src/HOL/Limits.thy
author huffman
Sun, 14 Aug 2011 07:54:24 -0700
changeset 44194 0639898074ae
parent 44081 730f7cced3a6
child 44195 f5363511b212
permissions -rw-r--r--
generalize lemmas about LIM and LIMSEQ to tendsto
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     1
(*  Title       : Limits.thy
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     2
    Author      : Brian Huffman
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     3
*)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     4
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     5
header {* Filters and Limits *}
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
theory Limits
36822
38a480e0346f minimize imports
huffman
parents: 36662
diff changeset
     8
imports RealVector
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
     9
begin
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    10
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    11
subsection {* Filters *}
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    12
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    13
text {*
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    14
  This definition also allows non-proper filters.
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    15
*}
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    16
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
    17
locale is_filter =
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    18
  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    19
  assumes True: "F (\<lambda>x. True)"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    20
  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    21
  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
    22
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    23
typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    24
proof
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    25
  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    26
qed
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    27
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    28
lemma is_filter_Rep_filter: "is_filter (Rep_filter A)"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    29
  using Rep_filter [of A] by simp
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    30
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    31
lemma Abs_filter_inverse':
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    32
  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    33
  using assms by (simp add: Abs_filter_inverse)
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    34
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    35
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
    36
subsection {* Eventually *}
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    37
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    38
definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    39
  where "eventually P A \<longleftrightarrow> Rep_filter A P"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
    40
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    41
lemma eventually_Abs_filter:
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    42
  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    43
  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    44
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    45
lemma filter_eq_iff:
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    46
  shows "A = B \<longleftrightarrow> (\<forall>P. eventually P A = eventually P B)"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    47
  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
    48
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    49
lemma eventually_True [simp]: "eventually (\<lambda>x. True) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    50
  unfolding eventually_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    51
  by (rule is_filter.True [OF is_filter_Rep_filter])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    52
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    53
lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P A"
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
    54
proof -
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
    55
  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    56
  thus "eventually P A" by simp
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
    57
qed
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
    58
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    59
lemma eventually_mono:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    60
  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P A \<Longrightarrow> eventually Q A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    61
  unfolding eventually_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    62
  by (rule is_filter.mono [OF is_filter_Rep_filter])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    63
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    64
lemma eventually_conj:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    65
  assumes P: "eventually (\<lambda>x. P x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    66
  assumes Q: "eventually (\<lambda>x. Q x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    67
  shows "eventually (\<lambda>x. P x \<and> Q x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    68
  using assms unfolding eventually_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    69
  by (rule is_filter.conj [OF is_filter_Rep_filter])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    70
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    71
lemma eventually_mp:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    72
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    73
  assumes "eventually (\<lambda>x. P x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    74
  shows "eventually (\<lambda>x. Q x) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    75
proof (rule eventually_mono)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    76
  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    77
  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    78
    using assms by (rule eventually_conj)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    79
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    80
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    81
lemma eventually_rev_mp:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    82
  assumes "eventually (\<lambda>x. P x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    83
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    84
  shows "eventually (\<lambda>x. Q x) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    85
using assms(2) assms(1) by (rule eventually_mp)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    86
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    87
lemma eventually_conj_iff:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    88
  "eventually (\<lambda>x. P x \<and> Q x) A \<longleftrightarrow> eventually P A \<and> eventually Q A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    89
  by (auto intro: eventually_conj elim: eventually_rev_mp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    90
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    91
lemma eventually_elim1:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    92
  assumes "eventually (\<lambda>i. P i) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    93
  assumes "\<And>i. P i \<Longrightarrow> Q i"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    94
  shows "eventually (\<lambda>i. Q i) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    95
  using assms by (auto elim!: eventually_rev_mp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    96
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
    97
lemma eventually_elim2:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    98
  assumes "eventually (\<lambda>i. P i) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
    99
  assumes "eventually (\<lambda>i. Q i) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   100
  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   101
  shows "eventually (\<lambda>i. R i) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   102
  using assms by (auto elim!: eventually_rev_mp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   103
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   104
subsection {* Finer-than relation *}
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   105
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   106
text {* @{term "A \<le> B"} means that filter @{term A} is finer than
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   107
filter @{term B}. *}
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   108
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   109
instantiation filter :: (type) complete_lattice
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   110
begin
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   111
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   112
definition le_filter_def:
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   113
  "A \<le> B \<longleftrightarrow> (\<forall>P. eventually P B \<longrightarrow> eventually P A)"
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   114
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   115
definition
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   116
  "(A :: 'a filter) < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   117
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   118
definition
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   119
  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   120
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   121
definition
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   122
  "bot = Abs_filter (\<lambda>P. True)"
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   123
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   124
definition
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   125
  "sup A B = Abs_filter (\<lambda>P. eventually P A \<and> eventually P B)"
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   126
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   127
definition
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   128
  "inf A B = Abs_filter
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   129
      (\<lambda>P. \<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   130
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   131
definition
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   132
  "Sup S = Abs_filter (\<lambda>P. \<forall>A\<in>S. eventually P A)"
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   133
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   134
definition
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   135
  "Inf S = Sup {A::'a filter. \<forall>B\<in>S. A \<le> B}"
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   136
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   137
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   138
  unfolding top_filter_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   139
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   140
36629
de62713aec6e swap ordering on nets, so x <= y means 'x is finer than y'
huffman
parents: 36360
diff changeset
   141
lemma eventually_bot [simp]: "eventually P bot"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   142
  unfolding bot_filter_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   143
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   144
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   145
lemma eventually_sup:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   146
  "eventually P (sup A B) \<longleftrightarrow> eventually P A \<and> eventually P B"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   147
  unfolding sup_filter_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   148
  by (rule eventually_Abs_filter, rule is_filter.intro)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   149
     (auto elim!: eventually_rev_mp)
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   150
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   151
lemma eventually_inf:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   152
  "eventually P (inf A B) \<longleftrightarrow>
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   153
   (\<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   154
  unfolding inf_filter_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   155
  apply (rule eventually_Abs_filter, rule is_filter.intro)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   156
  apply (fast intro: eventually_True)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   157
  apply clarify
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   158
  apply (intro exI conjI)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   159
  apply (erule (1) eventually_conj)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   160
  apply (erule (1) eventually_conj)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   161
  apply simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   162
  apply auto
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   163
  done
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   164
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   165
lemma eventually_Sup:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   166
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>A\<in>S. eventually P A)"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   167
  unfolding Sup_filter_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   168
  apply (rule eventually_Abs_filter, rule is_filter.intro)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   169
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   170
  done
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   171
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   172
instance proof
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   173
  fix A B :: "'a filter" show "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   174
    by (rule less_filter_def)
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   175
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   176
  fix A :: "'a filter" show "A \<le> A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   177
    unfolding le_filter_def by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   178
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   179
  fix A B C :: "'a filter" assume "A \<le> B" and "B \<le> C" thus "A \<le> C"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   180
    unfolding le_filter_def by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   181
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   182
  fix A B :: "'a filter" assume "A \<le> B" and "B \<le> A" thus "A = B"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   183
    unfolding le_filter_def filter_eq_iff by fast
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   184
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   185
  fix A :: "'a filter" show "A \<le> top"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   186
    unfolding le_filter_def eventually_top by (simp add: always_eventually)
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   187
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   188
  fix A :: "'a filter" show "bot \<le> A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   189
    unfolding le_filter_def by simp
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   190
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   191
  fix A B :: "'a filter" show "A \<le> sup A B" and "B \<le> sup A B"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   192
    unfolding le_filter_def eventually_sup by simp_all
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   193
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   194
  fix A B C :: "'a filter" assume "A \<le> C" and "B \<le> C" thus "sup A B \<le> C"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   195
    unfolding le_filter_def eventually_sup by simp
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   196
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   197
  fix A B :: "'a filter" show "inf A B \<le> A" and "inf A B \<le> B"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   198
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True)
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   199
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   200
  fix A B C :: "'a filter" assume "A \<le> B" and "A \<le> C" thus "A \<le> inf B C"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   201
    unfolding le_filter_def eventually_inf
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   202
    by (auto elim!: eventually_mono intro: eventually_conj)
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   203
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   204
  fix A :: "'a filter" and S assume "A \<in> S" thus "A \<le> Sup S"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   205
    unfolding le_filter_def eventually_Sup by simp
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   206
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   207
  fix S and B :: "'a filter" assume "\<And>A. A \<in> S \<Longrightarrow> A \<le> B" thus "Sup S \<le> B"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   208
    unfolding le_filter_def eventually_Sup by simp
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   209
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   210
  fix C :: "'a filter" and S assume "C \<in> S" thus "Inf S \<le> C"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   211
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp
36630
aa1f8acdcc1c complete_lattice instance for net type
huffman
parents: 36629
diff changeset
   212
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   213
  fix S and A :: "'a filter" assume "\<And>B. B \<in> S \<Longrightarrow> A \<le> B" thus "A \<le> Inf S"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   214
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   215
qed
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   216
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   217
end
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   218
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   219
lemma filter_leD:
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   220
  "A \<le> B \<Longrightarrow> eventually P B \<Longrightarrow> eventually P A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   221
  unfolding le_filter_def by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   222
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   223
lemma filter_leI:
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   224
  "(\<And>P. eventually P B \<Longrightarrow> eventually P A) \<Longrightarrow> A \<le> B"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   225
  unfolding le_filter_def by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   226
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   227
lemma eventually_False:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   228
  "eventually (\<lambda>x. False) A \<longleftrightarrow> A = bot"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   229
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   230
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   231
subsection {* Map function for filters *}
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   232
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   233
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   234
  where "filtermap f A = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) A)"
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   235
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   236
lemma eventually_filtermap:
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   237
  "eventually P (filtermap f A) = eventually (\<lambda>x. P (f x)) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   238
  unfolding filtermap_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   239
  apply (rule eventually_Abs_filter)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   240
  apply (rule is_filter.intro)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   241
  apply (auto elim!: eventually_rev_mp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   242
  done
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   243
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   244
lemma filtermap_ident: "filtermap (\<lambda>x. x) A = A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   245
  by (simp add: filter_eq_iff eventually_filtermap)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   246
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   247
lemma filtermap_filtermap:
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   248
  "filtermap f (filtermap g A) = filtermap (\<lambda>x. f (g x)) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   249
  by (simp add: filter_eq_iff eventually_filtermap)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   250
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   251
lemma filtermap_mono: "A \<le> B \<Longrightarrow> filtermap f A \<le> filtermap f B"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   252
  unfolding le_filter_def eventually_filtermap by simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   253
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   254
lemma filtermap_bot [simp]: "filtermap f bot = bot"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   255
  by (simp add: filter_eq_iff eventually_filtermap)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   256
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   257
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   258
subsection {* Sequentially *}
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   259
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   260
definition sequentially :: "nat filter"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   261
  where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   262
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   263
lemma eventually_sequentially:
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   264
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   265
unfolding sequentially_def
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   266
proof (rule eventually_Abs_filter, rule is_filter.intro)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   267
  fix P Q :: "nat \<Rightarrow> bool"
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   268
  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   269
  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   270
  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   271
  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   272
qed auto
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   273
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   274
lemma sequentially_bot [simp]: "sequentially \<noteq> bot"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   275
  unfolding filter_eq_iff eventually_sequentially by auto
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   276
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   277
lemma eventually_False_sequentially [simp]:
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   278
  "\<not> eventually (\<lambda>n. False) sequentially"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   279
  by (simp add: eventually_False)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   280
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   281
lemma le_sequentially:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   282
  "A \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) A)"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   283
  unfolding le_filter_def eventually_sequentially
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   284
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   285
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   286
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   287
definition trivial_limit :: "'a filter \<Rightarrow> bool"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   288
  where "trivial_limit A \<longleftrightarrow> eventually (\<lambda>x. False) A"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   289
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   290
lemma trivial_limit_sequentially [intro]: "\<not> trivial_limit sequentially"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   291
  by (auto simp add: trivial_limit_def eventually_sequentially)
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   292
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   293
subsection {* Standard filters *}
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   294
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   295
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   296
  where "A within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A)"
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   297
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   298
definition nhds :: "'a::topological_space \<Rightarrow> 'a filter"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   299
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   300
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   301
definition at :: "'a::topological_space \<Rightarrow> 'a filter"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   302
  where "at a = nhds a within - {a}"
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   303
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   304
lemma eventually_within:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   305
  "eventually P (A within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   306
  unfolding within_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   307
  by (rule eventually_Abs_filter, rule is_filter.intro)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   308
     (auto elim!: eventually_rev_mp)
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   309
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   310
lemma within_UNIV: "A within UNIV = A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   311
  unfolding filter_eq_iff eventually_within by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   312
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   313
lemma eventually_nhds:
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   314
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   315
unfolding nhds_def
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   316
proof (rule eventually_Abs_filter, rule is_filter.intro)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   317
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   318
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   319
next
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   320
  fix P Q
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   321
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   322
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   323
  then obtain S T where
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   324
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   325
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   326
  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   327
    by (simp add: open_Int)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   328
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   329
qed auto
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   330
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   331
lemma eventually_nhds_metric:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   332
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   333
unfolding eventually_nhds open_dist
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   334
apply safe
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   335
apply fast
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31488
diff changeset
   336
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   337
apply clarsimp
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   338
apply (rule_tac x="d - dist x a" in exI, clarsimp)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   339
apply (simp only: less_diff_eq)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   340
apply (erule le_less_trans [OF dist_triangle])
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   341
done
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   342
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   343
lemma eventually_at_topological:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   344
  "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   345
unfolding at_def eventually_within eventually_nhds by simp
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   346
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   347
lemma eventually_at:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   348
  fixes a :: "'a::metric_space"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   349
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   350
unfolding at_def eventually_within eventually_nhds_metric by auto
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   351
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   352
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   353
subsection {* Boundedness *}
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   354
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   355
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   356
  where "Bfun f A = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) A)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   357
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   358
lemma BfunI:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   359
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) A" shows "Bfun f A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   360
unfolding Bfun_def
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   361
proof (intro exI conjI allI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   362
  show "0 < max K 1" by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   363
next
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   364
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   365
    using K by (rule eventually_elim1, simp)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   366
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   367
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   368
lemma BfunE:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   369
  assumes "Bfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   370
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   371
using assms unfolding Bfun_def by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   372
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   373
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   374
subsection {* Convergence to Zero *}
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   375
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   376
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   377
  where "Zfun f A = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) A)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   378
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   379
lemma ZfunI:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   380
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A) \<Longrightarrow> Zfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   381
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   382
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   383
lemma ZfunD:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   384
  "\<lbrakk>Zfun f A; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   385
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   386
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   387
lemma Zfun_ssubst:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   388
  "eventually (\<lambda>x. f x = g x) A \<Longrightarrow> Zfun g A \<Longrightarrow> Zfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   389
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   390
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   391
lemma Zfun_zero: "Zfun (\<lambda>x. 0) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   392
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   393
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   394
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) A = Zfun (\<lambda>x. f x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   395
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   396
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   397
lemma Zfun_imp_Zfun:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   398
  assumes f: "Zfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   399
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   400
  shows "Zfun (\<lambda>x. g x) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   401
proof (cases)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   402
  assume K: "0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   403
  show ?thesis
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   404
  proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   405
    fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   406
    hence "0 < r / K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   407
      using K by (rule divide_pos_pos)
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   408
    then have "eventually (\<lambda>x. norm (f x) < r / K) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   409
      using ZfunD [OF f] by fast
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   410
    with g show "eventually (\<lambda>x. norm (g x) < r) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   411
    proof (rule eventually_elim2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   412
      fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   413
      assume *: "norm (g x) \<le> norm (f x) * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   414
      assume "norm (f x) < r / K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   415
      hence "norm (f x) * K < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   416
        by (simp add: pos_less_divide_eq K)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   417
      thus "norm (g x) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   418
        by (simp add: order_le_less_trans [OF *])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   419
    qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   420
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   421
next
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   422
  assume "\<not> 0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   423
  hence K: "K \<le> 0" by (simp only: not_less)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   424
  show ?thesis
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   425
  proof (rule ZfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   426
    fix r :: real
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   427
    assume "0 < r"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   428
    from g show "eventually (\<lambda>x. norm (g x) < r) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   429
    proof (rule eventually_elim1)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   430
      fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   431
      assume "norm (g x) \<le> norm (f x) * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   432
      also have "\<dots> \<le> norm (f x) * 0"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   433
        using K norm_ge_zero by (rule mult_left_mono)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   434
      finally show "norm (g x) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   435
        using `0 < r` by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   436
    qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   437
  qed
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   438
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   439
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   440
lemma Zfun_le: "\<lbrakk>Zfun g A; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   441
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   442
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   443
lemma Zfun_add:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   444
  assumes f: "Zfun f A" and g: "Zfun g A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   445
  shows "Zfun (\<lambda>x. f x + g x) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   446
proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   447
  fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   448
  hence r: "0 < r / 2" by simp
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   449
  have "eventually (\<lambda>x. norm (f x) < r/2) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   450
    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
   451
  moreover
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   452
  have "eventually (\<lambda>x. norm (g x) < r/2) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   453
    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
   454
  ultimately
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   455
  show "eventually (\<lambda>x. norm (f x + g x) < r) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   456
  proof (rule eventually_elim2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   457
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   458
    assume *: "norm (f x) < r/2" "norm (g x) < r/2"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   459
    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
   460
      by (rule norm_triangle_ineq)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   461
    also have "\<dots> < r/2 + r/2"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   462
      using * by (rule add_strict_mono)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   463
    finally show "norm (f x + g x) < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   464
      by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   465
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   466
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   467
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   468
lemma Zfun_minus: "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. - f x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   469
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   470
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   471
lemma Zfun_diff: "\<lbrakk>Zfun f A; Zfun g A\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   472
  by (simp only: diff_minus Zfun_add Zfun_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   473
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   474
lemma (in bounded_linear) Zfun:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   475
  assumes g: "Zfun g A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   476
  shows "Zfun (\<lambda>x. f (g x)) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   477
proof -
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   478
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   479
    using bounded by fast
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   480
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   481
    by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   482
  with g show ?thesis
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   483
    by (rule Zfun_imp_Zfun)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   484
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   485
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   486
lemma (in bounded_bilinear) Zfun:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   487
  assumes f: "Zfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   488
  assumes g: "Zfun g A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   489
  shows "Zfun (\<lambda>x. f x ** g x) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   490
proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   491
  fix r::real assume r: "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   492
  obtain K where K: "0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   493
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   494
    using pos_bounded by fast
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   495
  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
   496
    by (rule positive_imp_inverse_positive)
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   497
  have "eventually (\<lambda>x. norm (f x) < r) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   498
    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
   499
  moreover
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   500
  have "eventually (\<lambda>x. norm (g x) < inverse K) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   501
    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
   502
  ultimately
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   503
  show "eventually (\<lambda>x. norm (f x ** g x) < r) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   504
  proof (rule eventually_elim2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   505
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   506
    assume *: "norm (f x) < r" "norm (g x) < inverse K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   507
    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
   508
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   509
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   510
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   511
    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
   512
      by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   513
    finally show "norm (f x ** g x) < r" .
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   514
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   515
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   516
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   517
lemma (in bounded_bilinear) Zfun_left:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   518
  "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. f x ** a) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   519
  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
   520
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   521
lemma (in bounded_bilinear) Zfun_right:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   522
  "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. a ** f x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   523
  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
   524
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   525
lemmas Zfun_mult = mult.Zfun
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   526
lemmas Zfun_mult_right = mult.Zfun_right
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   527
lemmas Zfun_mult_left = mult.Zfun_left
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   528
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   529
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   530
subsection {* Limits *}
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   531
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   532
definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool"
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36822
diff changeset
   533
    (infixr "--->" 55) where
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   534
  "(f ---> l) A \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) A)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   535
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   536
ML {*
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   537
structure Tendsto_Intros = Named_Thms
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   538
(
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   539
  val name = "tendsto_intros"
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   540
  val description = "introduction rules for tendsto"
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   541
)
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   542
*}
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   543
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   544
setup Tendsto_Intros.setup
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   545
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   546
lemma tendsto_mono: "A \<le> A' \<Longrightarrow> (f ---> l) A' \<Longrightarrow> (f ---> l) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   547
  unfolding tendsto_def le_filter_def by fast
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   548
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   549
lemma topological_tendstoI:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   550
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   551
    \<Longrightarrow> (f ---> l) A"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   552
  unfolding tendsto_def by auto
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   553
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   554
lemma topological_tendstoD:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   555
  "(f ---> l) A \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   556
  unfolding tendsto_def by auto
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   557
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   558
lemma tendstoI:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   559
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   560
  shows "(f ---> l) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   561
  apply (rule topological_tendstoI)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   562
  apply (simp add: open_dist)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   563
  apply (drule (1) bspec, clarify)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   564
  apply (drule assms)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   565
  apply (erule eventually_elim1, simp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   566
  done
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   567
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   568
lemma tendstoD:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   569
  "(f ---> l) A \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   570
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   571
  apply (clarsimp simp add: open_dist)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   572
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   573
  apply (simp only: less_diff_eq)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   574
  apply (erule le_less_trans [OF dist_triangle])
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   575
  apply simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   576
  apply simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   577
  done
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   578
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   579
lemma tendsto_iff:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   580
  "(f ---> l) A \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) A)"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   581
  using tendstoI tendstoD by fast
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   582
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   583
lemma tendsto_Zfun_iff: "(f ---> a) A = Zfun (\<lambda>x. f x - a) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   584
  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
   585
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   586
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   587
  unfolding tendsto_def eventually_at_topological by auto
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   588
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   589
lemma tendsto_ident_at_within [tendsto_intros]:
36655
88f0125c3bd2 remove unneeded premise
huffman
parents: 36654
diff changeset
   590
  "((\<lambda>x. x) ---> a) (at a within S)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   591
  unfolding tendsto_def eventually_within eventually_at_topological by auto
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   592
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   593
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   594
  by (simp add: tendsto_def)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   595
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   596
lemma tendsto_const_iff:
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   597
  fixes k l :: "'a::metric_space"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   598
  assumes "A \<noteq> bot" shows "((\<lambda>n. k) ---> l) A \<longleftrightarrow> k = l"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   599
  apply (safe intro!: tendsto_const)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   600
  apply (rule ccontr)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   601
  apply (drule_tac e="dist k l" in tendstoD)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   602
  apply (simp add: zero_less_dist_iff)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   603
  apply (simp add: eventually_False assms)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   604
  done
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   605
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   606
lemma tendsto_dist [tendsto_intros]:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   607
  assumes f: "(f ---> l) A" and g: "(g ---> m) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   608
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) A"
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   609
proof (rule tendstoI)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   610
  fix e :: real assume "0 < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   611
  hence e2: "0 < e/2" by simp
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   612
  from tendstoD [OF f e2] tendstoD [OF g e2]
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   613
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) A"
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   614
  proof (rule eventually_elim2)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   615
    fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   616
    then show "dist (dist (f x) (g x)) (dist l m) < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   617
      unfolding dist_real_def
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   618
      using dist_triangle2 [of "f x" "g x" "l"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   619
      using dist_triangle2 [of "g x" "l" "m"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   620
      using dist_triangle3 [of "l" "m" "f x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   621
      using dist_triangle [of "f x" "m" "g x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   622
      by arith
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   623
  qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   624
qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   625
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   626
subsubsection {* Norms *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   627
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   628
lemma norm_conv_dist: "norm x = dist x 0"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   629
  unfolding dist_norm by simp
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   630
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   631
lemma tendsto_norm [tendsto_intros]:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   632
  "(f ---> a) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   633
  unfolding norm_conv_dist by (intro tendsto_intros)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   634
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   635
lemma tendsto_norm_zero:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   636
  "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   637
  by (drule tendsto_norm, simp)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   638
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   639
lemma tendsto_norm_zero_cancel:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   640
  "((\<lambda>x. norm (f x)) ---> 0) A \<Longrightarrow> (f ---> 0) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   641
  unfolding tendsto_iff dist_norm by simp
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   642
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   643
lemma tendsto_norm_zero_iff:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   644
  "((\<lambda>x. norm (f x)) ---> 0) A \<longleftrightarrow> (f ---> 0) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   645
  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
   646
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   647
lemma tendsto_rabs [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   648
  "(f ---> (l::real)) A \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   649
  by (fold real_norm_def, rule tendsto_norm)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   650
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   651
lemma tendsto_rabs_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   652
  "(f ---> (0::real)) A \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   653
  by (fold real_norm_def, rule tendsto_norm_zero)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   654
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   655
lemma tendsto_rabs_zero_cancel:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   656
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) A \<Longrightarrow> (f ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   657
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   658
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   659
lemma tendsto_rabs_zero_iff:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   660
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) A \<longleftrightarrow> (f ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   661
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   662
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   663
subsubsection {* Addition and subtraction *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   664
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   665
lemma tendsto_add [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   666
  fixes a b :: "'a::real_normed_vector"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   667
  shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   668
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   669
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   670
lemma tendsto_add_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   671
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   672
  shows "\<lbrakk>(f ---> 0) A; (g ---> 0) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   673
  by (drule (1) tendsto_add, simp)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   674
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   675
lemma tendsto_minus [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   676
  fixes a :: "'a::real_normed_vector"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   677
  shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   678
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   679
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   680
lemma tendsto_minus_cancel:
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   681
  fixes a :: "'a::real_normed_vector"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   682
  shows "((\<lambda>x. - f x) ---> - a) A \<Longrightarrow> (f ---> a) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   683
  by (drule tendsto_minus, simp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   684
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   685
lemma tendsto_diff [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   686
  fixes a b :: "'a::real_normed_vector"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   687
  shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   688
  by (simp add: diff_minus tendsto_add tendsto_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   689
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   690
lemma tendsto_setsum [tendsto_intros]:
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   691
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   692
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   693
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) A"
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   694
proof (cases "finite S")
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   695
  assume "finite S" thus ?thesis using assms
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   696
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   697
next
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   698
  assume "\<not> finite S" thus ?thesis
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   699
    by (simp add: tendsto_const)
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   700
qed
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   701
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   702
subsubsection {* Linear operators and multiplication *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   703
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   704
lemma (in bounded_linear) tendsto [tendsto_intros]:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   705
  "(g ---> a) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   706
  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
   707
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   708
lemma (in bounded_linear) tendsto_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   709
  "(g ---> 0) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   710
  by (drule tendsto, simp only: zero)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   711
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   712
lemma (in bounded_bilinear) tendsto [tendsto_intros]:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   713
  "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   714
  by (simp only: tendsto_Zfun_iff prod_diff_prod
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   715
                 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
   716
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   717
lemma (in bounded_bilinear) tendsto_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   718
  assumes f: "(f ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   719
  assumes g: "(g ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   720
  shows "((\<lambda>x. f x ** g x) ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   721
  using tendsto [OF f g] by (simp add: zero_left)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   722
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   723
lemma (in bounded_bilinear) tendsto_left_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   724
  "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   725
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   726
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   727
lemma (in bounded_bilinear) tendsto_right_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   728
  "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   729
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   730
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   731
lemmas tendsto_mult = mult.tendsto
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   732
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   733
lemma tendsto_power [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   734
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   735
  shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   736
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   737
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   738
lemma tendsto_setprod [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   739
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   740
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   741
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   742
proof (cases "finite S")
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   743
  assume "finite S" thus ?thesis using assms
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   744
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   745
next
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   746
  assume "\<not> finite S" thus ?thesis
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   747
    by (simp add: tendsto_const)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   748
qed
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   749
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   750
subsubsection {* Inverse and division *}
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   751
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   752
lemma (in bounded_bilinear) Zfun_prod_Bfun:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   753
  assumes f: "Zfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   754
  assumes g: "Bfun g A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   755
  shows "Zfun (\<lambda>x. f x ** g x) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   756
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   757
  obtain K where K: "0 \<le> K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   758
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   759
    using nonneg_bounded by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   760
  obtain B where B: "0 < B"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   761
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   762
    using g by (rule BfunE)
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   763
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   764
  using norm_g proof (rule eventually_elim1)
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   765
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   766
    assume *: "norm (g x) \<le> B"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   767
    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
   768
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   769
    also have "\<dots> \<le> norm (f x) * B * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   770
      by (intro mult_mono' order_refl norm_g norm_ge_zero
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   771
                mult_nonneg_nonneg K *)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   772
    also have "\<dots> = norm (f x) * (B * K)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   773
      by (rule mult_assoc)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   774
    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
   775
  qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   776
  with f show ?thesis
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   777
    by (rule Zfun_imp_Zfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   778
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   779
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   780
lemma (in bounded_bilinear) flip:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   781
  "bounded_bilinear (\<lambda>x y. y ** x)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   782
  apply default
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   783
  apply (rule add_right)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   784
  apply (rule add_left)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   785
  apply (rule scaleR_right)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   786
  apply (rule scaleR_left)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   787
  apply (subst mult_commute)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   788
  using bounded by fast
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   789
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   790
lemma (in bounded_bilinear) Bfun_prod_Zfun:
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   791
  assumes f: "Bfun f A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   792
  assumes g: "Zfun g A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   793
  shows "Zfun (\<lambda>x. f x ** g x) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   794
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   795
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   796
lemma Bfun_inverse_lemma:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   797
  fixes x :: "'a::real_normed_div_algebra"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   798
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   799
  apply (subst nonzero_norm_inverse, clarsimp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   800
  apply (erule (1) le_imp_inverse_le)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   801
  done
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   802
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   803
lemma Bfun_inverse:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   804
  fixes a :: "'a::real_normed_div_algebra"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   805
  assumes f: "(f ---> a) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   806
  assumes a: "a \<noteq> 0"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   807
  shows "Bfun (\<lambda>x. inverse (f x)) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   808
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   809
  from a have "0 < norm a" by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   810
  hence "\<exists>r>0. r < norm a" by (rule dense)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   811
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   812
  have "eventually (\<lambda>x. dist (f x) a < r) A"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   813
    using tendstoD [OF f r1] by fast
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   814
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   815
  proof (rule eventually_elim1)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   816
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   817
    assume "dist (f x) a < r"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   818
    hence 1: "norm (f x - a) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   819
      by (simp add: dist_norm)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   820
    hence 2: "f x \<noteq> 0" using r2 by auto
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   821
    hence "norm (inverse (f x)) = inverse (norm (f x))"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   822
      by (rule nonzero_norm_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   823
    also have "\<dots> \<le> inverse (norm a - r)"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   824
    proof (rule le_imp_inverse_le)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   825
      show "0 < norm a - r" using r2 by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   826
    next
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   827
      have "norm a - norm (f x) \<le> norm (a - f x)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   828
        by (rule norm_triangle_ineq2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   829
      also have "\<dots> = norm (f x - a)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   830
        by (rule norm_minus_commute)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   831
      also have "\<dots> < r" using 1 .
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   832
      finally show "norm a - r \<le> norm (f x)" by simp
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   833
    qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   834
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   835
  qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   836
  thus ?thesis by (rule BfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   837
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   838
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   839
lemma tendsto_inverse_lemma:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   840
  fixes a :: "'a::real_normed_div_algebra"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   841
  shows "\<lbrakk>(f ---> a) A; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) A\<rbrakk>
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   842
         \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   843
  apply (subst tendsto_Zfun_iff)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   844
  apply (rule Zfun_ssubst)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   845
  apply (erule eventually_elim1)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   846
  apply (erule (1) inverse_diff_inverse)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   847
  apply (rule Zfun_minus)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   848
  apply (rule Zfun_mult_left)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   849
  apply (rule mult.Bfun_prod_Zfun)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   850
  apply (erule (1) Bfun_inverse)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   851
  apply (simp add: tendsto_Zfun_iff)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   852
  done
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   853
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   854
lemma tendsto_inverse [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   855
  fixes a :: "'a::real_normed_div_algebra"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   856
  assumes f: "(f ---> a) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   857
  assumes a: "a \<noteq> 0"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   858
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   859
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   860
  from a have "0 < norm a" by simp
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   861
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   862
    by (rule tendstoD)
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   863
  then have "eventually (\<lambda>x. f x \<noteq> 0) A"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   864
    unfolding dist_norm by (auto elim!: eventually_elim1)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   865
  with f a show ?thesis
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   866
    by (rule tendsto_inverse_lemma)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   867
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   868
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   869
lemma tendsto_divide [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   870
  fixes a b :: "'a::real_normed_field"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   871
  shows "\<lbrakk>(f ---> a) A; (g ---> b) A; b \<noteq> 0\<rbrakk>
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   872
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   873
  by (simp add: mult.tendsto tendsto_inverse divide_inverse)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   874
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   875
lemma tendsto_sgn [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   876
  fixes l :: "'a::real_normed_vector"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   877
  shows "\<lbrakk>(f ---> l) A; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) A"
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   878
  unfolding sgn_div_norm by (simp add: tendsto_intros)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   879
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   880
subsubsection {* Uniqueness *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   881
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   882
lemma tendsto_unique:
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   883
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   884
  assumes "\<not> trivial_limit A"  "(f ---> l) A"  "(f ---> l') A"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   885
  shows "l = l'"
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   886
proof (rule ccontr)
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   887
  assume "l \<noteq> l'"
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   888
  obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   889
    using hausdorff [OF `l \<noteq> l'`] by fast
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   890
  have "eventually (\<lambda>x. f x \<in> U) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   891
    using `(f ---> l) A` `open U` `l \<in> U` by (rule topological_tendstoD)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   892
  moreover
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   893
  have "eventually (\<lambda>x. f x \<in> V) A"
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   894
    using `(f ---> l') A` `open V` `l' \<in> V` by (rule topological_tendstoD)
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   895
  ultimately
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   896
  have "eventually (\<lambda>x. False) A"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   897
  proof (rule eventually_elim2)
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   898
    fix x
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   899
    assume "f x \<in> U" "f x \<in> V"
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   900
    hence "f x \<in> U \<inter> V" by simp
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   901
    with `U \<inter> V = {}` show "False" by simp
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   902
  qed
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   903
  with `\<not> trivial_limit A` show "False"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   904
    by (simp add: trivial_limit_def)
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   905
qed
47d6e13d1710 generalize infinite sums
hoelzl
parents: 39302
diff changeset
   906
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   907
end