src/HOL/Library/Product_Vector.thy
author webertj
Fri, 19 Oct 2012 15:12:52 +0200
changeset 49962 a8cc904a6820
parent 44749 5b1e1432c320
child 51002 496013a6eb38
permissions -rw-r--r--
Renamed {left,right}_distrib to distrib_{right,left}.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     1
(*  Title:      HOL/Library/Product_Vector.thy
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     2
    Author:     Brian Huffman
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     3
*)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     4
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     5
header {* Cartesian Products as Vector Spaces *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     6
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     7
theory Product_Vector
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     8
imports Inner_Product Product_plus
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
     9
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    10
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    11
subsection {* Product is a real vector space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    12
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 36661
diff changeset
    13
instantiation prod :: (real_vector, real_vector) real_vector
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    14
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    15
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    16
definition scaleR_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    17
  "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    18
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    19
lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    20
  unfolding scaleR_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    21
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    22
lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    23
  unfolding scaleR_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    24
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    25
lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    26
  unfolding scaleR_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    27
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    28
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    29
  fix a b :: real and x y :: "'a \<times> 'b"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    30
  show "scaleR a (x + y) = scaleR a x + scaleR a y"
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 37678
diff changeset
    31
    by (simp add: prod_eq_iff scaleR_right_distrib)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    32
  show "scaleR (a + b) x = scaleR a x + scaleR b x"
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 37678
diff changeset
    33
    by (simp add: prod_eq_iff scaleR_left_distrib)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    34
  show "scaleR a (scaleR b x) = scaleR (a * b) x"
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 37678
diff changeset
    35
    by (simp add: prod_eq_iff)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    36
  show "scaleR 1 x = x"
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 37678
diff changeset
    37
    by (simp add: prod_eq_iff)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    38
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    39
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    40
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
    41
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    42
subsection {* Product is a topological space *}
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    43
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 36661
diff changeset
    44
instantiation prod :: (topological_space, topological_space) topological_space
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    45
begin
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    46
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    47
definition open_prod_def:
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    48
  "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    49
    (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    50
36332
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    51
lemma open_prod_elim:
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    52
  assumes "open S" and "x \<in> S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    53
  obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    54
using assms unfolding open_prod_def by fast
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    55
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    56
lemma open_prod_intro:
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    57
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    58
  shows "open S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    59
using assms unfolding open_prod_def by fast
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    60
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    61
instance proof
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    62
  show "open (UNIV :: ('a \<times> 'b) set)"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    63
    unfolding open_prod_def by auto
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    64
next
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    65
  fix S T :: "('a \<times> 'b) set"
36332
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    66
  assume "open S" "open T"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    67
  show "open (S \<inter> T)"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    68
  proof (rule open_prod_intro)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    69
    fix x assume x: "x \<in> S \<inter> T"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    70
    from x have "x \<in> S" by simp
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    71
    obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    72
      using `open S` and `x \<in> S` by (rule open_prod_elim)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    73
    from x have "x \<in> T" by simp
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    74
    obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    75
      using `open T` and `x \<in> T` by (rule open_prod_elim)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    76
    let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    77
    have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    78
      using A B by (auto simp add: open_Int)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    79
    thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    80
      by fast
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
    81
  qed
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    82
next
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    83
  fix K :: "('a \<times> 'b) set set"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    84
  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
    85
    unfolding open_prod_def by fast
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    86
qed
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    87
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    88
end
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
    89
31562
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    90
lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    91
unfolding open_prod_def by auto
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    92
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    93
lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    94
by auto
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    95
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    96
lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    97
by auto
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    98
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
    99
lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
   100
by (simp add: fst_vimage_eq_Times open_Times)
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
   101
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
   102
lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
   103
by (simp add: snd_vimage_eq_Times open_Times)
10d0fb526643 new lemmas
huffman
parents: 31492
diff changeset
   104
31568
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   105
lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   106
unfolding closed_open vimage_Compl [symmetric]
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   107
by (rule open_vimage_fst)
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   108
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   109
lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   110
unfolding closed_open vimage_Compl [symmetric]
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   111
by (rule open_vimage_snd)
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   112
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   113
lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   114
proof -
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   115
  have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   116
  thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   117
    by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   118
qed
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   119
34110
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   120
lemma openI: (* TODO: move *)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   121
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   122
  shows "open S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   123
proof -
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   124
  have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   125
  moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   126
  ultimately show "open S" by simp
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   127
qed
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   128
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   129
lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   130
  unfolding image_def subset_eq by force
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   131
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   132
lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   133
  unfolding image_def subset_eq by force
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   134
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   135
lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   136
proof (rule openI)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   137
  fix x assume "x \<in> fst ` S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   138
  then obtain y where "(x, y) \<in> S" by auto
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   139
  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   140
    using `open S` unfolding open_prod_def by auto
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   141
  from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   142
  with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   143
  then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   144
qed
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   145
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   146
lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   147
proof (rule openI)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   148
  fix y assume "y \<in> snd ` S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   149
  then obtain x where "(x, y) \<in> S" by auto
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   150
  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   151
    using `open S` unfolding open_prod_def by auto
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   152
  from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   153
  with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   154
  then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
4c113c744b86 add lemmas open_image_fst, open_image_snd
huffman
parents: 31590
diff changeset
   155
qed
31568
963caf6fa234 add lemmas about closed sets
huffman
parents: 31565
diff changeset
   156
44575
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   157
subsubsection {* Continuity of operations *}
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   158
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   159
lemma tendsto_fst [tendsto_intros]:
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   160
  assumes "(f ---> a) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   161
  shows "((\<lambda>x. fst (f x)) ---> fst a) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   162
proof (rule topological_tendstoI)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   163
  fix S assume "open S" and "fst a \<in> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   164
  then have "open (fst -` S)" and "a \<in> fst -` S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   165
    by (simp_all add: open_vimage_fst)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   166
  with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   167
    by (rule topological_tendstoD)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   168
  then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   169
    by simp
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   170
qed
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   171
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   172
lemma tendsto_snd [tendsto_intros]:
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   173
  assumes "(f ---> a) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   174
  shows "((\<lambda>x. snd (f x)) ---> snd a) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   175
proof (rule topological_tendstoI)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   176
  fix S assume "open S" and "snd a \<in> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   177
  then have "open (snd -` S)" and "a \<in> snd -` S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   178
    by (simp_all add: open_vimage_snd)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   179
  with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   180
    by (rule topological_tendstoD)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   181
  then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   182
    by simp
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   183
qed
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   184
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   185
lemma tendsto_Pair [tendsto_intros]:
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   186
  assumes "(f ---> a) F" and "(g ---> b) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   187
  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   188
proof (rule topological_tendstoI)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   189
  fix S assume "open S" and "(a, b) \<in> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   190
  then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   191
    unfolding open_prod_def by fast
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   192
  have "eventually (\<lambda>x. f x \<in> A) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   193
    using `(f ---> a) F` `open A` `a \<in> A`
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   194
    by (rule topological_tendstoD)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   195
  moreover
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   196
  have "eventually (\<lambda>x. g x \<in> B) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   197
    using `(g ---> b) F` `open B` `b \<in> B`
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   198
    by (rule topological_tendstoD)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   199
  ultimately
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   200
  show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   201
    by (rule eventually_elim2)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   202
       (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   203
qed
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   204
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   205
lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   206
  unfolding isCont_def by (rule tendsto_fst)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   207
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   208
lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   209
  unfolding isCont_def by (rule tendsto_snd)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   210
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   211
lemma isCont_Pair [simp]:
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   212
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   213
  unfolding isCont_def by (rule tendsto_Pair)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   214
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   215
subsubsection {* Separation axioms *}
44214
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   216
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   217
lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   218
  by (induct x) simp (* TODO: move elsewhere *)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   219
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   220
instance prod :: (t0_space, t0_space) t0_space
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   221
proof
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   222
  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   223
  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   224
    by (simp add: prod_eq_iff)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   225
  thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   226
    apply (rule disjE)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   227
    apply (drule t0_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   228
    apply (simp add: open_Times mem_Times_iff)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   229
    apply (drule t0_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   230
    apply (simp add: open_Times mem_Times_iff)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   231
    done
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   232
qed
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   233
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   234
instance prod :: (t1_space, t1_space) t1_space
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   235
proof
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   236
  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   237
  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   238
    by (simp add: prod_eq_iff)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   239
  thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   240
    apply (rule disjE)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   241
    apply (drule t1_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   242
    apply (simp add: open_Times mem_Times_iff)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   243
    apply (drule t1_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   244
    apply (simp add: open_Times mem_Times_iff)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   245
    done
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   246
qed
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   247
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   248
instance prod :: (t2_space, t2_space) t2_space
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   249
proof
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   250
  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   251
  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   252
    by (simp add: prod_eq_iff)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   253
  thus "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   254
    apply (rule disjE)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   255
    apply (drule hausdorff, clarify)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   256
    apply (rule_tac x="U \<times> UNIV" in exI, rule_tac x="V \<times> UNIV" in exI)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   257
    apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   258
    apply (drule hausdorff, clarify)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   259
    apply (rule_tac x="UNIV \<times> U" in exI, rule_tac x="UNIV \<times> V" in exI)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   260
    apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   261
    done
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   262
qed
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   263
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   264
subsection {* Product is a metric space *}
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   265
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 36661
diff changeset
   266
instantiation prod :: (metric_space, metric_space) metric_space
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   267
begin
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   268
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   269
definition dist_prod_def:
44214
1e0414bda9af Library/Product_Vector.thy: class instances for t0_space, t1_space, and t2_space
huffman
parents: 44127
diff changeset
   270
  "dist x y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   271
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   272
lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   273
  unfolding dist_prod_def by simp
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   274
36332
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   275
lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   276
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   277
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   278
lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   279
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   280
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   281
instance proof
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   282
  fix x y :: "'a \<times> 'b"
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   283
  show "dist x y = 0 \<longleftrightarrow> x = y"
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 37678
diff changeset
   284
    unfolding dist_prod_def prod_eq_iff by simp
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   285
next
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   286
  fix x y z :: "'a \<times> 'b"
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   287
  show "dist x y \<le> dist x z + dist y z"
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   288
    unfolding dist_prod_def
31563
ded2364d14d4 cleaned up some proofs
huffman
parents: 31562
diff changeset
   289
    by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
ded2364d14d4 cleaned up some proofs
huffman
parents: 31562
diff changeset
   290
        real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
31415
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
   291
next
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   292
  fix S :: "('a \<times> 'b) set"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   293
  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
31563
ded2364d14d4 cleaned up some proofs
huffman
parents: 31562
diff changeset
   294
  proof
36332
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   295
    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   296
    proof
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   297
      fix x assume "x \<in> S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   298
      obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   299
        using `open S` and `x \<in> S` by (rule open_prod_elim)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   300
      obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   301
        using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   302
      obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   303
        using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   304
      let ?e = "min r s"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   305
      have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   306
      proof (intro allI impI conjI)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   307
        show "0 < min r s" by (simp add: r(1) s(1))
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   308
      next
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   309
        fix y assume "dist y x < min r s"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   310
        hence "dist y x < r" and "dist y x < s"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   311
          by simp_all
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   312
        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   313
          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   314
        hence "fst y \<in> A" and "snd y \<in> B"
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   315
          by (simp_all add: r(2) s(2))
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   316
        hence "y \<in> A \<times> B" by (induct y, simp)
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   317
        with `A \<times> B \<subseteq> S` show "y \<in> S" ..
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   318
      qed
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   319
      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
3ddb2bc07784 convert proofs to Isar-style
huffman
parents: 34110
diff changeset
   320
    qed
31563
ded2364d14d4 cleaned up some proofs
huffman
parents: 31562
diff changeset
   321
  next
44575
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   322
    assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   323
    proof (rule open_prod_intro)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   324
      fix x assume "x \<in> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   325
      then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   326
        using * by fast
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   327
      def r \<equiv> "e / sqrt 2" and s \<equiv> "e / sqrt 2"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   328
      from `0 < e` have "0 < r" and "0 < s"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   329
        unfolding r_def s_def by (simp_all add: divide_pos_pos)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   330
      from `0 < e` have "e = sqrt (r\<twosuperior> + s\<twosuperior>)"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   331
        unfolding r_def s_def by (simp add: power_divide)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   332
      def A \<equiv> "{y. dist (fst x) y < r}" and B \<equiv> "{y. dist (snd x) y < s}"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   333
      have "open A" and "open B"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   334
        unfolding A_def B_def by (simp_all add: open_ball)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   335
      moreover have "x \<in> A \<times> B"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   336
        unfolding A_def B_def mem_Times_iff
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   337
        using `0 < r` and `0 < s` by simp
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   338
      moreover have "A \<times> B \<subseteq> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   339
      proof (clarify)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   340
        fix a b assume "a \<in> A" and "b \<in> B"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   341
        hence "dist a (fst x) < r" and "dist b (snd x) < s"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   342
          unfolding A_def B_def by (simp_all add: dist_commute)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   343
        hence "dist (a, b) x < e"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   344
          unfolding dist_prod_def `e = sqrt (r\<twosuperior> + s\<twosuperior>)`
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   345
          by (simp add: add_strict_mono power_strict_mono)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   346
        thus "(a, b) \<in> S"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   347
          by (simp add: S)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   348
      qed
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   349
      ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   350
    qed
31563
ded2364d14d4 cleaned up some proofs
huffman
parents: 31562
diff changeset
   351
  qed
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   352
qed
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   353
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   354
end
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   355
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   356
lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   357
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   358
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   359
lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   360
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   361
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   362
lemma Cauchy_Pair:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   363
  assumes "Cauchy X" and "Cauchy Y"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   364
  shows "Cauchy (\<lambda>n. (X n, Y n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   365
proof (rule metric_CauchyI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   366
  fix r :: real assume "0 < r"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   367
  then have "0 < r / sqrt 2" (is "0 < ?s")
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   368
    by (simp add: divide_pos_pos)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   369
  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   370
    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   371
  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   372
    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   373
  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   374
    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   375
  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   376
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   377
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   378
subsection {* Product is a complete metric space *}
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   379
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 36661
diff changeset
   380
instance prod :: (complete_space, complete_space) complete_space
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   381
proof
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   382
  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   383
  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   384
    using Cauchy_fst [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   385
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   386
  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   387
    using Cauchy_snd [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   388
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   389
  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
36660
1cc4ab4b7ff7 make (X ----> L) an abbreviation for (X ---> L) sequentially
huffman
parents: 36332
diff changeset
   390
    using tendsto_Pair [OF 1 2] by simp
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   391
  then show "convergent X"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   392
    by (rule convergentI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   393
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   394
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   395
subsection {* Product is a normed vector space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   396
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 36661
diff changeset
   397
instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   398
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   399
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   400
definition norm_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   401
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   402
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   403
definition sgn_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   404
  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   405
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   406
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   407
  unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   408
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   409
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   410
  fix r :: real and x y :: "'a \<times> 'b"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   411
  show "0 \<le> norm x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   412
    unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   413
  show "norm x = 0 \<longleftrightarrow> x = 0"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   414
    unfolding norm_prod_def
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 37678
diff changeset
   415
    by (simp add: prod_eq_iff)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   416
  show "norm (x + y) \<le> norm x + norm y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   417
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   418
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   419
    apply (simp add: add_mono power_mono norm_triangle_ineq)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   420
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   421
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   422
    unfolding norm_prod_def
31587
a7e187205fc0 remove simp add: norm_scaleR
huffman
parents: 31568
diff changeset
   423
    apply (simp add: power_mult_distrib)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44749
diff changeset
   424
    apply (simp add: distrib_left [symmetric])
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   425
    apply (simp add: real_sqrt_mult_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   426
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   427
  show "sgn x = scaleR (inverse (norm x)) x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   428
    by (rule sgn_prod_def)
31290
f41c023d90bc define dist for products
huffman
parents: 30729
diff changeset
   429
  show "dist x y = norm (x - y)"
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   430
    unfolding dist_prod_def norm_prod_def
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   431
    by (simp add: dist_norm)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   432
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   433
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   434
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   435
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 36661
diff changeset
   436
instance prod :: (banach, banach) banach ..
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   437
44575
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   438
subsubsection {* Pair operations are linear *}
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   439
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44233
diff changeset
   440
lemma bounded_linear_fst: "bounded_linear fst"
44127
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 44126
diff changeset
   441
  using fst_add fst_scaleR
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 44126
diff changeset
   442
  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   443
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44233
diff changeset
   444
lemma bounded_linear_snd: "bounded_linear snd"
44127
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 44126
diff changeset
   445
  using snd_add snd_scaleR
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 44126
diff changeset
   446
  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   447
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   448
text {* TODO: move to NthRoot *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   449
lemma sqrt_add_le_add_sqrt:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   450
  assumes x: "0 \<le> x" and y: "0 \<le> y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   451
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   452
apply (rule power2_le_imp_le)
44749
5b1e1432c320 remove redundant lemma real_sum_squared_expand in favor of power2_sum
huffman
parents: 44575
diff changeset
   453
apply (simp add: power2_sum x y)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   454
apply (simp add: mult_nonneg_nonneg x y)
44126
ce44e70d0c47 avoid duplicate rewrite warnings
huffman
parents: 44066
diff changeset
   455
apply (simp add: x y)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   456
done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   457
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   458
lemma bounded_linear_Pair:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   459
  assumes f: "bounded_linear f"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   460
  assumes g: "bounded_linear g"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   461
  shows "bounded_linear (\<lambda>x. (f x, g x))"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   462
proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   463
  interpret f: bounded_linear f by fact
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   464
  interpret g: bounded_linear g by fact
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   465
  fix x y and r :: real
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   466
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   467
    by (simp add: f.add g.add)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   468
  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   469
    by (simp add: f.scaleR g.scaleR)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   470
  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   471
    using f.pos_bounded by fast
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   472
  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   473
    using g.pos_bounded by fast
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   474
  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   475
    apply (rule allI)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   476
    apply (simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   477
    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44749
diff changeset
   478
    apply (simp add: distrib_left)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   479
    apply (rule add_mono [OF norm_f norm_g])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   480
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   481
  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   482
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   483
44575
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   484
subsubsection {* Frechet derivatives involving pairs *}
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   485
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   486
lemma FDERIV_Pair:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   487
  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   488
  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
44575
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   489
proof (rule FDERIV_I)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   490
  show "bounded_linear (\<lambda>h. (f' h, g' h))"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   491
    using f g by (intro bounded_linear_Pair FDERIV_bounded_linear)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   492
  let ?Rf = "\<lambda>h. f (x + h) - f x - f' h"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   493
  let ?Rg = "\<lambda>h. g (x + h) - g x - g' h"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   494
  let ?R = "\<lambda>h. ((f (x + h), g (x + h)) - (f x, g x) - (f' h, g' h))"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   495
  show "(\<lambda>h. norm (?R h) / norm h) -- 0 --> 0"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   496
  proof (rule real_LIM_sandwich_zero)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   497
    show "(\<lambda>h. norm (?Rf h) / norm h + norm (?Rg h) / norm h) -- 0 --> 0"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   498
      using f g by (intro tendsto_add_zero FDERIV_D)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   499
    fix h :: 'a assume "h \<noteq> 0"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   500
    thus "0 \<le> norm (?R h) / norm h"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   501
      by (simp add: divide_nonneg_pos)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   502
    show "norm (?R h) / norm h \<le> norm (?Rf h) / norm h + norm (?Rg h) / norm h"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   503
      unfolding add_divide_distrib [symmetric]
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   504
      by (simp add: norm_Pair divide_right_mono
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   505
        order_trans [OF sqrt_add_le_add_sqrt])
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   506
  qed
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   507
qed
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   508
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   509
subsection {* Product is an inner product space *}
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   510
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   511
instantiation prod :: (real_inner, real_inner) real_inner
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   512
begin
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   513
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   514
definition inner_prod_def:
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   515
  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   516
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   517
lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   518
  unfolding inner_prod_def by simp
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   519
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   520
instance proof
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   521
  fix r :: real
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   522
  fix x y z :: "'a::real_inner \<times> 'b::real_inner"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   523
  show "inner x y = inner y x"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   524
    unfolding inner_prod_def
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   525
    by (simp add: inner_commute)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   526
  show "inner (x + y) z = inner x z + inner y z"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   527
    unfolding inner_prod_def
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   528
    by (simp add: inner_add_left)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   529
  show "inner (scaleR r x) y = r * inner x y"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   530
    unfolding inner_prod_def
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44749
diff changeset
   531
    by (simp add: distrib_left)
44575
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   532
  show "0 \<le> inner x x"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   533
    unfolding inner_prod_def
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   534
    by (intro add_nonneg_nonneg inner_ge_zero)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   535
  show "inner x x = 0 \<longleftrightarrow> x = 0"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   536
    unfolding inner_prod_def prod_eq_iff
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   537
    by (simp add: add_nonneg_eq_0_iff)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   538
  show "norm x = sqrt (inner x x)"
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   539
    unfolding norm_prod_def inner_prod_def
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   540
    by (simp add: power2_norm_eq_inner)
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   541
qed
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   542
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   543
end
44575
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   544
c5e42b8590dd Product_Vector.thy: clean up some proofs
huffman
parents: 44568
diff changeset
   545
end