src/HOL/Library/Product_Vector.thy
author huffman
Sat Apr 24 09:37:24 2010 -0700 (2010-04-24)
changeset 36332 3ddb2bc07784
parent 34110 4c113c744b86
child 36660 1cc4ab4b7ff7
permissions -rw-r--r--
convert proofs to Isar-style
huffman@30019
     1
(*  Title:      HOL/Library/Product_Vector.thy
huffman@30019
     2
    Author:     Brian Huffman
huffman@30019
     3
*)
huffman@30019
     4
huffman@30019
     5
header {* Cartesian Products as Vector Spaces *}
huffman@30019
     6
huffman@30019
     7
theory Product_Vector
huffman@30019
     8
imports Inner_Product Product_plus
huffman@30019
     9
begin
huffman@30019
    10
huffman@30019
    11
subsection {* Product is a real vector space *}
huffman@30019
    12
huffman@30019
    13
instantiation "*" :: (real_vector, real_vector) real_vector
huffman@30019
    14
begin
huffman@30019
    15
huffman@30019
    16
definition scaleR_prod_def:
huffman@30019
    17
  "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
huffman@30019
    18
huffman@30019
    19
lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
huffman@30019
    20
  unfolding scaleR_prod_def by simp
huffman@30019
    21
huffman@30019
    22
lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
huffman@30019
    23
  unfolding scaleR_prod_def by simp
huffman@30019
    24
huffman@30019
    25
lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
huffman@30019
    26
  unfolding scaleR_prod_def by simp
huffman@30019
    27
huffman@30019
    28
instance proof
huffman@30019
    29
  fix a b :: real and x y :: "'a \<times> 'b"
huffman@30019
    30
  show "scaleR a (x + y) = scaleR a x + scaleR a y"
huffman@30019
    31
    by (simp add: expand_prod_eq scaleR_right_distrib)
huffman@30019
    32
  show "scaleR (a + b) x = scaleR a x + scaleR b x"
huffman@30019
    33
    by (simp add: expand_prod_eq scaleR_left_distrib)
huffman@30019
    34
  show "scaleR a (scaleR b x) = scaleR (a * b) x"
huffman@30019
    35
    by (simp add: expand_prod_eq)
huffman@30019
    36
  show "scaleR 1 x = x"
huffman@30019
    37
    by (simp add: expand_prod_eq)
huffman@30019
    38
qed
huffman@30019
    39
huffman@30019
    40
end
huffman@30019
    41
huffman@31415
    42
subsection {* Product is a topological space *}
huffman@31415
    43
huffman@31415
    44
instantiation
huffman@31415
    45
  "*" :: (topological_space, topological_space) topological_space
huffman@31415
    46
begin
huffman@31415
    47
huffman@31492
    48
definition open_prod_def:
huffman@31492
    49
  "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
huffman@31492
    50
    (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
huffman@31415
    51
huffman@36332
    52
lemma open_prod_elim:
huffman@36332
    53
  assumes "open S" and "x \<in> S"
huffman@36332
    54
  obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
huffman@36332
    55
using assms unfolding open_prod_def by fast
huffman@36332
    56
huffman@36332
    57
lemma open_prod_intro:
huffman@36332
    58
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
huffman@36332
    59
  shows "open S"
huffman@36332
    60
using assms unfolding open_prod_def by fast
huffman@36332
    61
huffman@31415
    62
instance proof
huffman@31492
    63
  show "open (UNIV :: ('a \<times> 'b) set)"
huffman@31492
    64
    unfolding open_prod_def by auto
huffman@31415
    65
next
huffman@31415
    66
  fix S T :: "('a \<times> 'b) set"
huffman@36332
    67
  assume "open S" "open T"
huffman@36332
    68
  show "open (S \<inter> T)"
huffman@36332
    69
  proof (rule open_prod_intro)
huffman@36332
    70
    fix x assume x: "x \<in> S \<inter> T"
huffman@36332
    71
    from x have "x \<in> S" by simp
huffman@36332
    72
    obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
huffman@36332
    73
      using `open S` and `x \<in> S` by (rule open_prod_elim)
huffman@36332
    74
    from x have "x \<in> T" by simp
huffman@36332
    75
    obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
huffman@36332
    76
      using `open T` and `x \<in> T` by (rule open_prod_elim)
huffman@36332
    77
    let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
huffman@36332
    78
    have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
huffman@36332
    79
      using A B by (auto simp add: open_Int)
huffman@36332
    80
    thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
huffman@36332
    81
      by fast
huffman@36332
    82
  qed
huffman@31415
    83
next
huffman@31492
    84
  fix K :: "('a \<times> 'b) set set"
huffman@31492
    85
  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
huffman@31492
    86
    unfolding open_prod_def by fast
huffman@31415
    87
qed
huffman@31415
    88
huffman@31415
    89
end
huffman@31415
    90
huffman@31562
    91
lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
huffman@31562
    92
unfolding open_prod_def by auto
huffman@31562
    93
huffman@31562
    94
lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
huffman@31562
    95
by auto
huffman@31562
    96
huffman@31562
    97
lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
huffman@31562
    98
by auto
huffman@31562
    99
huffman@31562
   100
lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
huffman@31562
   101
by (simp add: fst_vimage_eq_Times open_Times)
huffman@31562
   102
huffman@31562
   103
lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
huffman@31562
   104
by (simp add: snd_vimage_eq_Times open_Times)
huffman@31562
   105
huffman@31568
   106
lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
huffman@31568
   107
unfolding closed_open vimage_Compl [symmetric]
huffman@31568
   108
by (rule open_vimage_fst)
huffman@31568
   109
huffman@31568
   110
lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
huffman@31568
   111
unfolding closed_open vimage_Compl [symmetric]
huffman@31568
   112
by (rule open_vimage_snd)
huffman@31568
   113
huffman@31568
   114
lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
huffman@31568
   115
proof -
huffman@31568
   116
  have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
huffman@31568
   117
  thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
huffman@31568
   118
    by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
huffman@31568
   119
qed
huffman@31568
   120
huffman@34110
   121
lemma openI: (* TODO: move *)
huffman@34110
   122
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
huffman@34110
   123
  shows "open S"
huffman@34110
   124
proof -
huffman@34110
   125
  have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
huffman@34110
   126
  moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
huffman@34110
   127
  ultimately show "open S" by simp
huffman@34110
   128
qed
huffman@34110
   129
huffman@34110
   130
lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
huffman@34110
   131
  unfolding image_def subset_eq by force
huffman@34110
   132
huffman@34110
   133
lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
huffman@34110
   134
  unfolding image_def subset_eq by force
huffman@34110
   135
huffman@34110
   136
lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
huffman@34110
   137
proof (rule openI)
huffman@34110
   138
  fix x assume "x \<in> fst ` S"
huffman@34110
   139
  then obtain y where "(x, y) \<in> S" by auto
huffman@34110
   140
  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
huffman@34110
   141
    using `open S` unfolding open_prod_def by auto
huffman@34110
   142
  from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
huffman@34110
   143
  with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
huffman@34110
   144
  then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
huffman@34110
   145
qed
huffman@34110
   146
huffman@34110
   147
lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
huffman@34110
   148
proof (rule openI)
huffman@34110
   149
  fix y assume "y \<in> snd ` S"
huffman@34110
   150
  then obtain x where "(x, y) \<in> S" by auto
huffman@34110
   151
  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
huffman@34110
   152
    using `open S` unfolding open_prod_def by auto
huffman@34110
   153
  from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
huffman@34110
   154
  with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
huffman@34110
   155
  then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
huffman@34110
   156
qed
huffman@31568
   157
huffman@31339
   158
subsection {* Product is a metric space *}
huffman@31339
   159
huffman@31339
   160
instantiation
huffman@31339
   161
  "*" :: (metric_space, metric_space) metric_space
huffman@31339
   162
begin
huffman@31339
   163
huffman@31339
   164
definition dist_prod_def:
huffman@31339
   165
  "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
huffman@31339
   166
huffman@31339
   167
lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
huffman@31339
   168
  unfolding dist_prod_def by simp
huffman@31339
   169
huffman@36332
   170
lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
huffman@36332
   171
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
huffman@36332
   172
huffman@36332
   173
lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
huffman@36332
   174
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
huffman@36332
   175
huffman@31339
   176
instance proof
huffman@31339
   177
  fix x y :: "'a \<times> 'b"
huffman@31339
   178
  show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31563
   179
    unfolding dist_prod_def expand_prod_eq by simp
huffman@31339
   180
next
huffman@31339
   181
  fix x y z :: "'a \<times> 'b"
huffman@31339
   182
  show "dist x y \<le> dist x z + dist y z"
huffman@31339
   183
    unfolding dist_prod_def
huffman@31563
   184
    by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
huffman@31563
   185
        real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
huffman@31415
   186
next
huffman@31415
   187
  (* FIXME: long proof! *)
huffman@31415
   188
  (* Maybe it would be easier to define topological spaces *)
huffman@31415
   189
  (* in terms of neighborhoods instead of open sets? *)
huffman@31492
   190
  fix S :: "('a \<times> 'b) set"
huffman@31492
   191
  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31563
   192
  proof
huffman@36332
   193
    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
huffman@36332
   194
    proof
huffman@36332
   195
      fix x assume "x \<in> S"
huffman@36332
   196
      obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
huffman@36332
   197
        using `open S` and `x \<in> S` by (rule open_prod_elim)
huffman@36332
   198
      obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
huffman@36332
   199
        using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332
   200
      obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
huffman@36332
   201
        using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332
   202
      let ?e = "min r s"
huffman@36332
   203
      have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
huffman@36332
   204
      proof (intro allI impI conjI)
huffman@36332
   205
        show "0 < min r s" by (simp add: r(1) s(1))
huffman@36332
   206
      next
huffman@36332
   207
        fix y assume "dist y x < min r s"
huffman@36332
   208
        hence "dist y x < r" and "dist y x < s"
huffman@36332
   209
          by simp_all
huffman@36332
   210
        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
huffman@36332
   211
          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
huffman@36332
   212
        hence "fst y \<in> A" and "snd y \<in> B"
huffman@36332
   213
          by (simp_all add: r(2) s(2))
huffman@36332
   214
        hence "y \<in> A \<times> B" by (induct y, simp)
huffman@36332
   215
        with `A \<times> B \<subseteq> S` show "y \<in> S" ..
huffman@36332
   216
      qed
huffman@36332
   217
      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
huffman@36332
   218
    qed
huffman@31563
   219
  next
huffman@31563
   220
    assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" thus "open S"
huffman@31563
   221
    unfolding open_prod_def open_dist
huffman@31563
   222
    apply safe
huffman@31415
   223
    apply (drule (1) bspec)
huffman@31415
   224
    apply clarify
huffman@31415
   225
    apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
huffman@31415
   226
    apply clarify
huffman@31492
   227
    apply (rule_tac x="{y. dist y a < r}" in exI)
huffman@31492
   228
    apply (rule_tac x="{y. dist y b < s}" in exI)
huffman@31492
   229
    apply (rule conjI)
huffman@31415
   230
    apply clarify
huffman@31415
   231
    apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
huffman@31415
   232
    apply clarify
huffman@31563
   233
    apply (simp add: less_diff_eq)
huffman@31563
   234
    apply (erule le_less_trans [OF dist_triangle])
huffman@31492
   235
    apply (rule conjI)
huffman@31415
   236
    apply clarify
huffman@31415
   237
    apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
huffman@31415
   238
    apply clarify
huffman@31563
   239
    apply (simp add: less_diff_eq)
huffman@31563
   240
    apply (erule le_less_trans [OF dist_triangle])
huffman@31415
   241
    apply (rule conjI)
huffman@31415
   242
    apply simp
huffman@31415
   243
    apply (clarify, rename_tac c d)
huffman@31415
   244
    apply (drule spec, erule mp)
huffman@31415
   245
    apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
huffman@31415
   246
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
huffman@31415
   247
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
huffman@31415
   248
    apply (simp add: power_divide)
huffman@31415
   249
    done
huffman@31563
   250
  qed
huffman@31339
   251
qed
huffman@31339
   252
huffman@31339
   253
end
huffman@31339
   254
huffman@31405
   255
subsection {* Continuity of operations *}
huffman@31405
   256
huffman@31565
   257
lemma tendsto_fst [tendsto_intros]:
huffman@31491
   258
  assumes "(f ---> a) net"
huffman@31491
   259
  shows "((\<lambda>x. fst (f x)) ---> fst a) net"
huffman@31491
   260
proof (rule topological_tendstoI)
huffman@31492
   261
  fix S assume "open S" "fst a \<in> S"
huffman@31492
   262
  then have "open (fst -` S)" "a \<in> fst -` S"
huffman@31492
   263
    unfolding open_prod_def
huffman@31491
   264
    apply simp_all
huffman@31491
   265
    apply clarify
huffman@31492
   266
    apply (rule exI, erule conjI)
huffman@31492
   267
    apply (rule exI, rule conjI [OF open_UNIV])
huffman@31491
   268
    apply auto
huffman@31491
   269
    done
huffman@31491
   270
  with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
huffman@31491
   271
    by (rule topological_tendstoD)
huffman@31491
   272
  then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
huffman@31491
   273
    by simp
huffman@31405
   274
qed
huffman@31405
   275
huffman@31565
   276
lemma tendsto_snd [tendsto_intros]:
huffman@31491
   277
  assumes "(f ---> a) net"
huffman@31491
   278
  shows "((\<lambda>x. snd (f x)) ---> snd a) net"
huffman@31491
   279
proof (rule topological_tendstoI)
huffman@31492
   280
  fix S assume "open S" "snd a \<in> S"
huffman@31492
   281
  then have "open (snd -` S)" "a \<in> snd -` S"
huffman@31492
   282
    unfolding open_prod_def
huffman@31491
   283
    apply simp_all
huffman@31491
   284
    apply clarify
huffman@31492
   285
    apply (rule exI, rule conjI [OF open_UNIV])
huffman@31492
   286
    apply (rule exI, erule conjI)
huffman@31491
   287
    apply auto
huffman@31491
   288
    done
huffman@31491
   289
  with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
huffman@31491
   290
    by (rule topological_tendstoD)
huffman@31491
   291
  then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
huffman@31491
   292
    by simp
huffman@31405
   293
qed
huffman@31405
   294
huffman@31565
   295
lemma tendsto_Pair [tendsto_intros]:
huffman@31491
   296
  assumes "(f ---> a) net" and "(g ---> b) net"
huffman@31491
   297
  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
huffman@31491
   298
proof (rule topological_tendstoI)
huffman@31492
   299
  fix S assume "open S" "(a, b) \<in> S"
huffman@31492
   300
  then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
huffman@31492
   301
    unfolding open_prod_def by auto
huffman@31491
   302
  have "eventually (\<lambda>x. f x \<in> A) net"
huffman@31492
   303
    using `(f ---> a) net` `open A` `a \<in> A`
huffman@31491
   304
    by (rule topological_tendstoD)
huffman@31405
   305
  moreover
huffman@31491
   306
  have "eventually (\<lambda>x. g x \<in> B) net"
huffman@31492
   307
    using `(g ---> b) net` `open B` `b \<in> B`
huffman@31491
   308
    by (rule topological_tendstoD)
huffman@31405
   309
  ultimately
huffman@31491
   310
  show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
huffman@31405
   311
    by (rule eventually_elim2)
huffman@31491
   312
       (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
huffman@31405
   313
qed
huffman@31405
   314
huffman@31405
   315
lemma LIMSEQ_fst: "(X ----> a) \<Longrightarrow> (\<lambda>n. fst (X n)) ----> fst a"
huffman@31405
   316
unfolding LIMSEQ_conv_tendsto by (rule tendsto_fst)
huffman@31405
   317
huffman@31405
   318
lemma LIMSEQ_snd: "(X ----> a) \<Longrightarrow> (\<lambda>n. snd (X n)) ----> snd a"
huffman@31405
   319
unfolding LIMSEQ_conv_tendsto by (rule tendsto_snd)
huffman@31405
   320
huffman@31405
   321
lemma LIMSEQ_Pair:
huffman@31405
   322
  assumes "X ----> a" and "Y ----> b"
huffman@31405
   323
  shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
huffman@31405
   324
using assms unfolding LIMSEQ_conv_tendsto
huffman@31405
   325
by (rule tendsto_Pair)
huffman@31405
   326
huffman@31405
   327
lemma LIM_fst: "f -- x --> a \<Longrightarrow> (\<lambda>x. fst (f x)) -- x --> fst a"
huffman@31405
   328
unfolding LIM_conv_tendsto by (rule tendsto_fst)
huffman@31405
   329
huffman@31405
   330
lemma LIM_snd: "f -- x --> a \<Longrightarrow> (\<lambda>x. snd (f x)) -- x --> snd a"
huffman@31405
   331
unfolding LIM_conv_tendsto by (rule tendsto_snd)
huffman@31405
   332
huffman@31405
   333
lemma LIM_Pair:
huffman@31405
   334
  assumes "f -- x --> a" and "g -- x --> b"
huffman@31405
   335
  shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
huffman@31405
   336
using assms unfolding LIM_conv_tendsto
huffman@31405
   337
by (rule tendsto_Pair)
huffman@31405
   338
huffman@31405
   339
lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
huffman@31405
   340
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
huffman@31405
   341
huffman@31405
   342
lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
huffman@31405
   343
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
huffman@31405
   344
huffman@31405
   345
lemma Cauchy_Pair:
huffman@31405
   346
  assumes "Cauchy X" and "Cauchy Y"
huffman@31405
   347
  shows "Cauchy (\<lambda>n. (X n, Y n))"
huffman@31405
   348
proof (rule metric_CauchyI)
huffman@31405
   349
  fix r :: real assume "0 < r"
huffman@31405
   350
  then have "0 < r / sqrt 2" (is "0 < ?s")
huffman@31405
   351
    by (simp add: divide_pos_pos)
huffman@31405
   352
  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
huffman@31405
   353
    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
huffman@31405
   354
  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
huffman@31405
   355
    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
huffman@31405
   356
  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
huffman@31405
   357
    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
huffman@31405
   358
  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
huffman@31405
   359
qed
huffman@31405
   360
huffman@31405
   361
lemma isCont_Pair [simp]:
huffman@31405
   362
  "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
huffman@31405
   363
  unfolding isCont_def by (rule LIM_Pair)
huffman@31405
   364
huffman@31405
   365
subsection {* Product is a complete metric space *}
huffman@31405
   366
huffman@31405
   367
instance "*" :: (complete_space, complete_space) complete_space
huffman@31405
   368
proof
huffman@31405
   369
  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
huffman@31405
   370
  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
huffman@31405
   371
    using Cauchy_fst [OF `Cauchy X`]
huffman@31405
   372
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405
   373
  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
huffman@31405
   374
    using Cauchy_snd [OF `Cauchy X`]
huffman@31405
   375
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405
   376
  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
huffman@31405
   377
    using LIMSEQ_Pair [OF 1 2] by simp
huffman@31405
   378
  then show "convergent X"
huffman@31405
   379
    by (rule convergentI)
huffman@31405
   380
qed
huffman@31405
   381
huffman@30019
   382
subsection {* Product is a normed vector space *}
huffman@30019
   383
huffman@30019
   384
instantiation
huffman@30019
   385
  "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
huffman@30019
   386
begin
huffman@30019
   387
huffman@30019
   388
definition norm_prod_def:
huffman@30019
   389
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
huffman@30019
   390
huffman@30019
   391
definition sgn_prod_def:
huffman@30019
   392
  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
huffman@30019
   393
huffman@30019
   394
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
huffman@30019
   395
  unfolding norm_prod_def by simp
huffman@30019
   396
huffman@30019
   397
instance proof
huffman@30019
   398
  fix r :: real and x y :: "'a \<times> 'b"
huffman@30019
   399
  show "0 \<le> norm x"
huffman@30019
   400
    unfolding norm_prod_def by simp
huffman@30019
   401
  show "norm x = 0 \<longleftrightarrow> x = 0"
huffman@30019
   402
    unfolding norm_prod_def
huffman@30019
   403
    by (simp add: expand_prod_eq)
huffman@30019
   404
  show "norm (x + y) \<le> norm x + norm y"
huffman@30019
   405
    unfolding norm_prod_def
huffman@30019
   406
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
huffman@30019
   407
    apply (simp add: add_mono power_mono norm_triangle_ineq)
huffman@30019
   408
    done
huffman@30019
   409
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
huffman@30019
   410
    unfolding norm_prod_def
huffman@31587
   411
    apply (simp add: power_mult_distrib)
huffman@30019
   412
    apply (simp add: right_distrib [symmetric])
huffman@30019
   413
    apply (simp add: real_sqrt_mult_distrib)
huffman@30019
   414
    done
huffman@30019
   415
  show "sgn x = scaleR (inverse (norm x)) x"
huffman@30019
   416
    by (rule sgn_prod_def)
huffman@31290
   417
  show "dist x y = norm (x - y)"
huffman@31339
   418
    unfolding dist_prod_def norm_prod_def
huffman@31339
   419
    by (simp add: dist_norm)
huffman@30019
   420
qed
huffman@30019
   421
huffman@30019
   422
end
huffman@30019
   423
huffman@31405
   424
instance "*" :: (banach, banach) banach ..
huffman@31405
   425
huffman@30019
   426
subsection {* Product is an inner product space *}
huffman@30019
   427
huffman@30019
   428
instantiation "*" :: (real_inner, real_inner) real_inner
huffman@30019
   429
begin
huffman@30019
   430
huffman@30019
   431
definition inner_prod_def:
huffman@30019
   432
  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
huffman@30019
   433
huffman@30019
   434
lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
huffman@30019
   435
  unfolding inner_prod_def by simp
huffman@30019
   436
huffman@30019
   437
instance proof
huffman@30019
   438
  fix r :: real
huffman@30019
   439
  fix x y z :: "'a::real_inner * 'b::real_inner"
huffman@30019
   440
  show "inner x y = inner y x"
huffman@30019
   441
    unfolding inner_prod_def
huffman@30019
   442
    by (simp add: inner_commute)
huffman@30019
   443
  show "inner (x + y) z = inner x z + inner y z"
huffman@30019
   444
    unfolding inner_prod_def
huffman@31590
   445
    by (simp add: inner_add_left)
huffman@30019
   446
  show "inner (scaleR r x) y = r * inner x y"
huffman@30019
   447
    unfolding inner_prod_def
huffman@31590
   448
    by (simp add: right_distrib)
huffman@30019
   449
  show "0 \<le> inner x x"
huffman@30019
   450
    unfolding inner_prod_def
huffman@30019
   451
    by (intro add_nonneg_nonneg inner_ge_zero)
huffman@30019
   452
  show "inner x x = 0 \<longleftrightarrow> x = 0"
huffman@30019
   453
    unfolding inner_prod_def expand_prod_eq
huffman@30019
   454
    by (simp add: add_nonneg_eq_0_iff)
huffman@30019
   455
  show "norm x = sqrt (inner x x)"
huffman@30019
   456
    unfolding norm_prod_def inner_prod_def
huffman@30019
   457
    by (simp add: power2_norm_eq_inner)
huffman@30019
   458
qed
huffman@30019
   459
huffman@30019
   460
end
huffman@30019
   461
huffman@31405
   462
subsection {* Pair operations are linear *}
huffman@30019
   463
wenzelm@30729
   464
interpretation fst: bounded_linear fst
huffman@30019
   465
  apply (unfold_locales)
huffman@30019
   466
  apply (rule fst_add)
huffman@30019
   467
  apply (rule fst_scaleR)
huffman@30019
   468
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
huffman@30019
   469
  done
huffman@30019
   470
wenzelm@30729
   471
interpretation snd: bounded_linear snd
huffman@30019
   472
  apply (unfold_locales)
huffman@30019
   473
  apply (rule snd_add)
huffman@30019
   474
  apply (rule snd_scaleR)
huffman@30019
   475
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
huffman@30019
   476
  done
huffman@30019
   477
huffman@30019
   478
text {* TODO: move to NthRoot *}
huffman@30019
   479
lemma sqrt_add_le_add_sqrt:
huffman@30019
   480
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@30019
   481
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
huffman@30019
   482
apply (rule power2_le_imp_le)
huffman@30019
   483
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
huffman@30019
   484
apply (simp add: mult_nonneg_nonneg x y)
huffman@30019
   485
apply (simp add: add_nonneg_nonneg x y)
huffman@30019
   486
done
huffman@30019
   487
huffman@30019
   488
lemma bounded_linear_Pair:
huffman@30019
   489
  assumes f: "bounded_linear f"
huffman@30019
   490
  assumes g: "bounded_linear g"
huffman@30019
   491
  shows "bounded_linear (\<lambda>x. (f x, g x))"
huffman@30019
   492
proof
huffman@30019
   493
  interpret f: bounded_linear f by fact
huffman@30019
   494
  interpret g: bounded_linear g by fact
huffman@30019
   495
  fix x y and r :: real
huffman@30019
   496
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
huffman@30019
   497
    by (simp add: f.add g.add)
huffman@30019
   498
  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
huffman@30019
   499
    by (simp add: f.scaleR g.scaleR)
huffman@30019
   500
  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
huffman@30019
   501
    using f.pos_bounded by fast
huffman@30019
   502
  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
huffman@30019
   503
    using g.pos_bounded by fast
huffman@30019
   504
  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
huffman@30019
   505
    apply (rule allI)
huffman@30019
   506
    apply (simp add: norm_Pair)
huffman@30019
   507
    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
huffman@30019
   508
    apply (simp add: right_distrib)
huffman@30019
   509
    apply (rule add_mono [OF norm_f norm_g])
huffman@30019
   510
    done
huffman@30019
   511
  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
huffman@30019
   512
qed
huffman@30019
   513
huffman@30019
   514
subsection {* Frechet derivatives involving pairs *}
huffman@30019
   515
huffman@30019
   516
lemma FDERIV_Pair:
huffman@30019
   517
  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
huffman@30019
   518
  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
huffman@30019
   519
apply (rule FDERIV_I)
huffman@30019
   520
apply (rule bounded_linear_Pair)
huffman@30019
   521
apply (rule FDERIV_bounded_linear [OF f])
huffman@30019
   522
apply (rule FDERIV_bounded_linear [OF g])
huffman@30019
   523
apply (simp add: norm_Pair)
huffman@30019
   524
apply (rule real_LIM_sandwich_zero)
huffman@30019
   525
apply (rule LIM_add_zero)
huffman@30019
   526
apply (rule FDERIV_D [OF f])
huffman@30019
   527
apply (rule FDERIV_D [OF g])
huffman@30019
   528
apply (rename_tac h)
huffman@30019
   529
apply (simp add: divide_nonneg_pos)
huffman@30019
   530
apply (rename_tac h)
huffman@30019
   531
apply (subst add_divide_distrib [symmetric])
huffman@30019
   532
apply (rule divide_right_mono [OF _ norm_ge_zero])
huffman@30019
   533
apply (rule order_trans [OF sqrt_add_le_add_sqrt])
huffman@30019
   534
apply simp
huffman@30019
   535
apply simp
huffman@30019
   536
apply simp
huffman@30019
   537
done
huffman@30019
   538
huffman@30019
   539
end