src/HOL/Analysis/Topology_Euclidean_Space.thy
author paulson <lp15@cam.ac.uk>
Mon Mar 18 15:35:34 2019 +0000 (2 months ago)
changeset 69918 eddcc7c726f3
parent 69750 7d83b0abbfd7
child 69922 4a9167f377b0
permissions -rw-r--r--
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
lp15@63938
     1
(*  Author:     L C Paulson, University of Cambridge
himmelma@33175
     2
    Author:     Amine Chaieb, University of Cambridge
himmelma@33175
     3
    Author:     Robert Himmelmann, TU Muenchen
huffman@44075
     4
    Author:     Brian Huffman, Portland State University
himmelma@33175
     5
*)
himmelma@33175
     6
immler@69676
     7
chapter \<open>Vector Analysis\<close>
himmelma@33175
     8
himmelma@33175
     9
theory Topology_Euclidean_Space
immler@69516
    10
  imports
immler@69544
    11
    Elementary_Normed_Spaces
immler@69516
    12
    Linear_Algebra
immler@69516
    13
    Norm_Arith
hoelzl@51343
    14
begin
hoelzl@51343
    15
immler@69676
    16
section \<open>Elementary Topology in Euclidean Space\<close>
immler@69676
    17
hoelzl@50526
    18
lemma euclidean_dist_l2:
hoelzl@50526
    19
  fixes x y :: "'a :: euclidean_space"
nipkow@67155
    20
  shows "dist x y = L2_set (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
nipkow@67155
    21
  unfolding dist_norm norm_eq_sqrt_inner L2_set_def
hoelzl@50526
    22
  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
hoelzl@50526
    23
immler@67685
    24
lemma norm_nth_le: "norm (x \<bullet> i) \<le> norm x" if "i \<in> Basis"
immler@67685
    25
proof -
immler@67685
    26
  have "(x \<bullet> i)\<^sup>2 = (\<Sum>i\<in>{i}. (x \<bullet> i)\<^sup>2)"
immler@67685
    27
    by simp
immler@67685
    28
  also have "\<dots> \<le> (\<Sum>i\<in>Basis. (x \<bullet> i)\<^sup>2)"
immler@67685
    29
    by (intro sum_mono2) (auto simp: that)
immler@67685
    30
  finally show ?thesis
immler@67685
    31
    unfolding norm_conv_dist euclidean_dist_l2[of x] L2_set_def
immler@67685
    32
    by (auto intro!: real_le_rsqrt)
immler@67685
    33
qed
immler@67685
    34
immler@67685
    35
immler@69613
    36
subsection%unimportant\<open>Balls in Euclidean Space\<close>
immler@69613
    37
immler@69613
    38
lemma cball_subset_cball_iff:
immler@69613
    39
  fixes a :: "'a :: euclidean_space"
immler@69613
    40
  shows "cball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r < 0"
immler@69613
    41
    (is "?lhs \<longleftrightarrow> ?rhs")
immler@69613
    42
proof
immler@69613
    43
  assume ?lhs
immler@69613
    44
  then show ?rhs
immler@69613
    45
  proof (cases "r < 0")
immler@69613
    46
    case True
immler@69613
    47
    then show ?rhs by simp
immler@69613
    48
  next
immler@69613
    49
    case False
immler@69613
    50
    then have [simp]: "r \<ge> 0" by simp
immler@69613
    51
    have "norm (a - a') + r \<le> r'"
immler@69613
    52
    proof (cases "a = a'")
immler@69613
    53
      case True
immler@69613
    54
      then show ?thesis
immler@69613
    55
        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
immler@69613
    56
        by (force simp: SOME_Basis dist_norm)
immler@69613
    57
    next
immler@69613
    58
      case False
immler@69613
    59
      have "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = norm (a' - a - (r / norm (a - a')) *\<^sub>R (a - a'))"
immler@69613
    60
        by (simp add: algebra_simps)
immler@69613
    61
      also have "... = norm ((-1 - (r / norm (a - a'))) *\<^sub>R (a - a'))"
immler@69613
    62
        by (simp add: algebra_simps)
immler@69613
    63
      also from \<open>a \<noteq> a'\<close> have "... = \<bar>- norm (a - a') - r\<bar>"
immler@69613
    64
        by (simp add: abs_mult_pos field_simps)
immler@69613
    65
      finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \<bar>norm (a - a') + r\<bar>"
immler@69613
    66
        by linarith
immler@69613
    67
      from \<open>a \<noteq> a'\<close> show ?thesis
immler@69613
    68
        using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>]
immler@69613
    69
        by (simp add: dist_norm scaleR_add_left)
immler@69613
    70
    qed
immler@69613
    71
    then show ?rhs
immler@69613
    72
      by (simp add: dist_norm)
immler@69613
    73
  qed
immler@69613
    74
next
immler@69613
    75
  assume ?rhs
immler@69613
    76
  then show ?lhs
immler@69613
    77
    by (auto simp: ball_def dist_norm)
immler@69613
    78
      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 order_trans)
immler@69613
    79
qed
immler@69613
    80
immler@69613
    81
lemma cball_subset_ball_iff: "cball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r < r' \<or> r < 0"
immler@69613
    82
  (is "?lhs \<longleftrightarrow> ?rhs")
immler@69613
    83
  for a :: "'a::euclidean_space"
immler@69613
    84
proof
immler@69613
    85
  assume ?lhs
immler@69613
    86
  then show ?rhs
immler@69613
    87
  proof (cases "r < 0")
immler@69613
    88
    case True then
immler@69613
    89
    show ?rhs by simp
immler@69613
    90
  next
immler@69613
    91
    case False
immler@69613
    92
    then have [simp]: "r \<ge> 0" by simp
immler@69613
    93
    have "norm (a - a') + r < r'"
immler@69613
    94
    proof (cases "a = a'")
immler@69613
    95
      case True
immler@69613
    96
      then show ?thesis
immler@69613
    97
        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
immler@69613
    98
        by (force simp: SOME_Basis dist_norm)
immler@69613
    99
    next
immler@69613
   100
      case False
immler@69613
   101
      have False if "norm (a - a') + r \<ge> r'"
immler@69613
   102
      proof -
immler@69613
   103
        from that have "\<bar>r' - norm (a - a')\<bar> \<le> r"
immler@69613
   104
          by (simp split: abs_split)
immler@69613
   105
            (metis \<open>0 \<le> r\<close> \<open>?lhs\<close> centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
immler@69613
   106
        then show ?thesis
immler@69613
   107
          using subsetD [where c = "a + (r' / norm(a - a') - 1) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>
immler@69613
   108
          by (simp add: dist_norm field_simps)
immler@69613
   109
            (simp add: diff_divide_distrib scaleR_left_diff_distrib)
immler@69613
   110
      qed
immler@69613
   111
      then show ?thesis by force
immler@69613
   112
    qed
immler@69613
   113
    then show ?rhs by (simp add: dist_norm)
immler@69613
   114
  qed
immler@69613
   115
next
immler@69613
   116
  assume ?rhs
immler@69613
   117
  then show ?lhs
immler@69613
   118
    by (auto simp: ball_def dist_norm)
immler@69613
   119
      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 le_less_trans)
immler@69613
   120
qed
immler@69613
   121
immler@69613
   122
lemma ball_subset_cball_iff: "ball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
immler@69613
   123
  (is "?lhs = ?rhs")
immler@69613
   124
  for a :: "'a::euclidean_space"
immler@69613
   125
proof (cases "r \<le> 0")
immler@69613
   126
  case True
immler@69613
   127
  then show ?thesis
immler@69613
   128
    using dist_not_less_zero less_le_trans by force
immler@69613
   129
next
immler@69613
   130
  case False
immler@69613
   131
  show ?thesis
immler@69613
   132
  proof
immler@69613
   133
    assume ?lhs
immler@69613
   134
    then have "(cball a r \<subseteq> cball a' r')"
immler@69613
   135
      by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
immler@69613
   136
    with False show ?rhs
immler@69613
   137
      by (fastforce iff: cball_subset_cball_iff)
immler@69613
   138
  next
immler@69613
   139
    assume ?rhs
immler@69613
   140
    with False show ?lhs
immler@69613
   141
      using ball_subset_cball cball_subset_cball_iff by blast
immler@69613
   142
  qed
immler@69613
   143
qed
immler@69613
   144
immler@69613
   145
lemma ball_subset_ball_iff:
immler@69613
   146
  fixes a :: "'a :: euclidean_space"
immler@69613
   147
  shows "ball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
immler@69613
   148
        (is "?lhs = ?rhs")
immler@69613
   149
proof (cases "r \<le> 0")
immler@69613
   150
  case True then show ?thesis
immler@69613
   151
    using dist_not_less_zero less_le_trans by force
immler@69613
   152
next
immler@69613
   153
  case False show ?thesis
immler@69613
   154
  proof
immler@69613
   155
    assume ?lhs
immler@69613
   156
    then have "0 < r'"
lp15@69712
   157
      by (metis (no_types) False \<open>?lhs\<close> centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less subsetD)
immler@69613
   158
    then have "(cball a r \<subseteq> cball a' r')"
immler@69613
   159
      by (metis False\<open>?lhs\<close> closure_ball closure_mono not_less)
immler@69613
   160
    then show ?rhs
immler@69613
   161
      using False cball_subset_cball_iff by fastforce
immler@69613
   162
  next
immler@69613
   163
  assume ?rhs then show ?lhs
immler@69613
   164
    apply (auto simp: ball_def)
immler@69613
   165
    apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)
immler@69613
   166
    using dist_not_less_zero order.strict_trans2 apply blast
immler@69613
   167
    done
immler@69613
   168
  qed
immler@69613
   169
qed
immler@69613
   170
immler@69613
   171
immler@69613
   172
lemma ball_eq_ball_iff:
immler@69613
   173
  fixes x :: "'a :: euclidean_space"
immler@69613
   174
  shows "ball x d = ball y e \<longleftrightarrow> d \<le> 0 \<and> e \<le> 0 \<or> x=y \<and> d=e"
immler@69613
   175
        (is "?lhs = ?rhs")
immler@69613
   176
proof
immler@69613
   177
  assume ?lhs
immler@69613
   178
  then show ?rhs
immler@69613
   179
  proof (cases "d \<le> 0 \<or> e \<le> 0")
immler@69613
   180
    case True
immler@69613
   181
      with \<open>?lhs\<close> show ?rhs
immler@69613
   182
        by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
immler@69613
   183
  next
immler@69613
   184
    case False
immler@69613
   185
    with \<open>?lhs\<close> show ?rhs
immler@69613
   186
      apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
immler@69613
   187
      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
immler@69613
   188
      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
immler@69613
   189
      done
immler@69613
   190
  qed
immler@69613
   191
next
immler@69613
   192
  assume ?rhs then show ?lhs
immler@69613
   193
    by (auto simp: set_eq_subset ball_subset_ball_iff)
immler@69613
   194
qed
immler@69613
   195
immler@69613
   196
lemma cball_eq_cball_iff:
immler@69613
   197
  fixes x :: "'a :: euclidean_space"
immler@69613
   198
  shows "cball x d = cball y e \<longleftrightarrow> d < 0 \<and> e < 0 \<or> x=y \<and> d=e"
immler@69613
   199
        (is "?lhs = ?rhs")
immler@69613
   200
proof
immler@69613
   201
  assume ?lhs
immler@69613
   202
  then show ?rhs
immler@69613
   203
  proof (cases "d < 0 \<or> e < 0")
immler@69613
   204
    case True
immler@69613
   205
      with \<open>?lhs\<close> show ?rhs
immler@69613
   206
        by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
immler@69613
   207
  next
immler@69613
   208
    case False
immler@69613
   209
    with \<open>?lhs\<close> show ?rhs
immler@69613
   210
      apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
immler@69613
   211
      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
immler@69613
   212
      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
immler@69613
   213
      done
immler@69613
   214
  qed
immler@69613
   215
next
immler@69613
   216
  assume ?rhs then show ?lhs
immler@69613
   217
    by (auto simp: set_eq_subset cball_subset_cball_iff)
immler@69613
   218
qed
immler@69613
   219
immler@69613
   220
lemma ball_eq_cball_iff:
immler@69613
   221
  fixes x :: "'a :: euclidean_space"
immler@69613
   222
  shows "ball x d = cball y e \<longleftrightarrow> d \<le> 0 \<and> e < 0" (is "?lhs = ?rhs")
immler@69613
   223
proof
immler@69613
   224
  assume ?lhs
immler@69613
   225
  then show ?rhs
immler@69613
   226
    apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
immler@69613
   227
    apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
immler@69613
   228
    apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
immler@69613
   229
    using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
immler@69613
   230
    done
immler@69613
   231
next
immler@69613
   232
  assume ?rhs then show ?lhs by auto
immler@69613
   233
qed
immler@69613
   234
immler@69613
   235
lemma cball_eq_ball_iff:
immler@69613
   236
  fixes x :: "'a :: euclidean_space"
immler@69613
   237
  shows "cball x d = ball y e \<longleftrightarrow> d < 0 \<and> e \<le> 0"
immler@69613
   238
  using ball_eq_cball_iff by blast
immler@69613
   239
immler@69613
   240
lemma finite_ball_avoid:
immler@69613
   241
  fixes S :: "'a :: euclidean_space set"
immler@69613
   242
  assumes "open S" "finite X" "p \<in> S"
immler@69613
   243
  shows "\<exists>e>0. \<forall>w\<in>ball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
immler@69613
   244
proof -
immler@69613
   245
  obtain e1 where "0 < e1" and e1_b:"ball p e1 \<subseteq> S"
immler@69613
   246
    using open_contains_ball_eq[OF \<open>open S\<close>] assms by auto
immler@69613
   247
  obtain e2 where "0 < e2" and "\<forall>x\<in>X. x \<noteq> p \<longrightarrow> e2 \<le> dist p x"
immler@69613
   248
    using finite_set_avoid[OF \<open>finite X\<close>,of p] by auto
immler@69613
   249
  hence "\<forall>w\<in>ball p (min e1 e2). w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)" using e1_b by auto
immler@69613
   250
  thus "\<exists>e>0. \<forall>w\<in>ball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> \<open>e1>0\<close>
immler@69613
   251
    apply (rule_tac x="min e1 e2" in exI)
immler@69613
   252
    by auto
immler@69613
   253
qed
immler@69613
   254
immler@69613
   255
lemma finite_cball_avoid:
immler@69613
   256
  fixes S :: "'a :: euclidean_space set"
immler@69613
   257
  assumes "open S" "finite X" "p \<in> S"
immler@69613
   258
  shows "\<exists>e>0. \<forall>w\<in>cball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
immler@69613
   259
proof -
immler@69613
   260
  obtain e1 where "e1>0" and e1: "\<forall>w\<in>ball p e1. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
immler@69613
   261
    using finite_ball_avoid[OF assms] by auto
immler@69613
   262
  define e2 where "e2 \<equiv> e1/2"
immler@69613
   263
  have "e2>0" and "e2 < e1" unfolding e2_def using \<open>e1>0\<close> by auto
immler@69613
   264
  then have "cball p e2 \<subseteq> ball p e1" by (subst cball_subset_ball_iff,auto)
immler@69613
   265
  then show "\<exists>e>0. \<forall>w\<in>cball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> e1 by auto
immler@69613
   266
qed
immler@69613
   267
immler@69619
   268
lemma dim_cball:
immler@69619
   269
  assumes "e > 0"
immler@69619
   270
  shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
immler@69619
   271
proof -
immler@69619
   272
  {
immler@69619
   273
    fix x :: "'n::euclidean_space"
immler@69619
   274
    define y where "y = (e / norm x) *\<^sub>R x"
immler@69619
   275
    then have "y \<in> cball 0 e"
immler@69619
   276
      using assms by auto
immler@69619
   277
    moreover have *: "x = (norm x / e) *\<^sub>R y"
immler@69619
   278
      using y_def assms by simp
immler@69619
   279
    moreover from * have "x = (norm x/e) *\<^sub>R y"
immler@69619
   280
      by auto
immler@69619
   281
    ultimately have "x \<in> span (cball 0 e)"
immler@69619
   282
      using span_scale[of y "cball 0 e" "norm x/e"]
immler@69619
   283
        span_superset[of "cball 0 e"]
immler@69619
   284
      by (simp add: span_base)
immler@69619
   285
  }
immler@69619
   286
  then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
immler@69619
   287
    by auto
immler@69619
   288
  then show ?thesis
immler@69619
   289
    using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp: dim_UNIV)
immler@69619
   290
qed
immler@69619
   291
immler@69613
   292
wenzelm@60420
   293
subsection \<open>Boxes\<close>
immler@56189
   294
hoelzl@57447
   295
abbreviation One :: "'a::euclidean_space"
hoelzl@57447
   296
  where "One \<equiv> \<Sum>Basis"
hoelzl@57447
   297
lp15@63114
   298
lemma One_non_0: assumes "One = (0::'a::euclidean_space)" shows False
lp15@63114
   299
proof -
lp15@63114
   300
  have "dependent (Basis :: 'a set)"
lp15@63114
   301
    apply (simp add: dependent_finite)
lp15@63114
   302
    apply (rule_tac x="\<lambda>i. 1" in exI)
lp15@63114
   303
    using SOME_Basis apply (auto simp: assms)
lp15@63114
   304
    done
lp15@63114
   305
  with independent_Basis show False by force
lp15@63114
   306
qed
lp15@63114
   307
lp15@63114
   308
corollary One_neq_0[iff]: "One \<noteq> 0"
lp15@63114
   309
  by (metis One_non_0)
lp15@63114
   310
lp15@63114
   311
corollary Zero_neq_One[iff]: "0 \<noteq> One"
lp15@63114
   312
  by (metis One_non_0)
lp15@63114
   313
immler@67962
   314
definition%important (in euclidean_space) eucl_less (infix "<e" 50)
immler@54775
   315
  where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
immler@54775
   316
immler@67962
   317
definition%important box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"
immler@67962
   318
definition%important "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
immler@54775
   319
immler@54775
   320
lemma box_def: "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
immler@54775
   321
  and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"
immler@56188
   322
  and mem_box: "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"
immler@56188
   323
    "x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
immler@56188
   324
  by (auto simp: box_eucl_less eucl_less_def cbox_def)
immler@56188
   325
lp15@60615
   326
lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b \<times> cbox c d"
lp15@60615
   327
  by (force simp: cbox_def Basis_prod_def)
lp15@60615
   328
lp15@60615
   329
lemma cbox_Pair_iff [iff]: "(x, y) \<in> cbox (a, c) (b, d) \<longleftrightarrow> x \<in> cbox a b \<and> y \<in> cbox c d"
lp15@60615
   330
  by (force simp: cbox_Pair_eq)
lp15@60615
   331
lp15@65587
   332
lemma cbox_Complex_eq: "cbox (Complex a c) (Complex b d) = (\<lambda>(x,y). Complex x y) ` (cbox a b \<times> cbox c d)"
lp15@65587
   333
  apply (auto simp: cbox_def Basis_complex_def)
lp15@65587
   334
  apply (rule_tac x = "(Re x, Im x)" in image_eqI)
lp15@65587
   335
  using complex_eq by auto
lp15@65587
   336
lp15@60615
   337
lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} \<longleftrightarrow> cbox a b = {} \<or> cbox c d = {}"
lp15@60615
   338
  by (force simp: cbox_Pair_eq)
lp15@60615
   339
lp15@60615
   340
lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)"
lp15@60615
   341
  by auto
lp15@60615
   342
immler@56188
   343
lemma mem_box_real[simp]:
immler@56188
   344
  "(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b"
immler@56188
   345
  "(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b"
immler@56188
   346
  by (auto simp: mem_box)
immler@56188
   347
immler@56188
   348
lemma box_real[simp]:
immler@56188
   349
  fixes a b:: real
immler@56188
   350
  shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
immler@56188
   351
  by auto
hoelzl@50526
   352
hoelzl@57447
   353
lemma box_Int_box:
hoelzl@57447
   354
  fixes a :: "'a::euclidean_space"
hoelzl@57447
   355
  shows "box a b \<inter> box c d =
hoelzl@57447
   356
    box (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
hoelzl@57447
   357
  unfolding set_eq_iff and Int_iff and mem_box by auto
hoelzl@57447
   358
immler@50087
   359
lemma rational_boxes:
wenzelm@61076
   360
  fixes x :: "'a::euclidean_space"
wenzelm@53291
   361
  assumes "e > 0"
lp15@66643
   362
  shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat>) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
immler@50087
   363
proof -
wenzelm@63040
   364
  define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
wenzelm@53291
   365
  then have e: "e' > 0"
nipkow@56541
   366
    using assms by (auto simp: DIM_positive)
hoelzl@50526
   367
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
immler@50087
   368
  proof
wenzelm@53255
   369
    fix i
wenzelm@53255
   370
    from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
wenzelm@53255
   371
    show "?th i" by auto
immler@50087
   372
  qed
wenzelm@55522
   373
  from choice[OF this] obtain a where
wenzelm@55522
   374
    a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'" ..
hoelzl@50526
   375
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
immler@50087
   376
  proof
wenzelm@53255
   377
    fix i
wenzelm@53255
   378
    from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
wenzelm@53255
   379
    show "?th i" by auto
immler@50087
   380
  qed
wenzelm@55522
   381
  from choice[OF this] obtain b where
wenzelm@55522
   382
    b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'" ..
hoelzl@50526
   383
  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
hoelzl@50526
   384
  show ?thesis
hoelzl@50526
   385
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
wenzelm@53255
   386
    fix y :: 'a
wenzelm@53255
   387
    assume *: "y \<in> box ?a ?b"
wenzelm@53015
   388
    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
nipkow@67155
   389
      unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
hoelzl@50526
   390
    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
nipkow@64267
   391
    proof (rule real_sqrt_less_mono, rule sum_strict_mono)
wenzelm@53255
   392
      fix i :: "'a"
wenzelm@53255
   393
      assume i: "i \<in> Basis"
wenzelm@53255
   394
      have "a i < y\<bullet>i \<and> y\<bullet>i < b i"
wenzelm@53255
   395
        using * i by (auto simp: box_def)
wenzelm@53255
   396
      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
wenzelm@53255
   397
        using a by auto
wenzelm@53255
   398
      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
wenzelm@53255
   399
        using b by auto
wenzelm@53255
   400
      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
wenzelm@53255
   401
        by auto
hoelzl@50526
   402
      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
immler@50087
   403
        unfolding e'_def by (auto simp: dist_real_def)
wenzelm@53015
   404
      then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
immler@50087
   405
        by (rule power_strict_mono) auto
wenzelm@53015
   406
      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
immler@50087
   407
        by (simp add: power_divide)
immler@50087
   408
    qed auto
wenzelm@53255
   409
    also have "\<dots> = e"
lp15@61609
   410
      using \<open>0 < e\<close> by simp
wenzelm@53255
   411
    finally show "y \<in> ball x e"
wenzelm@53255
   412
      by (auto simp: ball_def)
hoelzl@50526
   413
  qed (insert a b, auto simp: box_def)
hoelzl@50526
   414
qed
immler@51103
   415
hoelzl@50526
   416
lemma open_UNION_box:
wenzelm@61076
   417
  fixes M :: "'a::euclidean_space set"
wenzelm@53282
   418
  assumes "open M"
hoelzl@50526
   419
  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
hoelzl@50526
   420
  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
wenzelm@53015
   421
  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
hoelzl@50526
   422
  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
wenzelm@52624
   423
proof -
wenzelm@60462
   424
  have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))" if "x \<in> M" for x
wenzelm@60462
   425
  proof -
wenzelm@52624
   426
    obtain e where e: "e > 0" "ball x e \<subseteq> M"
wenzelm@60420
   427
      using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
wenzelm@53282
   428
    moreover obtain a b where ab:
wenzelm@53282
   429
      "x \<in> box a b"
wenzelm@53282
   430
      "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
wenzelm@53282
   431
      "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"
wenzelm@53282
   432
      "box a b \<subseteq> ball x e"
wenzelm@52624
   433
      using rational_boxes[OF e(1)] by metis
wenzelm@60462
   434
    ultimately show ?thesis
wenzelm@52624
   435
       by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
wenzelm@52624
   436
          (auto simp: euclidean_representation I_def a'_def b'_def)
wenzelm@60462
   437
  qed
wenzelm@52624
   438
  then show ?thesis by (auto simp: I_def)
wenzelm@52624
   439
qed
wenzelm@52624
   440
lp15@66154
   441
corollary open_countable_Union_open_box:
lp15@66154
   442
  fixes S :: "'a :: euclidean_space set"
lp15@66154
   443
  assumes "open S"
lp15@66154
   444
  obtains \<D> where "countable \<D>" "\<D> \<subseteq> Pow S" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" "\<Union>\<D> = S"
lp15@66154
   445
proof -
lp15@66154
   446
  let ?a = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
lp15@66154
   447
  let ?b = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
lp15@66154
   448
  let ?I = "{f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (?a f) (?b f) \<subseteq> S}"
lp15@66154
   449
  let ?\<D> = "(\<lambda>f. box (?a f) (?b f)) ` ?I"
lp15@66154
   450
  show ?thesis
lp15@66154
   451
  proof
lp15@66154
   452
    have "countable ?I"
lp15@66154
   453
      by (simp add: countable_PiE countable_rat)
lp15@66154
   454
    then show "countable ?\<D>"
lp15@66154
   455
      by blast
lp15@66154
   456
    show "\<Union>?\<D> = S"
lp15@66154
   457
      using open_UNION_box [OF assms] by metis
lp15@66154
   458
  qed auto
lp15@66154
   459
qed
lp15@66154
   460
lp15@66154
   461
lemma rational_cboxes:
lp15@66154
   462
  fixes x :: "'a::euclidean_space"
lp15@66154
   463
  assumes "e > 0"
lp15@66154
   464
  shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat>) \<and> x \<in> cbox a b \<and> cbox a b \<subseteq> ball x e"
lp15@66154
   465
proof -
lp15@66154
   466
  define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
lp15@66154
   467
  then have e: "e' > 0"
lp15@66154
   468
    using assms by auto
lp15@66154
   469
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
lp15@66154
   470
  proof
lp15@66154
   471
    fix i
lp15@66154
   472
    from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
lp15@66154
   473
    show "?th i" by auto
lp15@66154
   474
  qed
lp15@66154
   475
  from choice[OF this] obtain a where
lp15@66154
   476
    a: "\<forall>u. a u \<in> \<rat> \<and> a u < x \<bullet> u \<and> x \<bullet> u - a u < e'" ..
lp15@66154
   477
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
lp15@66154
   478
  proof
lp15@66154
   479
    fix i
lp15@66154
   480
    from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
lp15@66154
   481
    show "?th i" by auto
lp15@66154
   482
  qed
lp15@66154
   483
  from choice[OF this] obtain b where
lp15@66154
   484
    b: "\<forall>u. b u \<in> \<rat> \<and> x \<bullet> u < b u \<and> b u - x \<bullet> u < e'" ..
lp15@66154
   485
  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
lp15@66154
   486
  show ?thesis
lp15@66154
   487
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
lp15@66154
   488
    fix y :: 'a
lp15@66154
   489
    assume *: "y \<in> cbox ?a ?b"
lp15@66154
   490
    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
nipkow@67155
   491
      unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
lp15@66154
   492
    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
lp15@66154
   493
    proof (rule real_sqrt_less_mono, rule sum_strict_mono)
lp15@66154
   494
      fix i :: "'a"
lp15@66154
   495
      assume i: "i \<in> Basis"
lp15@66154
   496
      have "a i \<le> y\<bullet>i \<and> y\<bullet>i \<le> b i"
lp15@66154
   497
        using * i by (auto simp: cbox_def)
lp15@66154
   498
      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
lp15@66154
   499
        using a by auto
lp15@66154
   500
      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
lp15@66154
   501
        using b by auto
lp15@66154
   502
      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
lp15@66154
   503
        by auto
lp15@66154
   504
      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
lp15@66154
   505
        unfolding e'_def by (auto simp: dist_real_def)
lp15@66154
   506
      then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
lp15@66154
   507
        by (rule power_strict_mono) auto
lp15@66154
   508
      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
lp15@66154
   509
        by (simp add: power_divide)
lp15@66154
   510
    qed auto
lp15@66154
   511
    also have "\<dots> = e"
lp15@66154
   512
      using \<open>0 < e\<close> by simp
lp15@66154
   513
    finally show "y \<in> ball x e"
lp15@66154
   514
      by (auto simp: ball_def)
lp15@66154
   515
  next
lp15@66154
   516
    show "x \<in> cbox (\<Sum>i\<in>Basis. a i *\<^sub>R i) (\<Sum>i\<in>Basis. b i *\<^sub>R i)"
lp15@66154
   517
      using a b less_imp_le by (auto simp: cbox_def)
lp15@66154
   518
  qed (use a b cbox_def in auto)
lp15@66154
   519
qed
lp15@66154
   520
lp15@66154
   521
lemma open_UNION_cbox:
lp15@66154
   522
  fixes M :: "'a::euclidean_space set"
lp15@66154
   523
  assumes "open M"
lp15@66154
   524
  defines "a' \<equiv> \<lambda>f. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
lp15@66154
   525
  defines "b' \<equiv> \<lambda>f. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
lp15@66154
   526
  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. cbox (a' f) (b' f) \<subseteq> M}"
lp15@66154
   527
  shows "M = (\<Union>f\<in>I. cbox (a' f) (b' f))"
lp15@66154
   528
proof -
lp15@66154
   529
  have "x \<in> (\<Union>f\<in>I. cbox (a' f) (b' f))" if "x \<in> M" for x
lp15@66154
   530
  proof -
lp15@66154
   531
    obtain e where e: "e > 0" "ball x e \<subseteq> M"
lp15@66154
   532
      using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
lp15@66154
   533
    moreover obtain a b where ab: "x \<in> cbox a b" "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
lp15@66154
   534
                                  "\<forall>i \<in> Basis. b \<bullet> i \<in> \<rat>" "cbox a b \<subseteq> ball x e"
lp15@66154
   535
      using rational_cboxes[OF e(1)] by metis
lp15@66154
   536
    ultimately show ?thesis
lp15@66154
   537
       by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
lp15@66154
   538
          (auto simp: euclidean_representation I_def a'_def b'_def)
lp15@66154
   539
  qed
lp15@66154
   540
  then show ?thesis by (auto simp: I_def)
lp15@66154
   541
qed
lp15@66154
   542
lp15@66154
   543
corollary open_countable_Union_open_cbox:
lp15@66154
   544
  fixes S :: "'a :: euclidean_space set"
lp15@66154
   545
  assumes "open S"
lp15@66154
   546
  obtains \<D> where "countable \<D>" "\<D> \<subseteq> Pow S" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" "\<Union>\<D> = S"
lp15@66154
   547
proof -
lp15@66154
   548
  let ?a = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
lp15@66154
   549
  let ?b = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
lp15@66154
   550
  let ?I = "{f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. cbox (?a f) (?b f) \<subseteq> S}"
lp15@66154
   551
  let ?\<D> = "(\<lambda>f. cbox (?a f) (?b f)) ` ?I"
lp15@66154
   552
  show ?thesis
lp15@66154
   553
  proof
lp15@66154
   554
    have "countable ?I"
lp15@66154
   555
      by (simp add: countable_PiE countable_rat)
lp15@66154
   556
    then show "countable ?\<D>"
lp15@66154
   557
      by blast
lp15@66154
   558
    show "\<Union>?\<D> = S"
lp15@66154
   559
      using open_UNION_cbox [OF assms] by metis
lp15@66154
   560
  qed auto
lp15@66154
   561
qed
lp15@66154
   562
immler@56189
   563
lemma box_eq_empty:
immler@56189
   564
  fixes a :: "'a::euclidean_space"
immler@56189
   565
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
immler@56189
   566
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
immler@56189
   567
proof -
immler@56189
   568
  {
immler@56189
   569
    fix i x
immler@56189
   570
    assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>box a b"
immler@56189
   571
    then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"
immler@56189
   572
      unfolding mem_box by (auto simp: box_def)
immler@56189
   573
    then have "a\<bullet>i < b\<bullet>i" by auto
immler@56189
   574
    then have False using as by auto
immler@56189
   575
  }
immler@56189
   576
  moreover
immler@56189
   577
  {
immler@56189
   578
    assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
immler@56189
   579
    let ?x = "(1/2) *\<^sub>R (a + b)"
immler@56189
   580
    {
immler@56189
   581
      fix i :: 'a
immler@56189
   582
      assume i: "i \<in> Basis"
immler@56189
   583
      have "a\<bullet>i < b\<bullet>i"
immler@56189
   584
        using as[THEN bspec[where x=i]] i by auto
immler@56189
   585
      then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
immler@56189
   586
        by (auto simp: inner_add_left)
immler@56189
   587
    }
immler@56189
   588
    then have "box a b \<noteq> {}"
immler@56189
   589
      using mem_box(1)[of "?x" a b] by auto
immler@56189
   590
  }
immler@56189
   591
  ultimately show ?th1 by blast
immler@56189
   592
immler@56189
   593
  {
immler@56189
   594
    fix i x
immler@56189
   595
    assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>cbox a b"
immler@56189
   596
    then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
immler@56189
   597
      unfolding mem_box by auto
immler@56189
   598
    then have "a\<bullet>i \<le> b\<bullet>i" by auto
immler@56189
   599
    then have False using as by auto
immler@56189
   600
  }
immler@56189
   601
  moreover
immler@56189
   602
  {
immler@56189
   603
    assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
immler@56189
   604
    let ?x = "(1/2) *\<^sub>R (a + b)"
immler@56189
   605
    {
immler@56189
   606
      fix i :: 'a
immler@56189
   607
      assume i:"i \<in> Basis"
immler@56189
   608
      have "a\<bullet>i \<le> b\<bullet>i"
immler@56189
   609
        using as[THEN bspec[where x=i]] i by auto
immler@56189
   610
      then have "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
immler@56189
   611
        by (auto simp: inner_add_left)
immler@56189
   612
    }
immler@56189
   613
    then have "cbox a b \<noteq> {}"
immler@56189
   614
      using mem_box(2)[of "?x" a b] by auto
immler@56189
   615
  }
immler@56189
   616
  ultimately show ?th2 by blast
immler@56189
   617
qed
immler@56189
   618
immler@56189
   619
lemma box_ne_empty:
immler@56189
   620
  fixes a :: "'a::euclidean_space"
immler@56189
   621
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"
immler@56189
   622
  and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
immler@56189
   623
  unfolding box_eq_empty[of a b] by fastforce+
immler@56189
   624
immler@56189
   625
lemma
immler@56189
   626
  fixes a :: "'a::euclidean_space"
lp15@66112
   627
  shows cbox_sing [simp]: "cbox a a = {a}"
lp15@66112
   628
    and box_sing [simp]: "box a a = {}"
immler@56189
   629
  unfolding set_eq_iff mem_box eq_iff [symmetric]
immler@56189
   630
  by (auto intro!: euclidean_eqI[where 'a='a])
immler@56189
   631
     (metis all_not_in_conv nonempty_Basis)
immler@56189
   632
immler@56189
   633
lemma subset_box_imp:
immler@56189
   634
  fixes a :: "'a::euclidean_space"
immler@56189
   635
  shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56189
   636
    and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56189
   637
    and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@56189
   638
     and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> box a b"
immler@56189
   639
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box
wenzelm@58757
   640
  by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
immler@56189
   641
immler@56189
   642
lemma box_subset_cbox:
immler@56189
   643
  fixes a :: "'a::euclidean_space"
immler@56189
   644
  shows "box a b \<subseteq> cbox a b"
immler@56189
   645
  unfolding subset_eq [unfolded Ball_def] mem_box
immler@56189
   646
  by (fast intro: less_imp_le)
immler@56189
   647
immler@56189
   648
lemma subset_box:
immler@56189
   649
  fixes a :: "'a::euclidean_space"
wenzelm@64539
   650
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1)
wenzelm@64539
   651
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2)
wenzelm@64539
   652
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3)
wenzelm@64539
   653
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
immler@56189
   654
proof -
lp15@68120
   655
  let ?lesscd = "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
lp15@68120
   656
  let ?lerhs = "\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
lp15@68120
   657
  show ?th1 ?th2
lp15@68120
   658
    by (fastforce simp: mem_box)+
lp15@68120
   659
  have acdb: "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
lp15@68120
   660
    if i: "i \<in> Basis" and box: "box c d \<subseteq> cbox a b" and cd: "\<And>i. i \<in> Basis \<Longrightarrow> c\<bullet>i < d\<bullet>i" for i
lp15@68120
   661
  proof -
lp15@68120
   662
    have "box c d \<noteq> {}"
lp15@68120
   663
      using that
lp15@68120
   664
      unfolding box_eq_empty by force
lp15@68120
   665
    { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
lp15@68120
   666
      assume *: "a\<bullet>i > c\<bullet>i"
lp15@68120
   667
      then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j" if "j \<in> Basis" for j
lp15@68120
   668
        using cd that by (fastforce simp add: i *)
lp15@68120
   669
      then have "?x \<in> box c d"
lp15@68120
   670
        unfolding mem_box by auto
lp15@68120
   671
      moreover have "?x \<notin> cbox a b"
lp15@68120
   672
        using i cd * by (force simp: mem_box)
lp15@68120
   673
      ultimately have False using box by auto
immler@56189
   674
    }
lp15@68120
   675
    then have "a\<bullet>i \<le> c\<bullet>i" by force
immler@56189
   676
    moreover
lp15@68120
   677
    { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
lp15@68120
   678
      assume *: "b\<bullet>i < d\<bullet>i"
lp15@68120
   679
      then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j" if "j \<in> Basis" for j
lp15@68120
   680
        using cd that by (fastforce simp add: i *)
lp15@68120
   681
      then have "?x \<in> box c d"
immler@56189
   682
        unfolding mem_box by auto
lp15@68120
   683
      moreover have "?x \<notin> cbox a b"
lp15@68120
   684
        using i cd * by (force simp: mem_box)
lp15@68120
   685
      ultimately have False using box by auto
immler@56189
   686
    }
immler@56189
   687
    then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto
lp15@68120
   688
    ultimately show ?thesis by auto
lp15@68120
   689
  qed
immler@56189
   690
  show ?th3
lp15@68120
   691
    using acdb by (fastforce simp add: mem_box)
lp15@68120
   692
  have acdb': "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
lp15@68120
   693
    if "i \<in> Basis" "box c d \<subseteq> box a b" "\<And>i. i \<in> Basis \<Longrightarrow> c\<bullet>i < d\<bullet>i" for i
lp15@68120
   694
      using box_subset_cbox[of a b] that acdb by auto
immler@56189
   695
  show ?th4
lp15@68120
   696
    using acdb' by (fastforce simp add: mem_box)
immler@56189
   697
qed
immler@56189
   698
lp15@63945
   699
lemma eq_cbox: "cbox a b = cbox c d \<longleftrightarrow> cbox a b = {} \<and> cbox c d = {} \<or> a = c \<and> b = d"
lp15@63945
   700
      (is "?lhs = ?rhs")
lp15@63945
   701
proof
lp15@63945
   702
  assume ?lhs
lp15@63945
   703
  then have "cbox a b \<subseteq> cbox c d" "cbox c d \<subseteq> cbox a b"
lp15@63945
   704
    by auto
lp15@63945
   705
  then show ?rhs
lp15@66643
   706
    by (force simp: subset_box box_eq_empty intro: antisym euclidean_eqI)
lp15@63945
   707
next
lp15@63945
   708
  assume ?rhs
lp15@63945
   709
  then show ?lhs
lp15@63945
   710
    by force
lp15@63945
   711
qed
lp15@63945
   712
lp15@63945
   713
lemma eq_cbox_box [simp]: "cbox a b = box c d \<longleftrightarrow> cbox a b = {} \<and> box c d = {}"
wenzelm@64539
   714
  (is "?lhs \<longleftrightarrow> ?rhs")
lp15@63945
   715
proof
lp15@68120
   716
  assume L: ?lhs
lp15@68120
   717
  then have "cbox a b \<subseteq> box c d" "box c d \<subseteq> cbox a b"
lp15@63945
   718
    by auto
lp15@63945
   719
  then show ?rhs
hoelzl@63957
   720
    apply (simp add: subset_box)
lp15@68120
   721
    using L box_ne_empty box_sing apply (fastforce simp add:)
lp15@63945
   722
    done
lp15@68120
   723
qed force
lp15@63945
   724
lp15@63945
   725
lemma eq_box_cbox [simp]: "box a b = cbox c d \<longleftrightarrow> box a b = {} \<and> cbox c d = {}"
lp15@63945
   726
  by (metis eq_cbox_box)
lp15@63945
   727
lp15@63945
   728
lemma eq_box: "box a b = box c d \<longleftrightarrow> box a b = {} \<and> box c d = {} \<or> a = c \<and> b = d"
wenzelm@64539
   729
  (is "?lhs \<longleftrightarrow> ?rhs")
lp15@63945
   730
proof
lp15@68120
   731
  assume L: ?lhs
lp15@63945
   732
  then have "box a b \<subseteq> box c d" "box c d \<subseteq> box a b"
lp15@63945
   733
    by auto
lp15@63945
   734
  then show ?rhs
lp15@63945
   735
    apply (simp add: subset_box)
lp15@68120
   736
    using box_ne_empty(2) L
lp15@63945
   737
    apply auto
lp15@63945
   738
     apply (meson euclidean_eqI less_eq_real_def not_less)+
lp15@63945
   739
    done
lp15@68120
   740
qed force
lp15@63945
   741
eberlm@66466
   742
lemma subset_box_complex:
lp15@66643
   743
   "cbox a b \<subseteq> cbox c d \<longleftrightarrow>
eberlm@66466
   744
      (Re a \<le> Re b \<and> Im a \<le> Im b) \<longrightarrow> Re a \<ge> Re c \<and> Im a \<ge> Im c \<and> Re b \<le> Re d \<and> Im b \<le> Im d"
lp15@66643
   745
   "cbox a b \<subseteq> box c d \<longleftrightarrow>
eberlm@66466
   746
      (Re a \<le> Re b \<and> Im a \<le> Im b) \<longrightarrow> Re a > Re c \<and> Im a > Im c \<and> Re b < Re d \<and> Im b < Im d"
eberlm@66466
   747
   "box a b \<subseteq> cbox c d \<longleftrightarrow>
eberlm@66466
   748
      (Re a < Re b \<and> Im a < Im b) \<longrightarrow> Re a \<ge> Re c \<and> Im a \<ge> Im c \<and> Re b \<le> Re d \<and> Im b \<le> Im d"
lp15@66643
   749
   "box a b \<subseteq> box c d \<longleftrightarrow>
eberlm@66466
   750
      (Re a < Re b \<and> Im a < Im b) \<longrightarrow> Re a \<ge> Re c \<and> Im a \<ge> Im c \<and> Re b \<le> Re d \<and> Im b \<le> Im d"
eberlm@66466
   751
  by (subst subset_box; force simp: Basis_complex_def)+
eberlm@66466
   752
lp15@63945
   753
lemma Int_interval:
immler@56189
   754
  fixes a :: "'a::euclidean_space"
immler@56189
   755
  shows "cbox a b \<inter> cbox c d =
immler@56189
   756
    cbox (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
immler@56189
   757
  unfolding set_eq_iff and Int_iff and mem_box
immler@56189
   758
  by auto
immler@56189
   759
immler@56189
   760
lemma disjoint_interval:
immler@56189
   761
  fixes a::"'a::euclidean_space"
immler@56189
   762
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1)
immler@56189
   763
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2)
immler@56189
   764
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3)
immler@56189
   765
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
immler@56189
   766
proof -
immler@56189
   767
  let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
immler@56189
   768
  have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
immler@56189
   769
      (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
immler@56189
   770
    by blast
immler@56189
   771
  note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
immler@56189
   772
  show ?th1 unfolding * by (intro **) auto
immler@56189
   773
  show ?th2 unfolding * by (intro **) auto
immler@56189
   774
  show ?th3 unfolding * by (intro **) auto
immler@56189
   775
  show ?th4 unfolding * by (intro **) auto
immler@56189
   776
qed
immler@56189
   777
hoelzl@57447
   778
lemma UN_box_eq_UNIV: "(\<Union>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"
hoelzl@57447
   779
proof -
wenzelm@61942
   780
  have "\<bar>x \<bullet> b\<bar> < real_of_int (\<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil> + 1)"
wenzelm@60462
   781
    if [simp]: "b \<in> Basis" for x b :: 'a
wenzelm@60462
   782
  proof -
wenzelm@61942
   783
    have "\<bar>x \<bullet> b\<bar> \<le> real_of_int \<lceil>\<bar>x \<bullet> b\<bar>\<rceil>"
lp15@61609
   784
      by (rule le_of_int_ceiling)
wenzelm@61942
   785
    also have "\<dots> \<le> real_of_int \<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil>"
nipkow@59587
   786
      by (auto intro!: ceiling_mono)
wenzelm@61942
   787
    also have "\<dots> < real_of_int (\<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil> + 1)"
hoelzl@57447
   788
      by simp
wenzelm@60462
   789
    finally show ?thesis .
wenzelm@60462
   790
  qed
wenzelm@60462
   791
  then have "\<exists>n::nat. \<forall>b\<in>Basis. \<bar>x \<bullet> b\<bar> < real n" for x :: 'a
nipkow@59587
   792
    by (metis order.strict_trans reals_Archimedean2)
hoelzl@57447
   793
  moreover have "\<And>x b::'a. \<And>n::nat.  \<bar>x \<bullet> b\<bar> < real n \<longleftrightarrow> - real n < x \<bullet> b \<and> x \<bullet> b < real n"
hoelzl@57447
   794
    by auto
hoelzl@57447
   795
  ultimately show ?thesis
nipkow@64267
   796
    by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
hoelzl@57447
   797
qed
hoelzl@57447
   798
immler@69613
   799
lemma image_affinity_cbox: fixes m::real
immler@69613
   800
  fixes a b c :: "'a::euclidean_space"
immler@69613
   801
  shows "(\<lambda>x. m *\<^sub>R x + c) ` cbox a b =
immler@69613
   802
    (if cbox a b = {} then {}
immler@69613
   803
     else (if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)
immler@69613
   804
     else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c)))"
immler@69613
   805
proof (cases "m = 0")
immler@69613
   806
  case True
immler@69613
   807
  {
immler@69613
   808
    fix x
immler@69613
   809
    assume "\<forall>i\<in>Basis. x \<bullet> i \<le> c \<bullet> i" "\<forall>i\<in>Basis. c \<bullet> i \<le> x \<bullet> i"
immler@69613
   810
    then have "x = c"
immler@69613
   811
      by (simp add: dual_order.antisym euclidean_eqI)
immler@69613
   812
  }
immler@69613
   813
  moreover have "c \<in> cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)"
immler@69613
   814
    unfolding True by (auto simp: cbox_sing)
immler@69613
   815
  ultimately show ?thesis using True by (auto simp: cbox_def)
immler@69613
   816
next
immler@69613
   817
  case False
immler@69613
   818
  {
immler@69613
   819
    fix y
immler@69613
   820
    assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m > 0"
immler@69613
   821
    then have "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
immler@69613
   822
      by (auto simp: inner_distrib)
immler@69613
   823
  }
immler@69613
   824
  moreover
immler@69613
   825
  {
immler@69613
   826
    fix y
immler@69613
   827
    assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m < 0"
immler@69613
   828
    then have "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i"
immler@69613
   829
      by (auto simp: mult_left_mono_neg inner_distrib)
immler@69613
   830
  }
immler@69613
   831
  moreover
immler@69613
   832
  {
immler@69613
   833
    fix y
immler@69613
   834
    assume "m > 0" and "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> y \<bullet> i" and "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
immler@69613
   835
    then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
immler@69613
   836
      unfolding image_iff Bex_def mem_box
immler@69613
   837
      apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
immler@69613
   838
      apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
immler@69613
   839
      done
immler@69613
   840
  }
immler@69613
   841
  moreover
immler@69613
   842
  {
immler@69613
   843
    fix y
immler@69613
   844
    assume "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i" "m < 0"
immler@69613
   845
    then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
immler@69613
   846
      unfolding image_iff Bex_def mem_box
immler@69613
   847
      apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
immler@69613
   848
      apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
immler@69613
   849
      done
immler@69613
   850
  }
immler@69613
   851
  ultimately show ?thesis using False by (auto simp: cbox_def)
immler@69613
   852
qed
immler@69613
   853
immler@69613
   854
lemma image_smult_cbox:"(\<lambda>x. m *\<^sub>R (x::_::euclidean_space)) ` cbox a b =
immler@69613
   855
  (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
immler@69613
   856
  using image_affinity_cbox[of m 0 a b] by auto
immler@69613
   857
immler@69619
   858
lemma swap_continuous:
immler@69619
   859
  assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
immler@69619
   860
    shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
immler@69619
   861
proof -
immler@69619
   862
  have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
immler@69619
   863
    by auto
immler@69619
   864
  then show ?thesis
immler@69619
   865
    apply (rule ssubst)
immler@69619
   866
    apply (rule continuous_on_compose)
immler@69619
   867
    apply (simp add: split_def)
immler@69619
   868
    apply (rule continuous_intros | simp add: assms)+
immler@69619
   869
    done
immler@69619
   870
qed
immler@69619
   871
immler@69516
   872
nipkow@69508
   873
subsection \<open>General Intervals\<close>
immler@67962
   874
immler@67962
   875
definition%important "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
immler@56189
   876
  (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"
immler@56189
   877
immler@67685
   878
lemma is_interval_1:
immler@67685
   879
  "is_interval (s::real set) \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> s)"
immler@67685
   880
  unfolding is_interval_def by auto
immler@67685
   881
immler@67685
   882
lemma is_interval_inter: "is_interval X \<Longrightarrow> is_interval Y \<Longrightarrow> is_interval (X \<inter> Y)"
immler@67685
   883
  unfolding is_interval_def
immler@67685
   884
  by blast
immler@67685
   885
lp15@66089
   886
lemma is_interval_cbox [simp]: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
lp15@66089
   887
  and is_interval_box [simp]: "is_interval (box a b)" (is ?th2)
immler@56189
   888
  unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
immler@56189
   889
  by (meson order_trans le_less_trans less_le_trans less_trans)+
immler@56189
   890
lp15@61609
   891
lemma is_interval_empty [iff]: "is_interval {}"
lp15@61609
   892
  unfolding is_interval_def  by simp
lp15@61609
   893
lp15@61609
   894
lemma is_interval_univ [iff]: "is_interval UNIV"
lp15@61609
   895
  unfolding is_interval_def  by simp
immler@56189
   896
immler@56189
   897
lemma mem_is_intervalI:
immler@56189
   898
  assumes "is_interval s"
wenzelm@64539
   899
    and "a \<in> s" "b \<in> s"
wenzelm@64539
   900
    and "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i"
immler@56189
   901
  shows "x \<in> s"
immler@56189
   902
  by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])
immler@56189
   903
immler@56189
   904
lemma interval_subst:
immler@56189
   905
  fixes S::"'a::euclidean_space set"
immler@56189
   906
  assumes "is_interval S"
wenzelm@64539
   907
    and "x \<in> S" "y j \<in> S"
wenzelm@64539
   908
    and "j \<in> Basis"
immler@56189
   909
  shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"
immler@56189
   910
  by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)
immler@56189
   911
immler@56189
   912
lemma mem_box_componentwiseI:
immler@56189
   913
  fixes S::"'a::euclidean_space set"
immler@56189
   914
  assumes "is_interval S"
immler@56189
   915
  assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i) ` S)"
immler@56189
   916
  shows "x \<in> S"
immler@56189
   917
proof -
immler@56189
   918
  from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"
immler@56189
   919
    by auto
wenzelm@64539
   920
  with finite_Basis obtain s and bs::"'a list"
wenzelm@64539
   921
    where s: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i = s i \<bullet> i" "\<And>i. i \<in> Basis \<Longrightarrow> s i \<in> S"
wenzelm@64539
   922
      and bs: "set bs = Basis" "distinct bs"
immler@56189
   923
    by (metis finite_distinct_list)
wenzelm@64539
   924
  from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S"
wenzelm@64539
   925
    by blast
wenzelm@63040
   926
  define y where
wenzelm@63040
   927
    "y = rec_list (s j) (\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"
immler@56189
   928
  have "x = (\<Sum>i\<in>Basis. (if i \<in> set bs then s i \<bullet> i else s j \<bullet> i) *\<^sub>R i)"
lp15@66643
   929
    using bs by (auto simp: s(1)[symmetric] euclidean_representation)
immler@56189
   930
  also have [symmetric]: "y bs = \<dots>"
immler@56189
   931
    using bs(2) bs(1)[THEN equalityD1]
immler@56189
   932
    by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
immler@56189
   933
  also have "y bs \<in> S"
immler@56189
   934
    using bs(1)[THEN equalityD1]
immler@56189
   935
    apply (induct bs)
wenzelm@64539
   936
     apply (auto simp: y_def j)
immler@56189
   937
    apply (rule interval_subst[OF assms(1)])
wenzelm@64539
   938
      apply (auto simp: s)
immler@56189
   939
    done
immler@56189
   940
  finally show ?thesis .
immler@56189
   941
qed
immler@56189
   942
lp15@63007
   943
lemma cbox01_nonempty [simp]: "cbox 0 One \<noteq> {}"
nipkow@64267
   944
  by (simp add: box_ne_empty inner_Basis inner_sum_left sum_nonneg)
lp15@63007
   945
lp15@63007
   946
lemma box01_nonempty [simp]: "box 0 One \<noteq> {}"
lp15@66089
   947
  by (simp add: box_ne_empty inner_Basis inner_sum_left)
lp15@63075
   948
lp15@64773
   949
lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
lp15@64773
   950
  using nonempty_Basis box01_nonempty box_eq_empty(1) box_ne_empty(1) by blast
lp15@64773
   951
lp15@66089
   952
lemma interval_subset_is_interval:
lp15@66089
   953
  assumes "is_interval S"
lp15@66089
   954
  shows "cbox a b \<subseteq> S \<longleftrightarrow> cbox a b = {} \<or> a \<in> S \<and> b \<in> S" (is "?lhs = ?rhs")
lp15@66089
   955
proof
lp15@66089
   956
  assume ?lhs
lp15@66089
   957
  then show ?rhs  using box_ne_empty(1) mem_box(2) by fastforce
lp15@66089
   958
next
lp15@66089
   959
  assume ?rhs
lp15@66089
   960
  have "cbox a b \<subseteq> S" if "a \<in> S" "b \<in> S"
lp15@66089
   961
    using assms unfolding is_interval_def
lp15@66089
   962
    apply (clarsimp simp add: mem_box)
lp15@66089
   963
    using that by blast
lp15@66089
   964
  with \<open>?rhs\<close> show ?lhs
lp15@66089
   965
    by blast
lp15@66089
   966
qed
lp15@66089
   967
immler@67685
   968
lemma is_real_interval_union:
immler@67685
   969
  "is_interval (X \<union> Y)"
immler@67685
   970
  if X: "is_interval X" and Y: "is_interval Y" and I: "(X \<noteq> {} \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> X \<inter> Y \<noteq> {})"
immler@67685
   971
  for X Y::"real set"
immler@67685
   972
proof -
immler@67685
   973
  consider "X \<noteq> {}" "Y \<noteq> {}" | "X = {}" | "Y = {}" by blast
immler@67685
   974
  then show ?thesis
immler@67685
   975
  proof cases
immler@67685
   976
    case 1
immler@67685
   977
    then obtain r where "r \<in> X \<or> X \<inter> Y = {}" "r \<in> Y \<or> X \<inter> Y = {}"
immler@67685
   978
      by blast
immler@67685
   979
    then show ?thesis
immler@67685
   980
      using I 1 X Y unfolding is_interval_1
immler@67685
   981
      by (metis (full_types) Un_iff le_cases)
immler@67685
   982
  qed (use that in auto)
immler@67685
   983
qed
immler@67685
   984
immler@67685
   985
lemma is_interval_translationI:
immler@67685
   986
  assumes "is_interval X"
immler@67685
   987
  shows "is_interval ((+) x ` X)"
immler@67685
   988
  unfolding is_interval_def
immler@67685
   989
proof safe
immler@67685
   990
  fix b d e
immler@67685
   991
  assume "b \<in> X" "d \<in> X"
immler@67685
   992
    "\<forall>i\<in>Basis. (x + b) \<bullet> i \<le> e \<bullet> i \<and> e \<bullet> i \<le> (x + d) \<bullet> i \<or>
immler@67685
   993
       (x + d) \<bullet> i \<le> e \<bullet> i \<and> e \<bullet> i \<le> (x + b) \<bullet> i"
immler@67685
   994
  hence "e - x \<in> X"
immler@67685
   995
    by (intro mem_is_intervalI[OF assms \<open>b \<in> X\<close> \<open>d \<in> X\<close>, of "e - x"])
immler@67685
   996
      (auto simp: algebra_simps)
immler@67685
   997
  thus "e \<in> (+) x ` X" by force
immler@67685
   998
qed
immler@67685
   999
immler@67685
  1000
lemma is_interval_uminusI:
immler@67685
  1001
  assumes "is_interval X"
immler@67685
  1002
  shows "is_interval (uminus ` X)"
immler@67685
  1003
  unfolding is_interval_def
immler@67685
  1004
proof safe
immler@67685
  1005
  fix b d e
immler@67685
  1006
  assume "b \<in> X" "d \<in> X"
immler@67685
  1007
    "\<forall>i\<in>Basis. (- b) \<bullet> i \<le> e \<bullet> i \<and> e \<bullet> i \<le> (- d) \<bullet> i \<or>
immler@67685
  1008
       (- d) \<bullet> i \<le> e \<bullet> i \<and> e \<bullet> i \<le> (- b) \<bullet> i"
immler@67685
  1009
  hence "- e \<in> X"
immler@67685
  1010
    by (intro mem_is_intervalI[OF assms \<open>b \<in> X\<close> \<open>d \<in> X\<close>, of "- e"])
immler@67685
  1011
      (auto simp: algebra_simps)
immler@67685
  1012
  thus "e \<in> uminus ` X" by force
immler@67685
  1013
qed
immler@67685
  1014
immler@67685
  1015
lemma is_interval_uminus[simp]: "is_interval (uminus ` x) = is_interval x"
immler@67685
  1016
  using is_interval_uminusI[of x] is_interval_uminusI[of "uminus ` x"]
immler@67685
  1017
  by (auto simp: image_image)
immler@67685
  1018
immler@67685
  1019
lemma is_interval_neg_translationI:
immler@67685
  1020
  assumes "is_interval X"
immler@67685
  1021
  shows "is_interval ((-) x ` X)"
immler@67685
  1022
proof -
immler@67685
  1023
  have "(-) x ` X = (+) x ` uminus ` X"
immler@67685
  1024
    by (force simp: algebra_simps)
immler@67685
  1025
  also have "is_interval \<dots>"
immler@67685
  1026
    by (metis is_interval_uminusI is_interval_translationI assms)
immler@67685
  1027
  finally show ?thesis .
immler@67685
  1028
qed
immler@67685
  1029
immler@67685
  1030
lemma is_interval_translation[simp]:
immler@67685
  1031
  "is_interval ((+) x ` X) = is_interval X"
immler@67685
  1032
  using is_interval_neg_translationI[of "(+) x ` X" x]
immler@67685
  1033
  by (auto intro!: is_interval_translationI simp: image_image)
immler@67685
  1034
immler@67685
  1035
lemma is_interval_minus_translation[simp]:
immler@67685
  1036
  shows "is_interval ((-) x ` X) = is_interval X"
immler@67685
  1037
proof -
immler@67685
  1038
  have "(-) x ` X = (+) x ` uminus ` X"
immler@67685
  1039
    by (force simp: algebra_simps)
immler@67685
  1040
  also have "is_interval \<dots> = is_interval X"
immler@67685
  1041
    by simp
immler@67685
  1042
  finally show ?thesis .
immler@67685
  1043
qed
immler@67685
  1044
immler@67685
  1045
lemma is_interval_minus_translation'[simp]:
immler@67685
  1046
  shows "is_interval ((\<lambda>x. x - c) ` X) = is_interval X"
immler@67685
  1047
  using is_interval_translation[of "-c" X]
immler@67685
  1048
  by (metis image_cong uminus_add_conv_diff)
immler@67685
  1049
immler@69611
  1050
immler@69611
  1051
subsection%unimportant \<open>Bounded Projections\<close>
immler@62127
  1052
immler@69611
  1053
lemma bounded_inner_imp_bdd_above:
immler@69611
  1054
  assumes "bounded s"
immler@69611
  1055
    shows "bdd_above ((\<lambda>x. x \<bullet> a) ` s)"
immler@69611
  1056
by (simp add: assms bounded_imp_bdd_above bounded_linear_image bounded_linear_inner_left)
himmelma@33175
  1057
immler@69611
  1058
lemma bounded_inner_imp_bdd_below:
immler@69611
  1059
  assumes "bounded s"
immler@69611
  1060
    shows "bdd_below ((\<lambda>x. x \<bullet> a) ` s)"
immler@69611
  1061
by (simp add: assms bounded_imp_bdd_below bounded_linear_image bounded_linear_inner_left)
huffman@44632
  1062
wenzelm@53282
  1063
immler@69611
  1064
subsection%unimportant \<open>Structural rules for pointwise continuity\<close>
himmelma@33175
  1065
hoelzl@51361
  1066
lemma continuous_infnorm[continuous_intros]:
wenzelm@53282
  1067
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
huffman@44647
  1068
  unfolding continuous_def by (rule tendsto_infnorm)
himmelma@33175
  1069
hoelzl@51361
  1070
lemma continuous_inner[continuous_intros]:
wenzelm@53282
  1071
  assumes "continuous F f"
wenzelm@53282
  1072
    and "continuous F g"
huffman@44647
  1073
  shows "continuous F (\<lambda>x. inner (f x) (g x))"
huffman@44647
  1074
  using assms unfolding continuous_def by (rule tendsto_inner)
huffman@44647
  1075
immler@69516
  1076
immler@69611
  1077
subsection%unimportant \<open>Structural rules for setwise continuity\<close>
himmelma@33175
  1078
hoelzl@56371
  1079
lemma continuous_on_infnorm[continuous_intros]:
wenzelm@53282
  1080
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
huffman@44647
  1081
  unfolding continuous_on by (fast intro: tendsto_infnorm)
huffman@44647
  1082
hoelzl@56371
  1083
lemma continuous_on_inner[continuous_intros]:
huffman@44531
  1084
  fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"
wenzelm@53282
  1085
  assumes "continuous_on s f"
wenzelm@53282
  1086
    and "continuous_on s g"
huffman@44531
  1087
  shows "continuous_on s (\<lambda>x. inner (f x) (g x))"
huffman@44531
  1088
  using bounded_bilinear_inner assms
huffman@44531
  1089
  by (rule bounded_bilinear.continuous_on)
huffman@44531
  1090
immler@69613
  1091
immler@69613
  1092
subsection%unimportant \<open>Openness of halfspaces.\<close>
himmelma@33175
  1093
himmelma@33175
  1094
lemma open_halfspace_lt: "open {x. inner a x < b}"
hoelzl@63332
  1095
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
himmelma@33175
  1096
himmelma@33175
  1097
lemma open_halfspace_gt: "open {x. inner a x > b}"
hoelzl@63332
  1098
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
himmelma@33175
  1099
wenzelm@53282
  1100
lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x\<bullet>i < a}"
hoelzl@63332
  1101
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
himmelma@33175
  1102
wenzelm@53282
  1103
lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x\<bullet>i > a}"
hoelzl@63332
  1104
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
himmelma@33175
  1105
immler@69611
  1106
lemma eucl_less_eq_halfspaces:
immler@69611
  1107
  fixes a :: "'a::euclidean_space"
immler@69611
  1108
  shows "{x. x <e a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"
immler@69611
  1109
        "{x. a <e x} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"
immler@69611
  1110
  by (auto simp: eucl_less_def)
immler@69611
  1111
immler@69611
  1112
lemma open_Collect_eucl_less[simp, intro]:
immler@69611
  1113
  fixes a :: "'a::euclidean_space"
immler@69611
  1114
  shows "open {x. x <e a}" "open {x. a <e x}"
immler@69611
  1115
  by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)
immler@69611
  1116
immler@69613
  1117
subsection%unimportant \<open>Closure of halfspaces and hyperplanes\<close>
immler@69613
  1118
immler@69613
  1119
lemma continuous_at_inner: "continuous (at x) (inner a)"
immler@69613
  1120
  unfolding continuous_at by (intro tendsto_intros)
immler@69613
  1121
immler@69613
  1122
lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
immler@69613
  1123
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
immler@69613
  1124
immler@69613
  1125
lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
immler@69613
  1126
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
immler@69613
  1127
immler@69613
  1128
lemma closed_hyperplane: "closed {x. inner a x = b}"
immler@69613
  1129
  by (simp add: closed_Collect_eq continuous_on_inner continuous_on_const continuous_on_id)
immler@69613
  1130
immler@69613
  1131
lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"
immler@69613
  1132
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
immler@69613
  1133
immler@69613
  1134
lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"
immler@69613
  1135
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
immler@69613
  1136
immler@69613
  1137
lemma closed_interval_left:
immler@69613
  1138
  fixes b :: "'a::euclidean_space"
immler@69613
  1139
  shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"
immler@69613
  1140
  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
immler@69613
  1141
immler@69613
  1142
lemma closed_interval_right:
immler@69613
  1143
  fixes a :: "'a::euclidean_space"
immler@69613
  1144
  shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"
immler@69613
  1145
  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
immler@69613
  1146
immler@69613
  1147
immler@69613
  1148
subsection%unimportant\<open>Some more convenient intermediate-value theorem formulations\<close>
immler@69613
  1149
immler@69613
  1150
lemma connected_ivt_hyperplane:
immler@69613
  1151
  assumes "connected S" and xy: "x \<in> S" "y \<in> S" and b: "inner a x \<le> b" "b \<le> inner a y"
immler@69613
  1152
  shows "\<exists>z \<in> S. inner a z = b"
immler@69613
  1153
proof (rule ccontr)
immler@69613
  1154
  assume as:"\<not> (\<exists>z\<in>S. inner a z = b)"
immler@69613
  1155
  let ?A = "{x. inner a x < b}"
immler@69613
  1156
  let ?B = "{x. inner a x > b}"
immler@69613
  1157
  have "open ?A" "open ?B"
immler@69613
  1158
    using open_halfspace_lt and open_halfspace_gt by auto
immler@69613
  1159
  moreover have "?A \<inter> ?B = {}" by auto
immler@69613
  1160
  moreover have "S \<subseteq> ?A \<union> ?B" using as by auto
immler@69613
  1161
  ultimately show False
immler@69613
  1162
    using \<open>connected S\<close>[unfolded connected_def not_ex,
immler@69613
  1163
      THEN spec[where x="?A"], THEN spec[where x="?B"]]
immler@69613
  1164
    using xy b by auto
immler@69613
  1165
qed
immler@69613
  1166
immler@69613
  1167
lemma connected_ivt_component:
immler@69613
  1168
  fixes x::"'a::euclidean_space"
immler@69613
  1169
  shows "connected S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>S.  z\<bullet>k = a)"
immler@69613
  1170
  using connected_ivt_hyperplane[of S x y "k::'a" a]
immler@69613
  1171
  by (auto simp: inner_commute)
immler@69613
  1172
immler@69611
  1173
immler@69611
  1174
subsection \<open>Limit Component Bounds\<close>
himmelma@33175
  1175
immler@69613
  1176
lemma Lim_component_le:
immler@69613
  1177
  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
immler@69613
  1178
  assumes "(f \<longlongrightarrow> l) net"
immler@69613
  1179
    and "\<not> (trivial_limit net)"
immler@69613
  1180
    and "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"
immler@69613
  1181
  shows "l\<bullet>i \<le> b"
immler@69613
  1182
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])
immler@69613
  1183
immler@69613
  1184
lemma Lim_component_ge:
immler@69613
  1185
  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
immler@69613
  1186
  assumes "(f \<longlongrightarrow> l) net"
immler@69613
  1187
    and "\<not> (trivial_limit net)"
immler@69613
  1188
    and "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"
immler@69613
  1189
  shows "b \<le> l\<bullet>i"
immler@69613
  1190
  by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])
immler@69613
  1191
immler@69613
  1192
lemma Lim_component_eq:
immler@69613
  1193
  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
immler@69613
  1194
  assumes net: "(f \<longlongrightarrow> l) net" "\<not> trivial_limit net"
immler@69613
  1195
    and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
immler@69613
  1196
  shows "l\<bullet>i = b"
immler@69613
  1197
  using ev[unfolded order_eq_iff eventually_conj_iff]
immler@69613
  1198
  using Lim_component_ge[OF net, of b i]
immler@69613
  1199
  using Lim_component_le[OF net, of i b]
immler@69613
  1200
  by auto
immler@69613
  1201
immler@56189
  1202
lemma open_box[intro]: "open (box a b)"
immler@56189
  1203
proof -
nipkow@67399
  1204
  have "open (\<Inter>i\<in>Basis. ((\<bullet>) i) -` {a \<bullet> i <..< b \<bullet> i})"
lp15@62533
  1205
    by (auto intro!: continuous_open_vimage continuous_inner continuous_ident continuous_const)
nipkow@67399
  1206
  also have "(\<Inter>i\<in>Basis. ((\<bullet>) i) -` {a \<bullet> i <..< b \<bullet> i}) = box a b"
lp15@66643
  1207
    by (auto simp: box_def inner_commute)
immler@56189
  1208
  finally show ?thesis .
immler@56189
  1209
qed
immler@56189
  1210
immler@56189
  1211
lemma closed_cbox[intro]:
immler@56189
  1212
  fixes a b :: "'a::euclidean_space"
immler@56189
  1213
  shows "closed (cbox a b)"
immler@56189
  1214
proof -
immler@56189
  1215
  have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i})"
immler@56189
  1216
    by (intro closed_INT ballI continuous_closed_vimage allI
immler@56189
  1217
      linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
immler@56189
  1218
  also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i}) = cbox a b"
lp15@66643
  1219
    by (auto simp: cbox_def)
immler@56189
  1220
  finally show "closed (cbox a b)" .
immler@56189
  1221
qed
immler@56189
  1222
lp15@62618
  1223
lemma interior_cbox [simp]:
immler@56189
  1224
  fixes a b :: "'a::euclidean_space"
immler@56189
  1225
  shows "interior (cbox a b) = box a b" (is "?L = ?R")
immler@56189
  1226
proof(rule subset_antisym)
immler@56189
  1227
  show "?R \<subseteq> ?L"
immler@56189
  1228
    using box_subset_cbox open_box
immler@56189
  1229
    by (rule interior_maximal)
immler@56189
  1230
  {
immler@56189
  1231
    fix x
immler@56189
  1232
    assume "x \<in> interior (cbox a b)"
immler@56189
  1233
    then obtain s where s: "open s" "x \<in> s" "s \<subseteq> cbox a b" ..
immler@56189
  1234
    then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> cbox a b"
immler@56189
  1235
      unfolding open_dist and subset_eq by auto
immler@56189
  1236
    {
immler@56189
  1237
      fix i :: 'a
immler@56189
  1238
      assume i: "i \<in> Basis"
immler@56189
  1239
      have "dist (x - (e / 2) *\<^sub>R i) x < e"
immler@56189
  1240
        and "dist (x + (e / 2) *\<^sub>R i) x < e"
immler@56189
  1241
        unfolding dist_norm
immler@56189
  1242
        apply auto
immler@56189
  1243
        unfolding norm_minus_cancel
wenzelm@60420
  1244
        using norm_Basis[OF i] \<open>e>0\<close>
immler@56189
  1245
        apply auto
immler@56189
  1246
        done
immler@56189
  1247
      then have "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i" and "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"
immler@56189
  1248
        using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]
immler@56189
  1249
          and e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
immler@56189
  1250
        unfolding mem_box
immler@56189
  1251
        using i
immler@56189
  1252
        by blast+
immler@56189
  1253
      then have "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
wenzelm@60420
  1254
        using \<open>e>0\<close> i
immler@56189
  1255
        by (auto simp: inner_diff_left inner_Basis inner_add_left)
immler@56189
  1256
    }
immler@56189
  1257
    then have "x \<in> box a b"
immler@56189
  1258
      unfolding mem_box by auto
immler@56189
  1259
  }
immler@56189
  1260
  then show "?L \<subseteq> ?R" ..
immler@56189
  1261
qed
immler@56189
  1262
lp15@63928
  1263
lemma bounded_cbox [simp]:
immler@56189
  1264
  fixes a :: "'a::euclidean_space"
immler@56189
  1265
  shows "bounded (cbox a b)"
immler@56189
  1266
proof -
immler@56189
  1267
  let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
immler@56189
  1268
  {
immler@56189
  1269
    fix x :: "'a"
lp15@68120
  1270
    assume "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
immler@56189
  1271
    then have "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b"
lp15@68120
  1272
      by (force simp: intro!: sum_mono)
immler@56189
  1273
    then have "norm x \<le> ?b"
immler@56189
  1274
      using norm_le_l1[of x] by auto
immler@56189
  1275
  }
immler@56189
  1276
  then show ?thesis
lp15@68120
  1277
    unfolding cbox_def bounded_iff by force
immler@56189
  1278
qed
immler@56189
  1279
lp15@62618
  1280
lemma bounded_box [simp]:
immler@56189
  1281
  fixes a :: "'a::euclidean_space"
immler@56189
  1282
  shows "bounded (box a b)"
lp15@68120
  1283
  using bounded_cbox[of a b] box_subset_cbox[of a b] bounded_subset[of "cbox a b" "box a b"]
immler@56189
  1284
  by simp
immler@56189
  1285
lp15@62618
  1286
lemma not_interval_UNIV [simp]:
immler@56189
  1287
  fixes a :: "'a::euclidean_space"
immler@56189
  1288
  shows "cbox a b \<noteq> UNIV" "box a b \<noteq> UNIV"
lp15@62618
  1289
  using bounded_box[of a b] bounded_cbox[of a b] by force+
lp15@62618
  1290
lp15@63945
  1291
lemma not_interval_UNIV2 [simp]:
lp15@63945
  1292
  fixes a :: "'a::euclidean_space"
lp15@63945
  1293
  shows "UNIV \<noteq> cbox a b" "UNIV \<noteq> box a b"
lp15@63945
  1294
  using bounded_box[of a b] bounded_cbox[of a b] by force+
lp15@63945
  1295
immler@56189
  1296
lemma box_midpoint:
immler@56189
  1297
  fixes a :: "'a::euclidean_space"
immler@56189
  1298
  assumes "box a b \<noteq> {}"
immler@56189
  1299
  shows "((1/2) *\<^sub>R (a + b)) \<in> box a b"
immler@56189
  1300
proof -
lp15@68120
  1301
  have "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i" if "i \<in> Basis" for i
lp15@68120
  1302
    using assms that by (auto simp: inner_add_left box_ne_empty)
immler@56189
  1303
  then show ?thesis unfolding mem_box by auto
immler@56189
  1304
qed
immler@56189
  1305
immler@56189
  1306
lemma open_cbox_convex:
immler@56189
  1307
  fixes x :: "'a::euclidean_space"
immler@56189
  1308
  assumes x: "x \<in> box a b"
immler@56189
  1309
    and y: "y \<in> cbox a b"
immler@56189
  1310
    and e: "0 < e" "e \<le> 1"
immler@56189
  1311
  shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> box a b"
immler@56189
  1312
proof -
immler@56189
  1313
  {
immler@56189
  1314
    fix i :: 'a
immler@56189
  1315
    assume i: "i \<in> Basis"
immler@56189
  1316
    have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)"
immler@56189
  1317
      unfolding left_diff_distrib by simp
immler@56189
  1318
    also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)"
lp15@68120
  1319
    proof (rule add_less_le_mono)
lp15@68120
  1320
      show "e * (a \<bullet> i) < e * (x \<bullet> i)"
lp15@68120
  1321
        using \<open>0 < e\<close> i mem_box(1) x by auto
lp15@68120
  1322
      show "(1 - e) * (a \<bullet> i) \<le> (1 - e) * (y \<bullet> i)"
lp15@68120
  1323
        by (meson diff_ge_0_iff_ge \<open>e \<le> 1\<close> i mem_box(2) mult_left_mono y)
lp15@68120
  1324
    qed
immler@56189
  1325
    finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i"
immler@56189
  1326
      unfolding inner_simps by auto
immler@56189
  1327
    moreover
immler@56189
  1328
    {
immler@56189
  1329
      have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)"
immler@56189
  1330
        unfolding left_diff_distrib by simp
immler@56189
  1331
      also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)"
lp15@66827
  1332
      proof (rule add_less_le_mono)
lp15@66827
  1333
        show "e * (x \<bullet> i) < e * (b \<bullet> i)"
lp15@68120
  1334
          using \<open>0 < e\<close> i mem_box(1) x by auto
lp15@66827
  1335
        show "(1 - e) * (y \<bullet> i) \<le> (1 - e) * (b \<bullet> i)"
lp15@68120
  1336
          by (meson diff_ge_0_iff_ge \<open>e \<le> 1\<close> i mem_box(2) mult_left_mono y)
lp15@66827
  1337
      qed
immler@56189
  1338
      finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
immler@56189
  1339
        unfolding inner_simps by auto
immler@56189
  1340
    }
immler@56189
  1341
    ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
immler@56189
  1342
      by auto
immler@56189
  1343
  }
immler@56189
  1344
  then show ?thesis
immler@56189
  1345
    unfolding mem_box by auto
immler@56189
  1346
qed
immler@56189
  1347
lp15@63881
  1348
lemma closure_cbox [simp]: "closure (cbox a b) = cbox a b"
lp15@63881
  1349
  by (simp add: closed_cbox)
lp15@63881
  1350
lp15@63881
  1351
lemma closure_box [simp]:
immler@56189
  1352
  fixes a :: "'a::euclidean_space"
immler@56189
  1353
   assumes "box a b \<noteq> {}"
immler@56189
  1354
  shows "closure (box a b) = cbox a b"
immler@56189
  1355
proof -
immler@56189
  1356
  have ab: "a <e b"
immler@56189
  1357
    using assms by (simp add: eucl_less_def box_ne_empty)
immler@56189
  1358
  let ?c = "(1 / 2) *\<^sub>R (a + b)"
immler@56189
  1359
  {
immler@56189
  1360
    fix x
immler@56189
  1361
    assume as:"x \<in> cbox a b"
wenzelm@63040
  1362
    define f where [abs_def]: "f n = x + (inverse (real n + 1)) *\<^sub>R (?c - x)" for n
immler@56189
  1363
    {
immler@56189
  1364
      fix n
immler@56189
  1365
      assume fn: "f n <e b \<longrightarrow> a <e f n \<longrightarrow> f n = x" and xc: "x \<noteq> ?c"
immler@56189
  1366
      have *: "0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1"
immler@56189
  1367
        unfolding inverse_le_1_iff by auto
immler@56189
  1368
      have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
immler@56189
  1369
        x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
lp15@66643
  1370
        by (auto simp: algebra_simps)
immler@56189
  1371
      then have "f n <e b" and "a <e f n"
immler@56189
  1372
        using open_cbox_convex[OF box_midpoint[OF assms] as *]
immler@56189
  1373
        unfolding f_def by (auto simp: box_def eucl_less_def)
immler@56189
  1374
      then have False
immler@56189
  1375
        using fn unfolding f_def using xc by auto
immler@56189
  1376
    }
immler@56189
  1377
    moreover
immler@56189
  1378
    {
wenzelm@61973
  1379
      assume "\<not> (f \<longlongrightarrow> x) sequentially"
immler@56189
  1380
      {
immler@56189
  1381
        fix e :: real
immler@56189
  1382
        assume "e > 0"
lp15@61609
  1383
        then obtain N :: nat where N: "inverse (real (N + 1)) < e"
lp15@68120
  1384
          using reals_Archimedean by auto
lp15@61609
  1385
        have "inverse (real n + 1) < e" if "N \<le> n" for n
lp15@61609
  1386
          by (auto intro!: that le_less_trans [OF _ N])
immler@56189
  1387
        then have "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto
immler@56189
  1388
      }
wenzelm@61973
  1389
      then have "((\<lambda>n. inverse (real n + 1)) \<longlongrightarrow> 0) sequentially"
lp15@66643
  1390
        unfolding lim_sequentially by(auto simp: dist_norm)
wenzelm@61973
  1391
      then have "(f \<longlongrightarrow> x) sequentially"
immler@56189
  1392
        unfolding f_def
immler@56189
  1393
        using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
immler@56189
  1394
        using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"]
immler@56189
  1395
        by auto
immler@56189
  1396
    }
immler@56189
  1397
    ultimately have "x \<in> closure (box a b)"
lp15@68120
  1398
      using as box_midpoint[OF assms]
lp15@68120
  1399
      unfolding closure_def islimpt_sequential
immler@56189
  1400
      by (cases "x=?c") (auto simp: in_box_eucl_less)
immler@56189
  1401
  }
immler@56189
  1402
  then show ?thesis
immler@56189
  1403
    using closure_minimal[OF box_subset_cbox, of a b] by blast
immler@56189
  1404
qed
immler@56189
  1405
immler@56189
  1406
lemma bounded_subset_box_symmetric:
lp15@68120
  1407
  fixes S :: "('a::euclidean_space) set"
lp15@68120
  1408
  assumes "bounded S"
lp15@68120
  1409
  obtains a where "S \<subseteq> box (-a) a"
immler@56189
  1410
proof -
lp15@68120
  1411
  obtain b where "b>0" and b: "\<forall>x\<in>S. norm x \<le> b"
immler@56189
  1412
    using assms[unfolded bounded_pos] by auto
wenzelm@63040
  1413
  define a :: 'a where "a = (\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)"
lp15@68120
  1414
  have "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i" if "x \<in> S" and i: "i \<in> Basis" for x i
lp15@68120
  1415
    using b Basis_le_norm[OF i, of x] that by (auto simp: a_def)
lp15@68120
  1416
  then have "S \<subseteq> box (-a) a"
lp15@68120
  1417
    by (auto simp: simp add: box_def)
lp15@68120
  1418
  then show ?thesis ..
immler@56189
  1419
qed
immler@56189
  1420
immler@56189
  1421
lemma bounded_subset_cbox_symmetric:
lp15@68120
  1422
  fixes S :: "('a::euclidean_space) set"
lp15@68120
  1423
  assumes "bounded S"
lp15@68120
  1424
  obtains a where "S \<subseteq> cbox (-a) a"
immler@56189
  1425
proof -
lp15@68120
  1426
  obtain a where "S \<subseteq> box (-a) a"
immler@56189
  1427
    using bounded_subset_box_symmetric[OF assms] by auto
immler@56189
  1428
  then show ?thesis
lp15@68120
  1429
    by (meson box_subset_cbox dual_order.trans that)
immler@56189
  1430
qed
immler@56189
  1431
immler@56189
  1432
lemma frontier_cbox:
immler@56189
  1433
  fixes a b :: "'a::euclidean_space"
immler@56189
  1434
  shows "frontier (cbox a b) = cbox a b - box a b"
immler@56189
  1435
  unfolding frontier_def unfolding interior_cbox and closure_closed[OF closed_cbox] ..
immler@56189
  1436
immler@56189
  1437
lemma frontier_box:
immler@56189
  1438
  fixes a b :: "'a::euclidean_space"
immler@56189
  1439
  shows "frontier (box a b) = (if box a b = {} then {} else cbox a b - box a b)"
immler@56189
  1440
proof (cases "box a b = {}")
immler@56189
  1441
  case True
immler@56189
  1442
  then show ?thesis
immler@56189
  1443
    using frontier_empty by auto
immler@56189
  1444
next
immler@56189
  1445
  case False
immler@56189
  1446
  then show ?thesis
immler@56189
  1447
    unfolding frontier_def and closure_box[OF False] and interior_open[OF open_box]
immler@56189
  1448
    by auto
immler@56189
  1449
qed
immler@56189
  1450
lp15@66884
  1451
lemma Int_interval_mixed_eq_empty:
immler@56189
  1452
  fixes a :: "'a::euclidean_space"
immler@56189
  1453
   assumes "box c d \<noteq> {}"
immler@56189
  1454
  shows "box a b \<inter> cbox c d = {} \<longleftrightarrow> box a b \<inter> box c d = {}"
immler@56189
  1455
  unfolding closure_box[OF assms, symmetric]
lp15@62843
  1456
  unfolding open_Int_closure_eq_empty[OF open_box] ..
immler@56189
  1457
immler@69611
  1458
subsection \<open>Class Instances\<close>
immler@69611
  1459
immler@69611
  1460
lemma compact_lemma:
immler@69611
  1461
  fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
immler@69611
  1462
  assumes "bounded (range f)"
immler@69611
  1463
  shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.
immler@69611
  1464
    strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
immler@69611
  1465
  by (rule compact_lemma_general[where unproj="\<lambda>e. \<Sum>i\<in>Basis. e i *\<^sub>R i"])
immler@69611
  1466
     (auto intro!: assms bounded_linear_inner_left bounded_linear_image
immler@69611
  1467
       simp: euclidean_representation)
immler@69611
  1468
immler@69611
  1469
instance%important euclidean_space \<subseteq> heine_borel
immler@69611
  1470
proof%unimportant
immler@69611
  1471
  fix f :: "nat \<Rightarrow> 'a"
immler@69611
  1472
  assume f: "bounded (range f)"
immler@69611
  1473
  then obtain l::'a and r where r: "strict_mono r"
immler@69611
  1474
    and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
immler@69611
  1475
    using compact_lemma [OF f] by blast
immler@69611
  1476
  {
immler@69611
  1477
    fix e::real
immler@69611
  1478
    assume "e > 0"
immler@69611
  1479
    hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive)
immler@69611
  1480
    with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))) sequentially"
immler@69611
  1481
      by simp
immler@69611
  1482
    moreover
immler@69611
  1483
    {
immler@69611
  1484
      fix n
immler@69611
  1485
      assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"
immler@69611
  1486
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"
immler@69611
  1487
        apply (subst euclidean_dist_l2)
immler@69611
  1488
        using zero_le_dist
immler@69611
  1489
        apply (rule L2_set_le_sum)
immler@69611
  1490
        done
immler@69611
  1491
      also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
immler@69611
  1492
        apply (rule sum_strict_mono)
immler@69611
  1493
        using n
immler@69611
  1494
        apply auto
immler@69611
  1495
        done
immler@69611
  1496
      finally have "dist (f (r n)) l < e"
immler@69611
  1497
        by auto
immler@69611
  1498
    }
immler@69611
  1499
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
immler@69611
  1500
      by (rule eventually_mono)
immler@69611
  1501
  }
immler@69611
  1502
  then have *: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
immler@69611
  1503
    unfolding o_def tendsto_iff by simp
immler@69611
  1504
  with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
immler@69611
  1505
    by auto
immler@69611
  1506
qed
immler@69611
  1507
immler@69611
  1508
instance%important euclidean_space \<subseteq> banach ..
immler@69611
  1509
immler@69611
  1510
instance euclidean_space \<subseteq> second_countable_topology
immler@69611
  1511
proof
immler@69611
  1512
  define a where "a f = (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i)" for f :: "'a \<Rightarrow> real \<times> real"
immler@69611
  1513
  then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f"
immler@69611
  1514
    by simp
immler@69611
  1515
  define b where "b f = (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i)" for f :: "'a \<Rightarrow> real \<times> real"
immler@69611
  1516
  then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f"
immler@69611
  1517
    by simp
immler@69611
  1518
  define B where "B = (\<lambda>f. box (a f) (b f)) ` (Basis \<rightarrow>\<^sub>E (\<rat> \<times> \<rat>))"
immler@69611
  1519
immler@69611
  1520
  have "Ball B open" by (simp add: B_def open_box)
immler@69611
  1521
  moreover have "(\<forall>A. open A \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = A))"
immler@69611
  1522
  proof safe
immler@69611
  1523
    fix A::"'a set"
immler@69611
  1524
    assume "open A"
immler@69611
  1525
    show "\<exists>B'\<subseteq>B. \<Union>B' = A"
immler@69611
  1526
      apply (rule exI[of _ "{b\<in>B. b \<subseteq> A}"])
immler@69611
  1527
      apply (subst (3) open_UNION_box[OF \<open>open A\<close>])
immler@69611
  1528
      apply (auto simp: a b B_def)
immler@69611
  1529
      done
immler@69611
  1530
  qed
immler@69611
  1531
  ultimately
immler@69611
  1532
  have "topological_basis B"
immler@69611
  1533
    unfolding topological_basis_def by blast
immler@69611
  1534
  moreover
immler@69611
  1535
  have "countable B"
immler@69611
  1536
    unfolding B_def
immler@69611
  1537
    by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)
immler@69611
  1538
  ultimately show "\<exists>B::'a set set. countable B \<and> open = generate_topology B"
immler@69611
  1539
    by (blast intro: topological_basis_imp_subbasis)
immler@69611
  1540
qed
immler@69611
  1541
immler@69611
  1542
instance euclidean_space \<subseteq> polish_space ..
immler@69611
  1543
immler@69611
  1544
immler@69611
  1545
subsection \<open>Compact Boxes\<close>
immler@69611
  1546
immler@69611
  1547
lemma compact_cbox [simp]:
wenzelm@61076
  1548
  fixes a :: "'a::euclidean_space"
immler@69611
  1549
  shows "compact (cbox a b)"
immler@69611
  1550
  using bounded_closed_imp_seq_compact[of "cbox a b"] using bounded_cbox[of a b]
immler@69611
  1551
  by (auto simp: compact_eq_seq_compact_metric)
immler@69611
  1552
immler@69611
  1553
proposition is_interval_compact:
immler@69611
  1554
   "is_interval S \<and> compact S \<longleftrightarrow> (\<exists>a b. S = cbox a b)"   (is "?lhs = ?rhs")
immler@69611
  1555
proof (cases "S = {}")
immler@69611
  1556
  case True
immler@69611
  1557
  with empty_as_interval show ?thesis by auto
immler@69611
  1558
next
immler@69611
  1559
  case False
immler@69611
  1560
  show ?thesis
immler@69611
  1561
  proof
immler@69611
  1562
    assume L: ?lhs
immler@69611
  1563
    then have "is_interval S" "compact S" by auto
immler@69611
  1564
    define a where "a \<equiv> \<Sum>i\<in>Basis. (INF x\<in>S. x \<bullet> i) *\<^sub>R i"
immler@69611
  1565
    define b where "b \<equiv> \<Sum>i\<in>Basis. (SUP x\<in>S. x \<bullet> i) *\<^sub>R i"
immler@69611
  1566
    have 1: "\<And>x i. \<lbrakk>x \<in> S; i \<in> Basis\<rbrakk> \<Longrightarrow> (INF x\<in>S. x \<bullet> i) \<le> x \<bullet> i"
immler@69611
  1567
      by (simp add: cInf_lower bounded_inner_imp_bdd_below compact_imp_bounded L)
immler@69611
  1568
    have 2: "\<And>x i. \<lbrakk>x \<in> S; i \<in> Basis\<rbrakk> \<Longrightarrow> x \<bullet> i \<le> (SUP x\<in>S. x \<bullet> i)"
immler@69611
  1569
      by (simp add: cSup_upper bounded_inner_imp_bdd_above compact_imp_bounded L)
immler@69611
  1570
    have 3: "x \<in> S" if inf: "\<And>i. i \<in> Basis \<Longrightarrow> (INF x\<in>S. x \<bullet> i) \<le> x \<bullet> i"
immler@69611
  1571
                   and sup: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<le> (SUP x\<in>S. x \<bullet> i)" for x
immler@69611
  1572
    proof (rule mem_box_componentwiseI [OF \<open>is_interval S\<close>])
immler@69611
  1573
      fix i::'a
immler@69611
  1574
      assume i: "i \<in> Basis"
immler@69611
  1575
      have cont: "continuous_on S (\<lambda>x. x \<bullet> i)"
immler@69611
  1576
        by (intro continuous_intros)
immler@69611
  1577
      obtain a where "a \<in> S" and a: "\<And>y. y\<in>S \<Longrightarrow> a \<bullet> i \<le> y \<bullet> i"
immler@69611
  1578
        using continuous_attains_inf [OF \<open>compact S\<close> False cont] by blast
immler@69611
  1579
      obtain b where "b \<in> S" and b: "\<And>y. y\<in>S \<Longrightarrow> y \<bullet> i \<le> b \<bullet> i"
immler@69611
  1580
        using continuous_attains_sup [OF \<open>compact S\<close> False cont] by blast
immler@69611
  1581
      have "a \<bullet> i \<le> (INF x\<in>S. x \<bullet> i)"
immler@69611
  1582
        by (simp add: False a cINF_greatest)
immler@69611
  1583
      also have "\<dots> \<le> x \<bullet> i"
immler@69611
  1584
        by (simp add: i inf)
immler@69611
  1585
      finally have ai: "a \<bullet> i \<le> x \<bullet> i" .
immler@69611
  1586
      have "x \<bullet> i \<le> (SUP x\<in>S. x \<bullet> i)"
immler@69611
  1587
        by (simp add: i sup)
immler@69611
  1588
      also have "(SUP x\<in>S. x \<bullet> i) \<le> b \<bullet> i"
immler@69611
  1589
        by (simp add: False b cSUP_least)
immler@69611
  1590
      finally have bi: "x \<bullet> i \<le> b \<bullet> i" .
immler@69611
  1591
      show "x \<bullet> i \<in> (\<lambda>x. x \<bullet> i) ` S"
immler@69611
  1592
        apply (rule_tac x="\<Sum>j\<in>Basis. (if j = i then x \<bullet> i else a \<bullet> j) *\<^sub>R j" in image_eqI)
immler@69611
  1593
        apply (simp add: i)
immler@69611
  1594
        apply (rule mem_is_intervalI [OF \<open>is_interval S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>])
immler@69611
  1595
        using i ai bi apply force
immler@69611
  1596
        done
immler@69611
  1597
    qed
immler@69611
  1598
    have "S = cbox a b"
immler@69611
  1599
      by (auto simp: a_def b_def mem_box intro: 1 2 3)
immler@69611
  1600
    then show ?rhs
immler@69611
  1601
      by blast
immler@69611
  1602
  next
immler@69611
  1603
    assume R: ?rhs
immler@69611
  1604
    then show ?lhs
immler@69611
  1605
      using compact_cbox is_interval_cbox by blast
immler@69611
  1606
  qed
immler@69611
  1607
qed
immler@69611
  1608
immler@69611
  1609
immler@69615
  1610
subsection%unimportant\<open>Componentwise limits and continuity\<close>
immler@69615
  1611
immler@69615
  1612
text\<open>But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}\<close>
immler@69615
  1613
lemma Euclidean_dist_upper: "i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y"
immler@69615
  1614
  by (metis (no_types) member_le_L2_set euclidean_dist_l2 finite_Basis)
immler@69615
  1615
immler@69615
  1616
text\<open>But is the premise \<^term>\<open>i \<in> Basis\<close> really necessary?\<close>
immler@69615
  1617
lemma open_preimage_inner:
immler@69615
  1618
  assumes "open S" "i \<in> Basis"
immler@69615
  1619
    shows "open {x. x \<bullet> i \<in> S}"
immler@69615
  1620
proof (rule openI, simp)
immler@69615
  1621
  fix x
immler@69615
  1622
  assume x: "x \<bullet> i \<in> S"
immler@69615
  1623
  with assms obtain e where "0 < e" and e: "ball (x \<bullet> i) e \<subseteq> S"
immler@69615
  1624
    by (auto simp: open_contains_ball_eq)
immler@69615
  1625
  have "\<exists>e>0. ball (y \<bullet> i) e \<subseteq> S" if dxy: "dist x y < e / 2" for y
immler@69615
  1626
  proof (intro exI conjI)
immler@69615
  1627
    have "dist (x \<bullet> i) (y \<bullet> i) < e / 2"
immler@69615
  1628
      by (meson \<open>i \<in> Basis\<close> dual_order.trans Euclidean_dist_upper not_le that)
immler@69615
  1629
    then have "dist (x \<bullet> i) z < e" if "dist (y \<bullet> i) z < e / 2" for z
immler@69615
  1630
      by (metis dist_commute dist_triangle_half_l that)
immler@69615
  1631
    then have "ball (y \<bullet> i) (e / 2) \<subseteq> ball (x \<bullet> i) e"
immler@69615
  1632
      using mem_ball by blast
immler@69615
  1633
      with e show "ball (y \<bullet> i) (e / 2) \<subseteq> S"
immler@69615
  1634
        by (metis order_trans)
immler@69615
  1635
  qed (simp add: \<open>0 < e\<close>)
immler@69615
  1636
  then show "\<exists>e>0. ball x e \<subseteq> {s. s \<bullet> i \<in> S}"
immler@69615
  1637
    by (metis (no_types, lifting) \<open>0 < e\<close> \<open>open S\<close> half_gt_zero_iff mem_Collect_eq mem_ball open_contains_ball_eq subsetI)
immler@69615
  1638
qed
immler@69615
  1639
immler@69615
  1640
proposition tendsto_componentwise_iff:
immler@69615
  1641
  fixes f :: "_ \<Rightarrow> 'b::euclidean_space"
immler@69615
  1642
  shows "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>i \<in> Basis. ((\<lambda>x. (f x \<bullet> i)) \<longlongrightarrow> (l \<bullet> i)) F)"
immler@69615
  1643
         (is "?lhs = ?rhs")
immler@69615
  1644
proof
immler@69615
  1645
  assume ?lhs
immler@69615
  1646
  then show ?rhs
immler@69615
  1647
    unfolding tendsto_def
immler@69615
  1648
    apply clarify
immler@69615
  1649
    apply (drule_tac x="{s. s \<bullet> i \<in> S}" in spec)
immler@69615
  1650
    apply (auto simp: open_preimage_inner)
immler@69615
  1651
    done
immler@69615
  1652
next
immler@69615
  1653
  assume R: ?rhs
immler@69615
  1654
  then have "\<And>e. e > 0 \<Longrightarrow> \<forall>i\<in>Basis. \<forall>\<^sub>F x in F. dist (f x \<bullet> i) (l \<bullet> i) < e"
immler@69615
  1655
    unfolding tendsto_iff by blast
immler@69615
  1656
  then have R': "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e"
immler@69615
  1657
      by (simp add: eventually_ball_finite_distrib [symmetric])
immler@69615
  1658
  show ?lhs
immler@69615
  1659
  unfolding tendsto_iff
immler@69615
  1660
  proof clarify
immler@69615
  1661
    fix e::real
immler@69615
  1662
    assume "0 < e"
immler@69615
  1663
    have *: "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e"
immler@69615
  1664
             if "\<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / real DIM('b)" for x
immler@69615
  1665
    proof -
immler@69615
  1666
      have "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis \<le> sum (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis"
immler@69615
  1667
        by (simp add: L2_set_le_sum)
immler@69615
  1668
      also have "... < DIM('b) * (e / real DIM('b))"
immler@69615
  1669
        apply (rule sum_bounded_above_strict)
immler@69615
  1670
        using that by auto
immler@69615
  1671
      also have "... = e"
immler@69615
  1672
        by (simp add: field_simps)
immler@69615
  1673
      finally show "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e" .
immler@69615
  1674
    qed
immler@69615
  1675
    have "\<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / DIM('b)"
immler@69615
  1676
      apply (rule R')
immler@69615
  1677
      using \<open>0 < e\<close> by simp
immler@69615
  1678
    then show "\<forall>\<^sub>F x in F. dist (f x) l < e"
immler@69615
  1679
      apply (rule eventually_mono)
immler@69615
  1680
      apply (subst euclidean_dist_l2)
immler@69615
  1681
      using * by blast
immler@69615
  1682
  qed
immler@69615
  1683
qed
immler@69615
  1684
immler@69615
  1685
immler@69615
  1686
corollary continuous_componentwise:
immler@69615
  1687
   "continuous F f \<longleftrightarrow> (\<forall>i \<in> Basis. continuous F (\<lambda>x. (f x \<bullet> i)))"
immler@69615
  1688
by (simp add: continuous_def tendsto_componentwise_iff [symmetric])
immler@69615
  1689
immler@69615
  1690
corollary continuous_on_componentwise:
immler@69615
  1691
  fixes S :: "'a :: t2_space set"
immler@69615
  1692
  shows "continuous_on S f \<longleftrightarrow> (\<forall>i \<in> Basis. continuous_on S (\<lambda>x. (f x \<bullet> i)))"
immler@69615
  1693
  apply (simp add: continuous_on_eq_continuous_within)
immler@69615
  1694
  using continuous_componentwise by blast
immler@69615
  1695
immler@69615
  1696
lemma linear_componentwise_iff:
immler@69615
  1697
     "(linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. linear (\<lambda>x. f' x \<bullet> i))"
immler@69615
  1698
  apply (auto simp: linear_iff inner_left_distrib)
immler@69615
  1699
   apply (metis inner_left_distrib euclidean_eq_iff)
immler@69615
  1700
  by (metis euclidean_eqI inner_scaleR_left)
immler@69615
  1701
immler@69615
  1702
lemma bounded_linear_componentwise_iff:
immler@69615
  1703
     "(bounded_linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. bounded_linear (\<lambda>x. f' x \<bullet> i))"
immler@69615
  1704
     (is "?lhs = ?rhs")
immler@69615
  1705
proof
immler@69615
  1706
  assume ?lhs then show ?rhs
immler@69615
  1707
    by (simp add: bounded_linear_inner_left_comp)
immler@69615
  1708
next
immler@69615
  1709
  assume ?rhs
immler@69615
  1710
  then have "(\<forall>i\<in>Basis. \<exists>K. \<forall>x. \<bar>f' x \<bullet> i\<bar> \<le> norm x * K)" "linear f'"
immler@69615
  1711
    by (auto simp: bounded_linear_def bounded_linear_axioms_def linear_componentwise_iff [symmetric] ball_conj_distrib)
immler@69615
  1712
  then obtain F where F: "\<And>i x. i \<in> Basis \<Longrightarrow> \<bar>f' x \<bullet> i\<bar> \<le> norm x * F i"
immler@69615
  1713
    by metis
immler@69615
  1714
  have "norm (f' x) \<le> norm x * sum F Basis" for x
immler@69615
  1715
  proof -
immler@69615
  1716
    have "norm (f' x) \<le> (\<Sum>i\<in>Basis. \<bar>f' x \<bullet> i\<bar>)"
immler@69615
  1717
      by (rule norm_le_l1)
immler@69615
  1718
    also have "... \<le> (\<Sum>i\<in>Basis. norm x * F i)"
immler@69615
  1719
      by (metis F sum_mono)
immler@69615
  1720
    also have "... = norm x * sum F Basis"
immler@69615
  1721
      by (simp add: sum_distrib_left)
immler@69615
  1722
    finally show ?thesis .
immler@69615
  1723
  qed
immler@69615
  1724
  then show ?lhs
immler@69615
  1725
    by (force simp: bounded_linear_def bounded_linear_axioms_def \<open>linear f'\<close>)
immler@69615
  1726
qed
immler@69615
  1727
immler@69615
  1728
subsection%unimportant \<open>Continuous Extension\<close>
immler@69615
  1729
immler@69615
  1730
definition clamp :: "'a::euclidean_space \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where
immler@69615
  1731
  "clamp a b x = (if (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)
immler@69615
  1732
    then (\<Sum>i\<in>Basis. (if x\<bullet>i < a\<bullet>i then a\<bullet>i else if x\<bullet>i \<le> b\<bullet>i then x\<bullet>i else b\<bullet>i) *\<^sub>R i)
immler@69615
  1733
    else a)"
immler@69615
  1734
immler@69615
  1735
lemma clamp_in_interval[simp]:
immler@69615
  1736
  assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
immler@69615
  1737
  shows "clamp a b x \<in> cbox a b"
immler@69615
  1738
  unfolding clamp_def
immler@69615
  1739
  using box_ne_empty(1)[of a b] assms by (auto simp: cbox_def)
immler@69615
  1740
immler@69615
  1741
lemma clamp_cancel_cbox[simp]:
immler@69615
  1742
  fixes x a b :: "'a::euclidean_space"
immler@69615
  1743
  assumes x: "x \<in> cbox a b"
immler@69615
  1744
  shows "clamp a b x = x"
immler@69615
  1745
  using assms
immler@69615
  1746
  by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a])
immler@69615
  1747
immler@69615
  1748
lemma clamp_empty_interval:
immler@69615
  1749
  assumes "i \<in> Basis" "a \<bullet> i > b \<bullet> i"
immler@69615
  1750
  shows "clamp a b = (\<lambda>_. a)"
immler@69615
  1751
  using assms
immler@69615
  1752
  by (force simp: clamp_def[abs_def] split: if_splits intro!: ext)
immler@69615
  1753
immler@69615
  1754
lemma dist_clamps_le_dist_args:
immler@69615
  1755
  fixes x :: "'a::euclidean_space"
immler@69615
  1756
  shows "dist (clamp a b y) (clamp a b x) \<le> dist y x"
immler@69615
  1757
proof cases
immler@69615
  1758
  assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
immler@69615
  1759
  then have "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2) \<le>
immler@69615
  1760
    (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
immler@69615
  1761
    by (auto intro!: sum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
immler@69615
  1762
  then show ?thesis
immler@69615
  1763
    by (auto intro: real_sqrt_le_mono
immler@69615
  1764
      simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] L2_set_def)
immler@69615
  1765
qed (auto simp: clamp_def)
immler@69615
  1766
immler@69615
  1767
lemma clamp_continuous_at:
immler@69615
  1768
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
immler@69615
  1769
    and x :: 'a
immler@69615
  1770
  assumes f_cont: "continuous_on (cbox a b) f"
immler@69615
  1771
  shows "continuous (at x) (\<lambda>x. f (clamp a b x))"
immler@69615
  1772
proof cases
immler@69615
  1773
  assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
immler@69615
  1774
  show ?thesis
immler@69615
  1775
    unfolding continuous_at_eps_delta
immler@69615
  1776
  proof safe
immler@69615
  1777
    fix x :: 'a
immler@69615
  1778
    fix e :: real
immler@69615
  1779
    assume "e > 0"
immler@69615
  1780
    moreover have "clamp a b x \<in> cbox a b"
immler@69615
  1781
      by (simp add: clamp_in_interval le)
immler@69615
  1782
    moreover note f_cont[simplified continuous_on_iff]
immler@69615
  1783
    ultimately
immler@69615
  1784
    obtain d where d: "0 < d"
immler@69615
  1785
      "\<And>x'. x' \<in> cbox a b \<Longrightarrow> dist x' (clamp a b x) < d \<Longrightarrow> dist (f x') (f (clamp a b x)) < e"
immler@69615
  1786
      by force
immler@69615
  1787
    show "\<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow>
immler@69615
  1788
      dist (f (clamp a b x')) (f (clamp a b x)) < e"
immler@69615
  1789
      using le
immler@69615
  1790
      by (auto intro!: d clamp_in_interval dist_clamps_le_dist_args[THEN le_less_trans])
immler@69615
  1791
  qed
immler@69615
  1792
qed (auto simp: clamp_empty_interval)
immler@69615
  1793
immler@69615
  1794
lemma clamp_continuous_on:
immler@69615
  1795
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
immler@69615
  1796
  assumes f_cont: "continuous_on (cbox a b) f"
immler@69615
  1797
  shows "continuous_on S (\<lambda>x. f (clamp a b x))"
immler@69615
  1798
  using assms
immler@69615
  1799
  by (auto intro: continuous_at_imp_continuous_on clamp_continuous_at)
immler@69615
  1800
immler@69615
  1801
lemma clamp_bounded:
immler@69615
  1802
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
immler@69615
  1803
  assumes bounded: "bounded (f ` (cbox a b))"
immler@69615
  1804
  shows "bounded (range (\<lambda>x. f (clamp a b x)))"
immler@69615
  1805
proof cases
immler@69615
  1806
  assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
immler@69615
  1807
  from bounded obtain c where f_bound: "\<forall>x\<in>f ` cbox a b. dist undefined x \<le> c"
immler@69615
  1808
    by (auto simp: bounded_any_center[where a=undefined])
immler@69615
  1809
  then show ?thesis
immler@69615
  1810
    by (auto intro!: exI[where x=c] clamp_in_interval[OF le[rule_format]]
immler@69615
  1811
        simp: bounded_any_center[where a=undefined])
immler@69615
  1812
qed (auto simp: clamp_empty_interval image_def)
immler@69615
  1813
immler@69615
  1814
immler@69615
  1815
definition ext_cont :: "('a::euclidean_space \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"
immler@69615
  1816
  where "ext_cont f a b = (\<lambda>x. f (clamp a b x))"
immler@69615
  1817
immler@69615
  1818
lemma ext_cont_cancel_cbox[simp]:
immler@69615
  1819
  fixes x a b :: "'a::euclidean_space"
immler@69615
  1820
  assumes x: "x \<in> cbox a b"
immler@69615
  1821
  shows "ext_cont f a b x = f x"
immler@69615
  1822
  using assms
immler@69615
  1823
  unfolding ext_cont_def
immler@69615
  1824
  by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a] arg_cong[where f=f])
immler@69615
  1825
immler@69615
  1826
lemma continuous_on_ext_cont[continuous_intros]:
immler@69615
  1827
  "continuous_on (cbox a b) f \<Longrightarrow> continuous_on S (ext_cont f a b)"
immler@69615
  1828
  by (auto intro!: clamp_continuous_on simp: ext_cont_def)
immler@69615
  1829
immler@69615
  1830
immler@69615
  1831
subsection \<open>Separability\<close>
immler@69615
  1832
immler@69615
  1833
lemma univ_second_countable_sequence:
immler@69615
  1834
  obtains B :: "nat \<Rightarrow> 'a::euclidean_space set"
immler@69615
  1835
    where "inj B" "\<And>n. open(B n)" "\<And>S. open S \<Longrightarrow> \<exists>k. S = \<Union>{B n |n. n \<in> k}"
immler@69615
  1836
proof -
immler@69615
  1837
  obtain \<B> :: "'a set set"
immler@69615
  1838
  where "countable \<B>"
immler@69615
  1839
    and opn: "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
immler@69615
  1840
    and Un: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
immler@69615
  1841
    using univ_second_countable by blast
immler@69615
  1842
  have *: "infinite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"
immler@69615
  1843
    apply (rule Infinite_Set.range_inj_infinite)
immler@69615
  1844
    apply (simp add: inj_on_def ball_eq_ball_iff)
immler@69615
  1845
    done
immler@69615
  1846
  have "infinite \<B>"
immler@69615
  1847
  proof
immler@69615
  1848
    assume "finite \<B>"
immler@69615
  1849
    then have "finite (Union ` (Pow \<B>))"
immler@69615
  1850
      by simp
immler@69615
  1851
    then have "finite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"
immler@69615
  1852
      apply (rule rev_finite_subset)
immler@69615
  1853
      by (metis (no_types, lifting) PowI image_eqI image_subset_iff Un [OF open_ball])
immler@69615
  1854
    with * show False by simp
immler@69615
  1855
  qed
immler@69615
  1856
  obtain f :: "nat \<Rightarrow> 'a set" where "\<B> = range f" "inj f"
immler@69615
  1857
    by (blast intro: countable_as_injective_image [OF \<open>countable \<B>\<close> \<open>infinite \<B>\<close>])
immler@69615
  1858
  have *: "\<exists>k. S = \<Union>{f n |n. n \<in> k}" if "open S" for S
immler@69615
  1859
    using Un [OF that]
immler@69615
  1860
    apply clarify
immler@69615
  1861
    apply (rule_tac x="f-`U" in exI)
immler@69615
  1862
    using \<open>inj f\<close> \<open>\<B> = range f\<close> apply force
immler@69615
  1863
    done
immler@69615
  1864
  show ?thesis
immler@69615
  1865
    apply (rule that [OF \<open>inj f\<close> _ *])
immler@69615
  1866
    apply (auto simp: \<open>\<B> = range f\<close> opn)
immler@69615
  1867
    done
immler@69615
  1868
qed
immler@69615
  1869
immler@69615
  1870
proposition separable:
immler@69615
  1871
  fixes S :: "'a::{metric_space, second_countable_topology} set"
immler@69615
  1872
  obtains T where "countable T" "T \<subseteq> S" "S \<subseteq> closure T"
immler@69615
  1873
proof -
immler@69615
  1874
  obtain \<B> :: "'a set set"
immler@69615
  1875
    where "countable \<B>"
immler@69615
  1876
      and "{} \<notin> \<B>"
immler@69615
  1877
      and ope: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
immler@69615
  1878
      and if_ope: "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
immler@69615
  1879
    by (meson subset_second_countable)
immler@69615
  1880
  then obtain f where f: "\<And>C. C \<in> \<B> \<Longrightarrow> f C \<in> C"
immler@69615
  1881
    by (metis equals0I)
immler@69615
  1882
  show ?thesis
immler@69615
  1883
  proof
immler@69615
  1884
    show "countable (f ` \<B>)"
immler@69615
  1885
      by (simp add: \<open>countable \<B>\<close>)
immler@69615
  1886
    show "f ` \<B> \<subseteq> S"
immler@69615
  1887
      using ope f openin_imp_subset by blast
immler@69615
  1888
    show "S \<subseteq> closure (f ` \<B>)"
immler@69615
  1889
    proof (clarsimp simp: closure_approachable)
immler@69615
  1890
      fix x and e::real
immler@69615
  1891
      assume "x \<in> S" "0 < e"
immler@69615
  1892
      have "openin (subtopology euclidean S) (S \<inter> ball x e)"
immler@69615
  1893
        by (simp add: openin_Int_open)
immler@69615
  1894
      with if_ope obtain \<U> where  \<U>: "\<U> \<subseteq> \<B>" "S \<inter> ball x e = \<Union>\<U>"
immler@69615
  1895
        by meson
immler@69615
  1896
      show "\<exists>C \<in> \<B>. dist (f C) x < e"
immler@69615
  1897
      proof (cases "\<U> = {}")
immler@69615
  1898
        case True
immler@69615
  1899
        then show ?thesis
immler@69615
  1900
          using \<open>0 < e\<close>  \<U> \<open>x \<in> S\<close> by auto
immler@69615
  1901
      next
immler@69615
  1902
        case False
immler@69615
  1903
        then obtain C where "C \<in> \<U>" by blast
immler@69615
  1904
        show ?thesis
immler@69615
  1905
        proof
immler@69615
  1906
          show "dist (f C) x < e"
immler@69615
  1907
            by (metis Int_iff Union_iff \<U> \<open>C \<in> \<U>\<close> dist_commute f mem_ball subsetCE)
immler@69615
  1908
          show "C \<in> \<B>"
immler@69615
  1909
            using \<open>\<U> \<subseteq> \<B>\<close> \<open>C \<in> \<U>\<close> by blast
immler@69615
  1910
        qed
immler@69615
  1911
      qed
immler@69615
  1912
    qed
immler@69615
  1913
  qed
immler@69615
  1914
qed
immler@69615
  1915
immler@69615
  1916
immler@69613
  1917
subsection%unimportant \<open>Diameter\<close>
immler@69613
  1918
immler@69613
  1919
lemma diameter_cball [simp]:
immler@69613
  1920
  fixes a :: "'a::euclidean_space"
immler@69613
  1921
  shows "diameter(cball a r) = (if r < 0 then 0 else 2*r)"
immler@69613
  1922
proof -
immler@69613
  1923
  have "diameter(cball a r) = 2*r" if "r \<ge> 0"
immler@69613
  1924
  proof (rule order_antisym)
immler@69613
  1925
    show "diameter (cball a r) \<le> 2*r"
immler@69613
  1926
    proof (rule diameter_le)
immler@69613
  1927
      fix x y assume "x \<in> cball a r" "y \<in> cball a r"
immler@69613
  1928
      then have "norm (x - a) \<le> r" "norm (a - y) \<le> r"
immler@69613
  1929
        by (auto simp: dist_norm norm_minus_commute)
immler@69613
  1930
      then have "norm (x - y) \<le> r+r"
immler@69613
  1931
        using norm_diff_triangle_le by blast
immler@69613
  1932
      then show "norm (x - y) \<le> 2*r" by simp
immler@69613
  1933
    qed (simp add: that)
immler@69613
  1934
    have "2*r = dist (a + r *\<^sub>R (SOME i. i \<in> Basis)) (a - r *\<^sub>R (SOME i. i \<in> Basis))"
immler@69613
  1935
      apply (simp add: dist_norm)
immler@69613
  1936
      by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)
immler@69613
  1937
    also have "... \<le> diameter (cball a r)"
immler@69613
  1938
      apply (rule diameter_bounded_bound)
immler@69613
  1939
      using that by (auto simp: dist_norm)
immler@69613
  1940
    finally show "2*r \<le> diameter (cball a r)" .
immler@69613
  1941
  qed
immler@69613
  1942
  then show ?thesis by simp
immler@69613
  1943
qed
immler@69613
  1944
immler@69613
  1945
lemma diameter_ball [simp]:
immler@69613
  1946
  fixes a :: "'a::euclidean_space"
immler@69613
  1947
  shows "diameter(ball a r) = (if r < 0 then 0 else 2*r)"
immler@69613
  1948
proof -
immler@69613
  1949
  have "diameter(ball a r) = 2*r" if "r > 0"
immler@69613
  1950
    by (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)
immler@69613
  1951
  then show ?thesis
immler@69613
  1952
    by (simp add: diameter_def)
immler@69613
  1953
qed
immler@69613
  1954
immler@69613
  1955
lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
immler@69613
  1956
proof -
immler@69613
  1957
  have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
immler@69613
  1958
    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
immler@69613
  1959
  then show ?thesis
immler@69613
  1960
    by simp
immler@69613
  1961
qed
immler@69613
  1962
immler@69613
  1963
lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
immler@69613
  1964
proof -
immler@69613
  1965
  have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
immler@69613
  1966
    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
immler@69613
  1967
  then show ?thesis
immler@69613
  1968
    by simp
immler@69613
  1969
qed
immler@69613
  1970
immler@69613
  1971
lemma diameter_cbox:
immler@69613
  1972
  fixes a b::"'a::euclidean_space"
immler@69613
  1973
  shows "(\<forall>i \<in> Basis. a \<bullet> i \<le> b \<bullet> i) \<Longrightarrow> diameter (cbox a b) = dist a b"
immler@69613
  1974
  by (force simp: diameter_def intro!: cSup_eq_maximum L2_set_mono
immler@69613
  1975
     simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)
immler@69613
  1976
immler@69613
  1977
immler@69617
  1978
subsection%unimportant\<open>Relating linear images to open/closed/interior/closure/connected\<close>
immler@56189
  1979
immler@69611
  1980
proposition open_surjective_linear_image:
immler@69611
  1981
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
immler@69611
  1982
  assumes "open A" "linear f" "surj f"
immler@69611
  1983
    shows "open(f ` A)"
immler@69611
  1984
unfolding open_dist
immler@69611
  1985
proof clarify
immler@69611
  1986
  fix x
immler@69611
  1987
  assume "x \<in> A"
immler@69611
  1988
  have "bounded (inv f ` Basis)"
immler@69611
  1989
    by (simp add: finite_imp_bounded)
immler@69611
  1990
  with bounded_pos obtain B where "B > 0" and B: "\<And>x. x \<in> inv f ` Basis \<Longrightarrow> norm x \<le> B"
immler@69611
  1991
    by metis
immler@69611
  1992
  obtain e where "e > 0" and e: "\<And>z. dist z x < e \<Longrightarrow> z \<in> A"
immler@69611
  1993
    by (metis open_dist \<open>x \<in> A\<close> \<open>open A\<close>)
immler@69611
  1994
  define \<delta> where "\<delta> \<equiv> e / B / DIM('b)"
immler@69611
  1995
  show "\<exists>e>0. \<forall>y. dist y (f x) < e \<longrightarrow> y \<in> f ` A"
immler@69611
  1996
  proof (intro exI conjI)
immler@69611
  1997
    show "\<delta> > 0"
immler@69611
  1998
      using \<open>e > 0\<close> \<open>B > 0\<close>  by (simp add: \<delta>_def divide_simps)
immler@69611
  1999
    have "y \<in> f ` A" if "dist y (f x) * (B * real DIM('b)) < e" for y
immler@69611
  2000
    proof -
immler@69611
  2001
      define u where "u \<equiv> y - f x"
immler@69611
  2002
      show ?thesis
immler@69611
  2003
      proof (rule image_eqI)
immler@69611
  2004
        show "y = f (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i))"
immler@69611
  2005
          apply (simp add: linear_add linear_sum linear.scaleR \<open>linear f\<close> surj_f_inv_f \<open>surj f\<close>)
immler@69611
  2006
          apply (simp add: euclidean_representation u_def)
immler@69611
  2007
          done
immler@69611
  2008
        have "dist (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i)) x \<le> (\<Sum>i\<in>Basis. norm ((u \<bullet> i) *\<^sub>R inv f i))"
immler@69611
  2009
          by (simp add: dist_norm sum_norm_le)
immler@69611
  2010
        also have "... = (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar> * norm (inv f i))"
immler@69611
  2011
          by simp
immler@69611
  2012
        also have "... \<le> (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar>) * B"
immler@69611
  2013
          by (simp add: B sum_distrib_right sum_mono mult_left_mono)
immler@69611
  2014
        also have "... \<le> DIM('b) * dist y (f x) * B"
immler@69611
  2015
          apply (rule mult_right_mono [OF sum_bounded_above])
immler@69611
  2016
          using \<open>0 < B\<close> by (auto simp: Basis_le_norm dist_norm u_def)
immler@69611
  2017
        also have "... < e"
immler@69611
  2018
          by (metis mult.commute mult.left_commute that)
immler@69611
  2019
        finally show "x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i) \<in> A"
immler@69611
  2020
          by (rule e)
immler@69611
  2021
      qed
immler@69611
  2022
    qed
immler@69611
  2023
    then show "\<forall>y. dist y (f x) < \<delta> \<longrightarrow> y \<in> f ` A"
immler@69611
  2024
      using \<open>e > 0\<close> \<open>B > 0\<close>
immler@69611
  2025
      by (auto simp: \<delta>_def divide_simps mult_less_0_iff)
immler@69611
  2026
  qed
immler@69611
  2027
qed
immler@69611
  2028
immler@69611
  2029
corollary open_bijective_linear_image_eq:
immler@69611
  2030
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
immler@69611
  2031
  assumes "linear f" "bij f"
immler@69611
  2032
    shows "open(f ` A) \<longleftrightarrow> open A"
immler@69611
  2033
proof
immler@69611
  2034
  assume "open(f ` A)"
immler@69611
  2035
  then have "open(f -` (f ` A))"
immler@69611
  2036
    using assms by (force simp: linear_continuous_at linear_conv_bounded_linear continuous_open_vimage)
immler@69611
  2037
  then show "open A"
immler@69611
  2038
    by (simp add: assms bij_is_inj inj_vimage_image_eq)
immler@69611
  2039
next
immler@69611
  2040
  assume "open A"
immler@69611
  2041
  then show "open(f ` A)"
immler@69611
  2042
    by (simp add: assms bij_is_surj open_surjective_linear_image)
immler@69611
  2043
qed
immler@69611
  2044
immler@69611
  2045
corollary interior_bijective_linear_image:
immler@69611
  2046
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
immler@69611
  2047
  assumes "linear f" "bij f"
immler@69611
  2048
  shows "interior (f ` S) = f ` interior S"  (is "?lhs = ?rhs")
immler@69611
  2049
proof safe
immler@69611
  2050
  fix x
immler@69611
  2051
  assume x: "x \<in> ?lhs"
immler@69611
  2052
  then obtain T where "open T" and "x \<in> T" and "T \<subseteq> f ` S"
immler@69611
  2053
    by (metis interiorE)
immler@69611
  2054
  then show "x \<in> ?rhs"
immler@69611
  2055
    by (metis (no_types, hide_lams) assms subsetD interior_maximal open_bijective_linear_image_eq subset_image_iff)
immler@69611
  2056
next
immler@69611
  2057
  fix x
immler@69611
  2058
  assume x: "x \<in> interior S"
immler@69611
  2059
  then show "f x \<in> interior (f ` S)"
immler@69611
  2060
    by (meson assms imageI image_mono interiorI interior_subset open_bijective_linear_image_eq open_interior)
immler@69611
  2061
qed
immler@69611
  2062
immler@69611
  2063
lemma interior_injective_linear_image:
immler@69611
  2064
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
immler@69611
  2065
  assumes "linear f" "inj f"
immler@69611
  2066
   shows "interior(f ` S) = f ` (interior S)"
immler@69611
  2067
  by (simp add: linear_injective_imp_surjective assms bijI interior_bijective_linear_image)
immler@69611
  2068
immler@69611
  2069
lemma interior_surjective_linear_image:
immler@69611
  2070
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
immler@69611
  2071
  assumes "linear f" "surj f"
immler@69611
  2072
   shows "interior(f ` S) = f ` (interior S)"
immler@69611
  2073
  by (simp add: assms interior_injective_linear_image linear_surjective_imp_injective)
immler@69611
  2074
immler@69611
  2075
lemma interior_negations:
immler@69611
  2076
  fixes S :: "'a::euclidean_space set"
immler@69611
  2077
  shows "interior(uminus ` S) = image uminus (interior S)"
immler@69611
  2078
  by (simp add: bij_uminus interior_bijective_linear_image linear_uminus)
immler@56189
  2079
immler@69617
  2080
lemma connected_linear_image:
immler@69617
  2081
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
immler@69617
  2082
  assumes "linear f" and "connected s"
immler@69617
  2083
  shows "connected (f ` s)"
immler@69617
  2084
using connected_continuous_image assms linear_continuous_on linear_conv_bounded_linear by blast
immler@69617
  2085
immler@69617
  2086
immler@69613
  2087
subsection%unimportant \<open>"Isometry" (up to constant bounds) of Injective Linear Map\<close>
immler@69613
  2088
immler@69613
  2089
proposition injective_imp_isometric:
immler@69613
  2090
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
immler@69613
  2091
  assumes s: "closed s" "subspace s"
immler@69613
  2092
    and f: "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0"
immler@69613
  2093
  shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm x"
immler@69613
  2094
proof (cases "s \<subseteq> {0::'a}")
immler@69613
  2095
  case True
immler@69613
  2096
  have "norm x \<le> norm (f x)" if "x \<in> s" for x
immler@69613
  2097
  proof -
immler@69613
  2098
    from True that have "x = 0" by auto
immler@69613
  2099
    then show ?thesis by simp
immler@69613
  2100
  qed
immler@69613
  2101
  then show ?thesis
immler@69613
  2102
    by (auto intro!: exI[where x=1])
immler@69613
  2103
next
immler@69613
  2104
  case False
immler@69613
  2105
  interpret f: bounded_linear f by fact
immler@69613
  2106
  from False obtain a where a: "a \<noteq> 0" "a \<in> s"
immler@69613
  2107
    by auto
immler@69613
  2108
  from False have "s \<noteq> {}"
immler@69613
  2109
    by auto
immler@69613
  2110
  let ?S = "{f x| x. x \<in> s \<and> norm x = norm a}"
immler@69613
  2111
  let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
immler@69613
  2112
  let ?S'' = "{x::'a. norm x = norm a}"
immler@69613
  2113
immler@69613
  2114
  have "?S'' = frontier (cball 0 (norm a))"
immler@69613
  2115
    by (simp add: sphere_def dist_norm)
immler@69613
  2116
  then have "compact ?S''" by (metis compact_cball compact_frontier)
immler@69613
  2117
  moreover have "?S' = s \<inter> ?S''" by auto
immler@69613
  2118
  ultimately have "compact ?S'"
immler@69613
  2119
    using closed_Int_compact[of s ?S''] using s(1) by auto
immler@69613
  2120
  moreover have *:"f ` ?S' = ?S" by auto
immler@69613
  2121
  ultimately have "compact ?S"
immler@69613
  2122
    using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
immler@69613
  2123
  then have "closed ?S"
immler@69613
  2124
    using compact_imp_closed by auto
immler@69613
  2125
  moreover from a have "?S \<noteq> {}" by auto
immler@69613
  2126
  ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y"
immler@69613
  2127
    using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
immler@69613
  2128
  then obtain b where "b\<in>s"
immler@69613
  2129
    and ba: "norm b = norm a"
immler@69613
  2130
    and b: "\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"
immler@69613
  2131
    unfolding *[symmetric] unfolding image_iff by auto
immler@69613
  2132
immler@69613
  2133
  let ?e = "norm (f b) / norm b"
immler@69613
  2134
  have "norm b > 0"
immler@69613
  2135
    using ba and a and norm_ge_zero by auto
immler@69613
  2136
  moreover have "norm (f b) > 0"
immler@69613
  2137
    using f(2)[THEN bspec[where x=b], OF \<open>b\<in>s\<close>]
immler@69613
  2138
    using \<open>norm b >0\<close> by simp
immler@69613
  2139
  ultimately have "0 < norm (f b) / norm b" by simp
immler@69613
  2140
  moreover
immler@69613
  2141
  have "norm (f b) / norm b * norm x \<le> norm (f x)" if "x\<in>s" for x
immler@69613
  2142
  proof (cases "x = 0")
immler@69613
  2143
    case True
immler@69613
  2144
    then show "norm (f b) / norm b * norm x \<le> norm (f x)"
immler@69613
  2145
      by auto
immler@69613
  2146
  next
immler@69613
  2147
    case False
immler@69613
  2148
    with \<open>a \<noteq> 0\<close> have *: "0 < norm a / norm x"
immler@69613
  2149
      unfolding zero_less_norm_iff[symmetric] by simp
immler@69613
  2150
    have "\<forall>x\<in>s. c *\<^sub>R x \<in> s" for c
immler@69613
  2151
      using s[unfolded subspace_def] by simp
immler@69613
  2152
    with \<open>x \<in> s\<close> \<open>x \<noteq> 0\<close> have "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}"
immler@69613
  2153
      by simp
immler@69613
  2154
    with \<open>x \<noteq> 0\<close> \<open>a \<noteq> 0\<close> show "norm (f b) / norm b * norm x \<le> norm (f x)"
immler@69613
  2155
      using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
immler@69613
  2156
      unfolding f.scaleR and ba
immler@69613
  2157
      by (auto simp: mult.commute pos_le_divide_eq pos_divide_le_eq)
immler@69613
  2158
  qed
immler@69613
  2159
  ultimately show ?thesis by auto
immler@69613
  2160
qed
immler@69613
  2161
immler@69613
  2162
proposition closed_injective_image_subspace:
immler@69613
  2163
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
immler@69613
  2164
  assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0" "closed s"
immler@69613
  2165
  shows "closed(f ` s)"
immler@69613
  2166
proof -
immler@69613
  2167
  obtain e where "e > 0" and e: "\<forall>x\<in>s. e * norm x \<le> norm (f x)"
immler@69613
  2168
    using injective_imp_isometric[OF assms(4,1,2,3)] by auto
immler@69613
  2169
  show ?thesis
immler@69613
  2170
    using complete_isometric_image[OF \<open>e>0\<close> assms(1,2) e] and assms(4)
immler@69613
  2171
    unfolding complete_eq_closed[symmetric] by auto
immler@69613
  2172
qed
immler@69613
  2173
immler@69613
  2174
immler@69613
  2175
subsection%unimportant \<open>Some properties of a canonical subspace\<close>
immler@69613
  2176
immler@69613
  2177
lemma closed_substandard: "closed {x::'a::euclidean_space. \<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0}"
immler@69613
  2178
  (is "closed ?A")
immler@69613
  2179
proof -
immler@69613
  2180
  let ?D = "{i\<in>Basis. P i}"
immler@69613
  2181
  have "closed (\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0})"
immler@69613
  2182
    by (simp add: closed_INT closed_Collect_eq continuous_on_inner
immler@69613
  2183
        continuous_on_const continuous_on_id)
immler@69613
  2184
  also have "(\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0}) = ?A"
immler@69613
  2185
    by auto
immler@69613
  2186
  finally show "closed ?A" .
immler@69613
  2187
qed
immler@69613
  2188
immler@69613
  2189
lemma closed_subspace:
immler@69613
  2190
  fixes s :: "'a::euclidean_space set"
immler@69613
  2191
  assumes "subspace s"
immler@69613
  2192
  shows "closed s"
immler@69613
  2193
proof -
immler@69613
  2194
  have "dim s \<le> card (Basis :: 'a set)"
immler@69613
  2195
    using dim_subset_UNIV by auto
immler@69613
  2196
  with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d \<subseteq> Basis"
immler@69613
  2197
    by auto
immler@69613
  2198
  let ?t = "{x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
immler@69613
  2199
  have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s \<and>
immler@69613
  2200
      inj_on f {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
immler@69613
  2201
    using dim_substandard[of d] t d assms
immler@69613
  2202
    by (intro subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]]) (auto simp: inner_Basis)
immler@69613
  2203
  then obtain f where f:
immler@69613
  2204
      "linear f"
immler@69613
  2205
      "f ` {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s"
immler@69613
  2206
      "inj_on f {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
immler@69613
  2207
    by blast
immler@69613
  2208
  interpret f: bounded_linear f
immler@69613
  2209
    using f by (simp add: linear_conv_bounded_linear)
immler@69613
  2210
  have "x \<in> ?t \<Longrightarrow> f x = 0 \<Longrightarrow> x = 0" for x
immler@69613
  2211
    using f.zero d f(3)[THEN inj_onD, of x 0] by auto
immler@69613
  2212
  moreover have "closed ?t" by (rule closed_substandard)
immler@69613
  2213
  moreover have "subspace ?t" by (rule subspace_substandard)
immler@69613
  2214
  ultimately show ?thesis
immler@69613
  2215
    using closed_injective_image_subspace[of ?t f]
immler@69613
  2216
    unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
immler@69613
  2217
qed
immler@69613
  2218
immler@69613
  2219
lemma complete_subspace: "subspace s \<Longrightarrow> complete s"
immler@69613
  2220
  for s :: "'a::euclidean_space set"
immler@69613
  2221
  using complete_eq_closed closed_subspace by auto
immler@69613
  2222
immler@69613
  2223
lemma closed_span [iff]: "closed (span s)"
immler@69613
  2224
  for s :: "'a::euclidean_space set"
immler@69613
  2225
  by (simp add: closed_subspace subspace_span)
immler@69613
  2226
immler@69613
  2227
lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")
immler@69613
  2228
  for s :: "'a::euclidean_space set"
immler@69613
  2229
proof -
immler@69613
  2230
  have "?dc \<le> ?d"
immler@69613
  2231
    using closure_minimal[OF span_superset, of s]
immler@69613
  2232
    using closed_subspace[OF subspace_span, of s]
immler@69613
  2233
    using dim_subset[of "closure s" "span s"]
immler@69613
  2234
    by simp
immler@69613
  2235
  then show ?thesis
immler@69613
  2236
    using dim_subset[OF closure_subset, of s]
immler@69613
  2237
    by simp
immler@69613
  2238
qed
immler@69613
  2239
immler@69613
  2240
immler@69618
  2241
subsection \<open>Set Distance\<close>
immler@69618
  2242
immler@69618
  2243
lemma setdist_compact_closed:
lp15@69918
  2244
  fixes A :: "'a::heine_borel set"
lp15@69918
  2245
  assumes A: "compact A" and B: "closed B"
lp15@69918
  2246
    and "A \<noteq> {}" "B \<noteq> {}"
lp15@69918
  2247
  shows "\<exists>x \<in> A. \<exists>y \<in> B. dist x y = setdist A B"
immler@69618
  2248
proof -
lp15@69918
  2249
  obtain x where "x \<in> A" "setdist A B = infdist x B"
lp15@69918
  2250
    by (metis A assms(3) setdist_attains_inf setdist_sym)
lp15@69918
  2251
  moreover
lp15@69918
  2252
  obtain y where"y \<in> B" "infdist x B = dist x y"
lp15@69918
  2253
    using B \<open>B \<noteq> {}\<close> infdist_attains_inf by blast
lp15@69918
  2254
  ultimately show ?thesis
lp15@69918
  2255
    using \<open>x \<in> A\<close> \<open>y \<in> B\<close> by auto
immler@69618
  2256
qed
immler@69618
  2257
immler@69618
  2258
lemma setdist_closed_compact:
lp15@69918
  2259
  fixes S :: "'a::heine_borel set"
immler@69618
  2260
  assumes S: "closed S" and T: "compact T"
immler@69618
  2261
      and "S \<noteq> {}" "T \<noteq> {}"
immler@69618
  2262
    shows "\<exists>x \<in> S. \<exists>y \<in> T. dist x y = setdist S T"
immler@69618
  2263
  using setdist_compact_closed [OF T S \<open>T \<noteq> {}\<close> \<open>S \<noteq> {}\<close>]
immler@69618
  2264
  by (metis dist_commute setdist_sym)
immler@69618
  2265
immler@69618
  2266
lemma setdist_eq_0_compact_closed:
immler@69618
  2267
  assumes S: "compact S" and T: "closed T"
immler@69618
  2268
    shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {}"
lp15@69918
  2269
proof (cases "S = {} \<or> T = {}")
lp15@69918
  2270
  case True
lp15@69918
  2271
  then show ?thesis