_Zfun_iff)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2561
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2562
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2563
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2564
63556
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2565
lemma
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2566
  tendsto_power_zero:
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2567
  fixes x::"'a::real_normed_algebra_1"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2568
  assumes "filterlim f at_top F"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2569
  assumes "norm x < 1"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2570
  shows "((\<lambda>y. x ^ (f y)) \<longlongrightarrow> 0) F"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2571
proof (rule tendstoI)
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2572
  fix e::real assume "0 < e"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2573
  from tendstoD[OF LIMSEQ_power_zero[OF \<open>norm x < 1\<close>] \<open>0 < e\<close>]
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2574
  have "\<forall>\<^sub>F xa in sequentially. norm (x ^ xa) < e"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2575
    by simp
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2576
  then obtain N where N: "norm (x ^ n) < e" if "n \<ge> N" for n
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2577
    by (auto simp: eventually_sequentially)
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2578
  have "\<forall>\<^sub>F i in F. f i \<ge> N"
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2579
    using \<open>filterlim f sequentially F\<close>
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2580
    by (simp add: filterlim_at_top)
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2581
  then show "\<forall>\<^sub>F i in F. dist (x ^ f i) 0 < e"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2582
    by eventually_elim (auto simp: N)
63556
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2583
qed
36e9732988ce numerical bounds on pi
immler
parents: 63548
diff changeset
  2584
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69272
diff changeset
  2585
text \<open>Limit of \<^term>\<open>c^n\<close> for \<^term>\<open>\<bar>c\<bar> < 1\<close>.\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2586
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
  2587
lemma LIMSEQ_abs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2588
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2589
68614
3cb44b0abc5c more de-applying
paulson <lp15@cam.ac.uk>
parents: 68611
diff changeset
  2590
lemma LIMSEQ_abs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) \<longlonglongrightarrow> 0"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2591
  by (rule LIMSEQ_power_zero) simp
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2592
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2593
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2594
subsection \<open>Limits of Functions\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2595
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2596
lemma LIM_eq: "f \<midarrow>a\<rightarrow> L = (\<forall>r>0. \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2597
  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2598
  by (simp add: LIM_def dist_norm)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2599
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2600
lemma LIM_I:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2601
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r) \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2602
  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2603
  by (simp add: LIM_eq)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2604
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2605
lemma LIM_D: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2606
  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2607
  by (simp add: LIM_eq)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2608
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2609
lemma LIM_offset: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2610
  for a :: "'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2611
  by (simp add: filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2612
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2613
lemma LIM_offset_zero: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2614
  for a :: "'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2615
  by (drule LIM_offset [where k = a]) (simp add: add.commute)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2616
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2617
lemma LIM_offset_zero_cancel: "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2618
  for a :: "'a::real_normed_vector"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2619
  by (drule LIM_offset [where k = "- a"]) simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2620
72245
cbe7aa1c2bdc tidying and de-applying
paulson <lp15@cam.ac.uk>
parents: 72220
diff changeset
  2621
lemma LIM_offset_zero_iff: "NO_MATCH 0 a \<Longrightarrow> f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2622
  for f :: "'a :: real_normed_vector \<Rightarrow> _"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  2623
  using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  2624
70999
5b753486c075 Inverse function theorem + lemmas
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2625
lemma tendsto_offset_zero_iff:
5b753486c075 Inverse function theorem + lemmas
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2626
  fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
72245
cbe7aa1c2bdc tidying and de-applying
paulson <lp15@cam.ac.uk>
parents: 72220
diff changeset
  2627
  assumes " NO_MATCH 0 a" "a \<in> S" "open S"
70999
5b753486c075 Inverse function theorem + lemmas
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2628
  shows "(f \<longlongrightarrow> L) (at a within S) \<longleftrightarrow> ((\<lambda>h. f (a + h)) \<longlongrightarrow> L) (at 0)"
72245
cbe7aa1c2bdc tidying and de-applying
paulson <lp15@cam.ac.uk>
parents: 72220
diff changeset
  2629
  using assms by (simp add: tendsto_within_open_NO_MATCH LIM_offset_zero_iff)
70999
5b753486c075 Inverse function theorem + lemmas
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2630
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2631
lemma LIM_zero: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65204
diff changeset
  2632
  for f :: "'a \<Rightarrow> 'b::real_normed_vector"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2633
  unfolding tendsto_iff dist_norm by simp
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2634
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2635
lemma LIM_zero_cancel:
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65204
diff changeset
  2636
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2637
  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2638
unfolding tendsto_iff dist_norm by simp
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2639
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2640
lemma LIM_zero_iff: "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65204
diff changeset
  2641
  for f :: "'a \<Rightarrow> 'b::real_normed_vector"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2642
  unfolding tendsto_iff dist_norm by simp
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2643
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2644
lemma LIM_imp_LIM:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2645
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2646
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2647
  assumes f: "f \<midarrow>a\<rightarrow> l"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2648
    and le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2649
  shows "g \<midarrow>a\<rightarrow> m"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2650
  by (rule metric_LIM_imp_LIM [OF f]) (simp add: dist_norm le)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2651
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2652
lemma LIM_equal2:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2653
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2654
  assumes "0 < R"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2655
    and "\<And>x. x \<noteq> a \<Longrightarrow> norm (x - a) < R \<Longrightarrow> f x = g x"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2656
  shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l"
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68532
diff changeset
  2657
  by (rule metric_LIM_equal2 [OF _ assms]) (simp_all add: dist_norm)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2658
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2659
lemma LIM_compose2:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2660
  fixes a :: "'a::real_normed_vector"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2661
  assumes f: "f \<midarrow>a\<rightarrow> b"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2662
    and g: "g \<midarrow>b\<rightarrow> c"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2663
    and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2664
  shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2665
  by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2666
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2667
lemma real_LIM_sandwich_zero:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2668
  fixes f g :: "'a::topological_space \<Rightarrow> real"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2669
  assumes f: "f \<midarrow>a\<rightarrow> 0"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2670
    and 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2671
    and 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2672
  shows "g \<midarrow>a\<rightarrow> 0"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2673
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2674
  fix x
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2675
  assume x: "x \<noteq> a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2676
  with 1 have "norm (g x - 0) = g x" by simp
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2677
  also have "g x \<le> f x" by (rule 2 [OF x])
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2678
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2679
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2680
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2681
qed
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2682
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2683
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2684
subsection \<open>Continuity\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2685
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2686
lemma LIM_isCont_iff: "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2687
  for f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2688
  by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2689
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2690
lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2691
  for f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2692
  by (simp add: isCont_def LIM_isCont_iff)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2693
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2694
lemma isCont_LIM_compose2:
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2695
  fixes a :: "'a::real_normed_vector"
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2696
  assumes f [unfolded isCont_def]: "isCont f a"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2697
    and g: "g \<midarrow>f a\<rightarrow> l"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2698
    and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
61976
3a27957ac658 more symbols;
wenzelm
parents: 61973
diff changeset
  2699
  shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2700
  by (rule LIM_compose2 [OF f g inj])
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2701
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2702
lemma isCont_norm [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2703
  for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2704
  by (fact continuous_norm)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2705
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2706
lemma isCont_rabs [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2707
  for f :: "'a::t2_space \<Rightarrow> real"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2708
  by (fact continuous_rabs)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2709
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2710
lemma isCont_add [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2711
  for f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2712
  by (fact continuous_add)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2713
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2714
lemma isCont_minus [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2715
  for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2716
  by (fact continuous_minus)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2717
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2718
lemma isCont_diff [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2719
  for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2720
  by (fact continuous_diff)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2721
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2722
lemma isCont_mult [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2723
  for f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2724
  by (fact continuous_mult)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2725
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2726
lemma (in bounded_linear) isCont: "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2727
  by (fact continuous)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2728
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2729
lemma (in bounded_bilinear) isCont: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2730
  by (fact continuous)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2731
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2732
lemmas isCont_scaleR [simp] =
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2733
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2734
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2735
lemmas isCont_of_real [simp] =
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2736
  bounded_linear.isCont [OF bounded_linear_of_real]
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2737
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2738
lemma isCont_power [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2739
  for f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2740
  by (fact continuous_power)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2741
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
  2742
lemma isCont_sum [simp]: "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2743
  for f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63952
diff changeset
  2744
  by (auto intro: continuous_sum)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2745
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2746
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2747
subsection \<open>Uniform Continuity\<close>
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2748
63104
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2749
lemma uniformly_continuous_on_def:
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2750
  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2751
  shows "uniformly_continuous_on s f \<longleftrightarrow>
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2752
    (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2753
  unfolding uniformly_continuous_on_uniformity
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2754
    uniformity_dist filterlim_INF filterlim_principal eventually_inf_principal
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2755
  by (force simp: Ball_def uniformity_dist[symmetric] eventually_uniformity_metric)
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2756
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2757
abbreviation isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2758
  where "isUCont f \<equiv> uniformly_continuous_on UNIV f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2759
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2760
lemma isUCont_def: "isUCont f \<longleftrightarrow> (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
63104
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2761
  by (auto simp: uniformly_continuous_on_def dist_commute)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
  2762
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2763
lemma isUCont_isCont: "isUCont f \<Longrightarrow> isCont f x"
63104
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2764
  by (drule uniformly_continuous_imp_continuous) (simp add: continuous_on_eq_continuous_at)
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2765
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2766
lemma uniformly_continuous_on_Cauchy:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2767
  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
63104
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2768
  assumes "uniformly_continuous_on S f" "Cauchy X" "\<And>n. X n \<in> S"
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2769
  shows "Cauchy (\<lambda>n. f (X n))"
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2770
  using assms
68594
5b05ede597b8 de-applying
paulson <lp15@cam.ac.uk>
parents: 68532
diff changeset
  2771
  unfolding uniformly_continuous_on_def  by (meson Cauchy_def)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
  2772
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2773
lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
63104
9505a883403c reduce isUCont to uniformly_continuous_on
immler
parents: 63081
diff changeset
  2774
  by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  2775
64287
d85d88722745 more from moretop.ml
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  2776
lemma uniformly_continuous_imp_Cauchy_continuous:
d85d88722745 more from moretop.ml
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  2777
  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
67091
1393c2340eec more symbols;
wenzelm
parents: 66827
diff changeset
  2778
  shows "\<lbrakk>uniformly_continuous_on S f; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f \<circ> \<sigma>)"
64287
d85d88722745 more from moretop.ml
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  2779
  by (simp add: uniformly_continuous_on_def Cauchy_def) meson
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
  2780
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2781
lemma (in bounded_linear) isUCont: "isUCont f"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2782
  unfolding isUCont_def dist_norm
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2783
proof (intro allI impI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2784
  fix r :: real
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2785
  assume r: "0 < r"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2786
  obtain K where K: "0 < K" and norm_le: "norm (f x) \<le> norm x * K" for x
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
  2787
    using pos_bounded by blast
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2788
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2789
  proof (rule exI, safe)
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
  2790
    from r K show "0 < r / K" by simp
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2791
  next
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2792
    fix x y :: 'a
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2793
    assume xy: "norm (x - y) < r / K"
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2794
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2795
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2796
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2797
    finally show "norm (f x - f y) < r" .
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2798
  qed
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2799
qed
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2800
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2801
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2802
  by (rule isUCont [THEN isUCont_Cauchy])
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2803
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2804
lemma LIM_less_bound:
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2805
  fixes f :: "real \<Rightarrow> real"
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2806
  assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2807
  shows "0 \<le> f x"
63952
354808e9f44b new material connected with HOL Light measure theory, plus more rationalisation
paulson <lp15@cam.ac.uk>
parents: 63915
diff changeset
  2808
proof (rule tendsto_lowerbound)
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
  2809
  show "(f \<longlongrightarrow> f x) (at_left x)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2810
    using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2811
  show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  2812
    using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
51526
155263089e7b move SEQ.thy and Lim.thy to Limits.thy
hoelzl
parents: 51524
diff changeset
  2813
qed simp
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents: 51360
diff changeset
  2814
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2815
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2816
subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2817
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2818
lemma nested_sequence_unique:
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2819
  assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0"
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2820
  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f \<longlonglongrightarrow> l) \<and> ((\<forall>n. l \<le> g n) \<and> g \<longlonglongrightarrow> l)"
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2821
proof -
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2822
  have "incseq f" unfolding incseq_Suc_iff by fact
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2823
  have "decseq g" unfolding decseq_Suc_iff by fact
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2824
  have "f n \<le> g 0" for n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2825
  proof -
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2826
    from \<open>decseq g\<close> have "g n \<le> g 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2827
      by (rule decseqD) simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2828
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] show ?thesis
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2829
      by auto
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2830
  qed
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2831
  then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2832
    using incseq_convergent[OF \<open>incseq f\<close>] by auto
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2833
  moreover have "f 0 \<le> g n" for n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2834
  proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2835
    from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2836
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] show ?thesis
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2837
      by simp
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2838
  qed
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2839
  then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2840
    using decseq_convergent[OF \<open>decseq g\<close>] by auto
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2841
  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2842
  ultimately show ?thesis by auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2843
qed
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2844
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2845
lemma Bolzano[consumes 1, case_names trans local]:
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2846
  fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2847
  assumes [arith]: "a \<le> b"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2848
    and trans: "\<And>a b c. P a b \<Longrightarrow> P b c \<Longrightarrow> a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> P a c"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2849
    and local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2850
  shows "P a b"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2851
proof -
74513
67d87d224e00 A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents: 74475
diff changeset
  2852
  define bisect where "bisect \<equiv> \<lambda>(x,y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2)"
67d87d224e00 A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents: 74475
diff changeset
  2853
  define l u where "l n \<equiv> fst ((bisect^^n)(a,b))" and "u n \<equiv> snd ((bisect^^n)(a,b))" for n
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2854
  have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2855
    and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2856
    by (simp_all add: l_def u_def bisect_def split: prod.split)
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2857
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2858
  have [simp]: "l n \<le> u n" for n by (induct n) auto
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2859
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2860
  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2861
  proof (safe intro!: nested_sequence_unique)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2862
    show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" for n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2863
      by (induct n) auto
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2864
  next
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2865
    have "l n - u n = (a - b) / 2^n" for n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2866
      by (induct n) (auto simp: field_simps)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2867
    then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2868
      by (simp add: LIMSEQ_divide_realpow_zero)
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2869
  qed fact
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2870
  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2871
    by auto
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2872
  obtain d where "0 < d" and d: "a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b" for a b
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2873
    using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2874
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2875
  show "P a b"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2876
  proof (rule ccontr)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  2877
    assume "\<not> P a b"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2878
    have "\<not> P (l n) (u n)" for n
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2879
    proof (induct n)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2880
      case 0
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2881
      then show ?case
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2882
        by (simp add: \<open>\<not> P a b\<close>)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2883
    next
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2884
      case (Suc n)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2885
      with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2886
        by auto
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2887
    qed
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2888
    moreover
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2889
    {
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2890
      have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2891
        using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2892
      moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
61969
e01015e49041 more symbols;
wenzelm
parents: 61916
diff changeset
  2893
        using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2894
      ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2895
      proof eventually_elim
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2896
        case (elim n)
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2897
        from add_strict_mono[OF this] have "u n - l n < d" by simp
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2898
        with x show "P (l n) (u n)" by (rule d)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2899
      qed
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2900
    }
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2901
    ultimately show False by simp
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2902
  qed
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2903
qed
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2904
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2905
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2906
proof (cases "a \<le> b", rule compactI)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2907
  fix C
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2908
  assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62393
diff changeset
  2909
  define T where "T = {a .. b}"
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2910
  from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2911
  proof (induct rule: Bolzano)
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2912
    case (trans a b c)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2913
    then have *: "{a..c} = {a..b} \<union> {b..c}"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2914
      by auto
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2915
    with trans obtain C1 C2
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2916
      where "C1\<subseteq>C" "finite C1" "{a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C" "finite C2" "{b..c} \<subseteq> \<Union>C2"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2917
      by auto
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2918
    with trans show ?case
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2919
      unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2920
  next
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2921
    case (local x)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2922
    with C have "x \<in> \<Union>C" by auto
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2923
    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2924
      by auto
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2925
    then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 62087
diff changeset
  2926
      by (auto simp: open_dist dist_real_def subset_eq Ball_def abs_less_iff)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2927
    with \<open>c \<in> C\<close> show ?case
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2928
      by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2929
  qed
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2930
qed simp
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2931
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2932
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2933
lemma continuous_image_closed_interval:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2934
  fixes a b and f :: "real \<Rightarrow> real"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2935
  defines "S \<equiv> {a..b}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2936
  assumes "a \<le> b" and f: "continuous_on S f"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2937
  shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2938
proof -
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2939
  have S: "compact S" "S \<noteq> {}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2940
    using \<open>a \<le> b\<close> by (auto simp: S_def)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2941
  obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2942
    using continuous_attains_sup[OF S f] by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2943
  moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2944
    using continuous_attains_inf[OF S f] by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2945
  moreover have "connected (f`S)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2946
    using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2947
  ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2948
    by (auto simp: connected_iff_interval)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2949
  then show ?thesis
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2950
    by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2951
qed
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57276
diff changeset
  2952
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60758
diff changeset
  2953
lemma open_Collect_positive:
67958
732c0b059463 tuned proofs and generalized some lemmas about limits
huffman
parents: 67950
diff changeset
  2954
  fixes f :: "'a::topological_space \<Rightarrow> real"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2955
  assumes f: "continuous_on s f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2956
  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2957
  using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2958
  by (auto simp: Int_def field_simps)
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60758
diff changeset
  2959
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60758
diff changeset
  2960
lemma open_Collect_less_Int:
67958
732c0b059463 tuned proofs and generalized some lemmas about limits
huffman
parents: 67950
diff changeset
  2961
  fixes f g :: "'a::topological_space \<Rightarrow> real"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2962
  assumes f: "continuous_on s f"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2963
    and g: "continuous_on s g"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2964
  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2965
  using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60758
diff changeset
  2966
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60758
diff changeset
  2967
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2968
subsection \<open>Boundedness of continuous functions\<close>
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2969
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60721
diff changeset
  2970
text\<open>By bisection, function continuous on closed interval is bounded above\<close>
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2971
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2972
lemma isCont_eq_Ub:
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2973
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2974
  shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2975
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2976
  using continuous_attains_sup[of "{a..b}" f]
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2977
  by (auto simp: continuous_at_imp_continuous_on Ball_def Bex_def)
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2978
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2979
lemma isCont_eq_Lb:
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2980
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2981
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2982
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  2983
  using continuous_attains_inf[of "{a..b}" f]
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  2984
  by (auto simp: continuous_at_imp_continuous_on Ball_def Bex_def)
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2985
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2986
lemma isCont_bounded:
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2987
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2988
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2989
  using isCont_eq_Ub[of a b f] by auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2990
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2991
lemma isCont_has_Ub:
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2992
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2993
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2994
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2995
  using isCont_eq_Ub[of a b f] by auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2996
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2997
lemma isCont_Lb_Ub:
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2998
  fixes f :: "real \<Rightarrow> real"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  2999
  assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  3000
  shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3001
    (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3002
proof -
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3003
  obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3004
    using isCont_eq_Ub[OF assms] by auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3005
  obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3006
    using isCont_eq_Lb[OF assms] by auto
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3007
  have "(\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x \<and> f x \<le> f M)"
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3008
    using M L by simp
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3009
  moreover
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3010
  have "(\<forall>y. f L \<le> y \<and> y \<le> f M \<longrightarrow> (\<exists>x\<ge>a. x \<le> b \<and> f x = y))"
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3011
  proof (cases "L \<le> M")
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3012
    case True then show ?thesis
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3013
    using IVT[of f L _ M] M L assms by (metis order.trans)
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3014
  next
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3015
    case False then show ?thesis
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3016
    using IVT2[of f L _ M]
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3017
    by (metis L(2) M(1) assms(2) le_cases order.trans)
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3018
qed
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3019
  ultimately show ?thesis
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3020
    by blast
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3021
qed
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3022
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3023
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3024
text \<open>Continuity of inverse function.\<close>
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3025
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3026
lemma isCont_inverse_function:
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3027
  fixes f g :: "real \<Rightarrow> real"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3028
  assumes d: "0 < d"
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3029
    and inj: "\<And>z. \<bar>z-x\<bar> \<le> d \<Longrightarrow> g (f z) = z"
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3030
    and cont: "\<And>z. \<bar>z-x\<bar> \<le> d \<Longrightarrow> isCont f z"
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3031
  shows "isCont g (f x)"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3032
proof -
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3033
  let ?A = "f (x - d)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3034
  let ?B = "f (x + d)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3035
  let ?D = "{x - d..x + d}"
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3036
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3037
  have f: "continuous_on ?D f"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3038
    using cont by (intro continuous_at_imp_continuous_on ballI) auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3039
  then have g: "continuous_on (f`?D) g"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3040
    using inj by (intro continuous_on_inv) auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3041
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3042
  from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3043
    by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3044
  with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3045
    by (rule continuous_on_subset)
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3046
  moreover
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3047
  have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3048
    using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3049
  then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3050
    by auto
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3051
  ultimately show ?thesis
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3052
    by (simp add: continuous_on_eq_continuous_at)
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3053
qed
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3054
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3055
lemma isCont_inverse_function2:
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3056
  fixes f g :: "real \<Rightarrow> real"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3057
  shows
68611
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3058
    "\<lbrakk>a < x; x < b;
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3059
      \<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> g (f z) = z;
4bc4b5c0ccfc de-applying, etc.
paulson <lp15@cam.ac.uk>
parents: 68594
diff changeset
  3060
      \<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> isCont f z\<rbrakk> \<Longrightarrow> isCont g (f x)"
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3061
  apply (rule isCont_inverse_function [where f=f and d="min (x - a) (b - x)"])
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3062
  apply (simp_all add: abs_le_iff)
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3063
  done
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  3064
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3065
text \<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110.\<close>
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3066
lemma LIM_fun_gt_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3067
  for f :: "real \<Rightarrow> real"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3068
  by (force simp: dest: LIM_D)
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3069
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3070
lemma LIM_fun_less_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3071
  for f :: "real \<Rightarrow> real"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3072
  by (drule LIM_D [where r="-l"]) force+
63546
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3073
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3074
lemma LIM_fun_not_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
5f097087fa1e misc tuning and modernization;
wenzelm
parents: 63301
diff changeset
  3075
  for f :: "real \<Rightarrow> real"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 68614
diff changeset
  3076
  using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp: neq_iff)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51529
diff changeset
  3077
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
  3078
end