src/HOL/Library/Product_Vector.thy
author huffman
Mon Aug 08 10:32:55 2011 -0700 (2011-08-08)
changeset 44066 d74182c93f04
parent 37678 0040bafffdef
child 44126 ce44e70d0c47
permissions -rw-r--r--
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
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
haftmann@37678
    13
instantiation prod :: (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@44066
    31
    by (simp add: prod_eq_iff scaleR_right_distrib)
huffman@30019
    32
  show "scaleR (a + b) x = scaleR a x + scaleR b x"
huffman@44066
    33
    by (simp add: prod_eq_iff scaleR_left_distrib)
huffman@30019
    34
  show "scaleR a (scaleR b x) = scaleR (a * b) x"
huffman@44066
    35
    by (simp add: prod_eq_iff)
huffman@30019
    36
  show "scaleR 1 x = x"
huffman@44066
    37
    by (simp add: prod_eq_iff)
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
haftmann@37678
    44
instantiation prod :: (topological_space, topological_space) topological_space
huffman@31415
    45
begin
huffman@31415
    46
huffman@31492
    47
definition open_prod_def:
huffman@31492
    48
  "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
huffman@31492
    49
    (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
huffman@31415
    50
huffman@36332
    51
lemma open_prod_elim:
huffman@36332
    52
  assumes "open S" and "x \<in> S"
huffman@36332
    53
  obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
huffman@36332
    54
using assms unfolding open_prod_def by fast
huffman@36332
    55
huffman@36332
    56
lemma open_prod_intro:
huffman@36332
    57
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
huffman@36332
    58
  shows "open S"
huffman@36332
    59
using assms unfolding open_prod_def by fast
huffman@36332
    60
huffman@31415
    61
instance proof
huffman@31492
    62
  show "open (UNIV :: ('a \<times> 'b) set)"
huffman@31492
    63
    unfolding open_prod_def by auto
huffman@31415
    64
next
huffman@31415
    65
  fix S T :: "('a \<times> 'b) set"
huffman@36332
    66
  assume "open S" "open T"
huffman@36332
    67
  show "open (S \<inter> T)"
huffman@36332
    68
  proof (rule open_prod_intro)
huffman@36332
    69
    fix x assume x: "x \<in> S \<inter> T"
huffman@36332
    70
    from x have "x \<in> S" by simp
huffman@36332
    71
    obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
huffman@36332
    72
      using `open S` and `x \<in> S` by (rule open_prod_elim)
huffman@36332
    73
    from x have "x \<in> T" by simp
huffman@36332
    74
    obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
huffman@36332
    75
      using `open T` and `x \<in> T` by (rule open_prod_elim)
huffman@36332
    76
    let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
huffman@36332
    77
    have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
huffman@36332
    78
      using A B by (auto simp add: open_Int)
huffman@36332
    79
    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
    80
      by fast
huffman@36332
    81
  qed
huffman@31415
    82
next
huffman@31492
    83
  fix K :: "('a \<times> 'b) set set"
huffman@31492
    84
  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
huffman@31492
    85
    unfolding open_prod_def by fast
huffman@31415
    86
qed
huffman@31415
    87
huffman@31415
    88
end
huffman@31415
    89
huffman@31562
    90
lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
huffman@31562
    91
unfolding open_prod_def by auto
huffman@31562
    92
huffman@31562
    93
lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
huffman@31562
    94
by auto
huffman@31562
    95
huffman@31562
    96
lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
huffman@31562
    97
by auto
huffman@31562
    98
huffman@31562
    99
lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
huffman@31562
   100
by (simp add: fst_vimage_eq_Times open_Times)
huffman@31562
   101
huffman@31562
   102
lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
huffman@31562
   103
by (simp add: snd_vimage_eq_Times open_Times)
huffman@31562
   104
huffman@31568
   105
lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
huffman@31568
   106
unfolding closed_open vimage_Compl [symmetric]
huffman@31568
   107
by (rule open_vimage_fst)
huffman@31568
   108
huffman@31568
   109
lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
huffman@31568
   110
unfolding closed_open vimage_Compl [symmetric]
huffman@31568
   111
by (rule open_vimage_snd)
huffman@31568
   112
huffman@31568
   113
lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
huffman@31568
   114
proof -
huffman@31568
   115
  have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
huffman@31568
   116
  thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
huffman@31568
   117
    by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
huffman@31568
   118
qed
huffman@31568
   119
huffman@34110
   120
lemma openI: (* TODO: move *)
huffman@34110
   121
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
huffman@34110
   122
  shows "open S"
huffman@34110
   123
proof -
huffman@34110
   124
  have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
huffman@34110
   125
  moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
huffman@34110
   126
  ultimately show "open S" by simp
huffman@34110
   127
qed
huffman@34110
   128
huffman@34110
   129
lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
huffman@34110
   130
  unfolding image_def subset_eq by force
huffman@34110
   131
huffman@34110
   132
lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
huffman@34110
   133
  unfolding image_def subset_eq by force
huffman@34110
   134
huffman@34110
   135
lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
huffman@34110
   136
proof (rule openI)
huffman@34110
   137
  fix x assume "x \<in> fst ` S"
huffman@34110
   138
  then obtain y where "(x, y) \<in> S" by auto
huffman@34110
   139
  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
huffman@34110
   140
    using `open S` unfolding open_prod_def by auto
huffman@34110
   141
  from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
huffman@34110
   142
  with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
huffman@34110
   143
  then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
huffman@34110
   144
qed
huffman@34110
   145
huffman@34110
   146
lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
huffman@34110
   147
proof (rule openI)
huffman@34110
   148
  fix y assume "y \<in> snd ` S"
huffman@34110
   149
  then obtain x where "(x, y) \<in> S" by auto
huffman@34110
   150
  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
huffman@34110
   151
    using `open S` unfolding open_prod_def by auto
huffman@34110
   152
  from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
huffman@34110
   153
  with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
huffman@34110
   154
  then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
huffman@34110
   155
qed
huffman@31568
   156
huffman@31339
   157
subsection {* Product is a metric space *}
huffman@31339
   158
haftmann@37678
   159
instantiation prod :: (metric_space, metric_space) metric_space
huffman@31339
   160
begin
huffman@31339
   161
huffman@31339
   162
definition dist_prod_def:
huffman@31339
   163
  "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
huffman@31339
   164
huffman@31339
   165
lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
huffman@31339
   166
  unfolding dist_prod_def by simp
huffman@31339
   167
huffman@36332
   168
lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
huffman@36332
   169
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
huffman@36332
   170
huffman@36332
   171
lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
huffman@36332
   172
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
huffman@36332
   173
huffman@31339
   174
instance proof
huffman@31339
   175
  fix x y :: "'a \<times> 'b"
huffman@31339
   176
  show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@44066
   177
    unfolding dist_prod_def prod_eq_iff by simp
huffman@31339
   178
next
huffman@31339
   179
  fix x y z :: "'a \<times> 'b"
huffman@31339
   180
  show "dist x y \<le> dist x z + dist y z"
huffman@31339
   181
    unfolding dist_prod_def
huffman@31563
   182
    by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
huffman@31563
   183
        real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
huffman@31415
   184
next
huffman@31415
   185
  (* FIXME: long proof! *)
huffman@31415
   186
  (* Maybe it would be easier to define topological spaces *)
huffman@31415
   187
  (* in terms of neighborhoods instead of open sets? *)
huffman@31492
   188
  fix S :: "('a \<times> 'b) set"
huffman@31492
   189
  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31563
   190
  proof
huffman@36332
   191
    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
huffman@36332
   192
    proof
huffman@36332
   193
      fix x assume "x \<in> S"
huffman@36332
   194
      obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
huffman@36332
   195
        using `open S` and `x \<in> S` by (rule open_prod_elim)
huffman@36332
   196
      obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
huffman@36332
   197
        using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332
   198
      obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
huffman@36332
   199
        using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332
   200
      let ?e = "min r s"
huffman@36332
   201
      have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
huffman@36332
   202
      proof (intro allI impI conjI)
huffman@36332
   203
        show "0 < min r s" by (simp add: r(1) s(1))
huffman@36332
   204
      next
huffman@36332
   205
        fix y assume "dist y x < min r s"
huffman@36332
   206
        hence "dist y x < r" and "dist y x < s"
huffman@36332
   207
          by simp_all
huffman@36332
   208
        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
huffman@36332
   209
          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
huffman@36332
   210
        hence "fst y \<in> A" and "snd y \<in> B"
huffman@36332
   211
          by (simp_all add: r(2) s(2))
huffman@36332
   212
        hence "y \<in> A \<times> B" by (induct y, simp)
huffman@36332
   213
        with `A \<times> B \<subseteq> S` show "y \<in> S" ..
huffman@36332
   214
      qed
huffman@36332
   215
      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
huffman@36332
   216
    qed
huffman@31563
   217
  next
huffman@31563
   218
    assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" thus "open S"
huffman@31563
   219
    unfolding open_prod_def open_dist
huffman@31563
   220
    apply safe
huffman@31415
   221
    apply (drule (1) bspec)
huffman@31415
   222
    apply clarify
huffman@31415
   223
    apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
huffman@31415
   224
    apply clarify
huffman@31492
   225
    apply (rule_tac x="{y. dist y a < r}" in exI)
huffman@31492
   226
    apply (rule_tac x="{y. dist y b < s}" in exI)
huffman@31492
   227
    apply (rule conjI)
huffman@31415
   228
    apply clarify
huffman@31415
   229
    apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
huffman@31415
   230
    apply clarify
huffman@31563
   231
    apply (simp add: less_diff_eq)
huffman@31563
   232
    apply (erule le_less_trans [OF dist_triangle])
huffman@31492
   233
    apply (rule conjI)
huffman@31415
   234
    apply clarify
huffman@31415
   235
    apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
huffman@31415
   236
    apply clarify
huffman@31563
   237
    apply (simp add: less_diff_eq)
huffman@31563
   238
    apply (erule le_less_trans [OF dist_triangle])
huffman@31415
   239
    apply (rule conjI)
huffman@31415
   240
    apply simp
huffman@31415
   241
    apply (clarify, rename_tac c d)
huffman@31415
   242
    apply (drule spec, erule mp)
huffman@31415
   243
    apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
huffman@31415
   244
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
huffman@31415
   245
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
huffman@31415
   246
    apply (simp add: power_divide)
huffman@31415
   247
    done
huffman@31563
   248
  qed
huffman@31339
   249
qed
huffman@31339
   250
huffman@31339
   251
end
huffman@31339
   252
huffman@31405
   253
subsection {* Continuity of operations *}
huffman@31405
   254
huffman@31565
   255
lemma tendsto_fst [tendsto_intros]:
huffman@31491
   256
  assumes "(f ---> a) net"
huffman@31491
   257
  shows "((\<lambda>x. fst (f x)) ---> fst a) net"
huffman@31491
   258
proof (rule topological_tendstoI)
huffman@31492
   259
  fix S assume "open S" "fst a \<in> S"
huffman@31492
   260
  then have "open (fst -` S)" "a \<in> fst -` S"
huffman@31492
   261
    unfolding open_prod_def
huffman@31491
   262
    apply simp_all
huffman@31491
   263
    apply clarify
huffman@31492
   264
    apply (rule exI, erule conjI)
huffman@31492
   265
    apply (rule exI, rule conjI [OF open_UNIV])
huffman@31491
   266
    apply auto
huffman@31491
   267
    done
huffman@31491
   268
  with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
huffman@31491
   269
    by (rule topological_tendstoD)
huffman@31491
   270
  then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
huffman@31491
   271
    by simp
huffman@31405
   272
qed
huffman@31405
   273
huffman@31565
   274
lemma tendsto_snd [tendsto_intros]:
huffman@31491
   275
  assumes "(f ---> a) net"
huffman@31491
   276
  shows "((\<lambda>x. snd (f x)) ---> snd a) net"
huffman@31491
   277
proof (rule topological_tendstoI)
huffman@31492
   278
  fix S assume "open S" "snd a \<in> S"
huffman@31492
   279
  then have "open (snd -` S)" "a \<in> snd -` S"
huffman@31492
   280
    unfolding open_prod_def
huffman@31491
   281
    apply simp_all
huffman@31491
   282
    apply clarify
huffman@31492
   283
    apply (rule exI, rule conjI [OF open_UNIV])
huffman@31492
   284
    apply (rule exI, erule conjI)
huffman@31491
   285
    apply auto
huffman@31491
   286
    done
huffman@31491
   287
  with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
huffman@31491
   288
    by (rule topological_tendstoD)
huffman@31491
   289
  then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
huffman@31491
   290
    by simp
huffman@31405
   291
qed
huffman@31405
   292
huffman@31565
   293
lemma tendsto_Pair [tendsto_intros]:
huffman@31491
   294
  assumes "(f ---> a) net" and "(g ---> b) net"
huffman@31491
   295
  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
huffman@31491
   296
proof (rule topological_tendstoI)
huffman@31492
   297
  fix S assume "open S" "(a, b) \<in> S"
huffman@31492
   298
  then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
huffman@31492
   299
    unfolding open_prod_def by auto
huffman@31491
   300
  have "eventually (\<lambda>x. f x \<in> A) net"
huffman@31492
   301
    using `(f ---> a) net` `open A` `a \<in> A`
huffman@31491
   302
    by (rule topological_tendstoD)
huffman@31405
   303
  moreover
huffman@31491
   304
  have "eventually (\<lambda>x. g x \<in> B) net"
huffman@31492
   305
    using `(g ---> b) net` `open B` `b \<in> B`
huffman@31491
   306
    by (rule topological_tendstoD)
huffman@31405
   307
  ultimately
huffman@31491
   308
  show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
huffman@31405
   309
    by (rule eventually_elim2)
huffman@31491
   310
       (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
huffman@31405
   311
qed
huffman@31405
   312
huffman@31405
   313
lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
huffman@31405
   314
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
huffman@31405
   315
huffman@31405
   316
lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
huffman@31405
   317
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
huffman@31405
   318
huffman@31405
   319
lemma Cauchy_Pair:
huffman@31405
   320
  assumes "Cauchy X" and "Cauchy Y"
huffman@31405
   321
  shows "Cauchy (\<lambda>n. (X n, Y n))"
huffman@31405
   322
proof (rule metric_CauchyI)
huffman@31405
   323
  fix r :: real assume "0 < r"
huffman@31405
   324
  then have "0 < r / sqrt 2" (is "0 < ?s")
huffman@31405
   325
    by (simp add: divide_pos_pos)
huffman@31405
   326
  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
huffman@31405
   327
    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
huffman@31405
   328
  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
huffman@31405
   329
    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
huffman@31405
   330
  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
   331
    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
huffman@31405
   332
  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
huffman@31405
   333
qed
huffman@31405
   334
huffman@31405
   335
lemma isCont_Pair [simp]:
huffman@31405
   336
  "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
huffman@36661
   337
  unfolding isCont_def by (rule tendsto_Pair)
huffman@31405
   338
huffman@31405
   339
subsection {* Product is a complete metric space *}
huffman@31405
   340
haftmann@37678
   341
instance prod :: (complete_space, complete_space) complete_space
huffman@31405
   342
proof
huffman@31405
   343
  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
huffman@31405
   344
  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
huffman@31405
   345
    using Cauchy_fst [OF `Cauchy X`]
huffman@31405
   346
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405
   347
  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
huffman@31405
   348
    using Cauchy_snd [OF `Cauchy X`]
huffman@31405
   349
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405
   350
  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
huffman@36660
   351
    using tendsto_Pair [OF 1 2] by simp
huffman@31405
   352
  then show "convergent X"
huffman@31405
   353
    by (rule convergentI)
huffman@31405
   354
qed
huffman@31405
   355
huffman@30019
   356
subsection {* Product is a normed vector space *}
huffman@30019
   357
haftmann@37678
   358
instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
huffman@30019
   359
begin
huffman@30019
   360
huffman@30019
   361
definition norm_prod_def:
huffman@30019
   362
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
huffman@30019
   363
huffman@30019
   364
definition sgn_prod_def:
huffman@30019
   365
  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
huffman@30019
   366
huffman@30019
   367
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
huffman@30019
   368
  unfolding norm_prod_def by simp
huffman@30019
   369
huffman@30019
   370
instance proof
huffman@30019
   371
  fix r :: real and x y :: "'a \<times> 'b"
huffman@30019
   372
  show "0 \<le> norm x"
huffman@30019
   373
    unfolding norm_prod_def by simp
huffman@30019
   374
  show "norm x = 0 \<longleftrightarrow> x = 0"
huffman@30019
   375
    unfolding norm_prod_def
huffman@44066
   376
    by (simp add: prod_eq_iff)
huffman@30019
   377
  show "norm (x + y) \<le> norm x + norm y"
huffman@30019
   378
    unfolding norm_prod_def
huffman@30019
   379
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
huffman@30019
   380
    apply (simp add: add_mono power_mono norm_triangle_ineq)
huffman@30019
   381
    done
huffman@30019
   382
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
huffman@30019
   383
    unfolding norm_prod_def
huffman@31587
   384
    apply (simp add: power_mult_distrib)
huffman@30019
   385
    apply (simp add: right_distrib [symmetric])
huffman@30019
   386
    apply (simp add: real_sqrt_mult_distrib)
huffman@30019
   387
    done
huffman@30019
   388
  show "sgn x = scaleR (inverse (norm x)) x"
huffman@30019
   389
    by (rule sgn_prod_def)
huffman@31290
   390
  show "dist x y = norm (x - y)"
huffman@31339
   391
    unfolding dist_prod_def norm_prod_def
huffman@31339
   392
    by (simp add: dist_norm)
huffman@30019
   393
qed
huffman@30019
   394
huffman@30019
   395
end
huffman@30019
   396
haftmann@37678
   397
instance prod :: (banach, banach) banach ..
huffman@31405
   398
huffman@30019
   399
subsection {* Product is an inner product space *}
huffman@30019
   400
haftmann@37678
   401
instantiation prod :: (real_inner, real_inner) real_inner
huffman@30019
   402
begin
huffman@30019
   403
huffman@30019
   404
definition inner_prod_def:
huffman@30019
   405
  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
huffman@30019
   406
huffman@30019
   407
lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
huffman@30019
   408
  unfolding inner_prod_def by simp
huffman@30019
   409
huffman@30019
   410
instance proof
huffman@30019
   411
  fix r :: real
huffman@30019
   412
  fix x y z :: "'a::real_inner * 'b::real_inner"
huffman@30019
   413
  show "inner x y = inner y x"
huffman@30019
   414
    unfolding inner_prod_def
huffman@30019
   415
    by (simp add: inner_commute)
huffman@30019
   416
  show "inner (x + y) z = inner x z + inner y z"
huffman@30019
   417
    unfolding inner_prod_def
huffman@31590
   418
    by (simp add: inner_add_left)
huffman@30019
   419
  show "inner (scaleR r x) y = r * inner x y"
huffman@30019
   420
    unfolding inner_prod_def
huffman@31590
   421
    by (simp add: right_distrib)
huffman@30019
   422
  show "0 \<le> inner x x"
huffman@30019
   423
    unfolding inner_prod_def
huffman@30019
   424
    by (intro add_nonneg_nonneg inner_ge_zero)
huffman@30019
   425
  show "inner x x = 0 \<longleftrightarrow> x = 0"
huffman@44066
   426
    unfolding inner_prod_def prod_eq_iff
huffman@30019
   427
    by (simp add: add_nonneg_eq_0_iff)
huffman@30019
   428
  show "norm x = sqrt (inner x x)"
huffman@30019
   429
    unfolding norm_prod_def inner_prod_def
huffman@30019
   430
    by (simp add: power2_norm_eq_inner)
huffman@30019
   431
qed
huffman@30019
   432
huffman@30019
   433
end
huffman@30019
   434
huffman@31405
   435
subsection {* Pair operations are linear *}
huffman@30019
   436
wenzelm@30729
   437
interpretation fst: bounded_linear fst
huffman@30019
   438
  apply (unfold_locales)
huffman@30019
   439
  apply (rule fst_add)
huffman@30019
   440
  apply (rule fst_scaleR)
huffman@30019
   441
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
huffman@30019
   442
  done
huffman@30019
   443
wenzelm@30729
   444
interpretation snd: bounded_linear snd
huffman@30019
   445
  apply (unfold_locales)
huffman@30019
   446
  apply (rule snd_add)
huffman@30019
   447
  apply (rule snd_scaleR)
huffman@30019
   448
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
huffman@30019
   449
  done
huffman@30019
   450
huffman@30019
   451
text {* TODO: move to NthRoot *}
huffman@30019
   452
lemma sqrt_add_le_add_sqrt:
huffman@30019
   453
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@30019
   454
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
huffman@30019
   455
apply (rule power2_le_imp_le)
huffman@30019
   456
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
huffman@30019
   457
apply (simp add: mult_nonneg_nonneg x y)
huffman@30019
   458
apply (simp add: add_nonneg_nonneg x y)
huffman@30019
   459
done
huffman@30019
   460
huffman@30019
   461
lemma bounded_linear_Pair:
huffman@30019
   462
  assumes f: "bounded_linear f"
huffman@30019
   463
  assumes g: "bounded_linear g"
huffman@30019
   464
  shows "bounded_linear (\<lambda>x. (f x, g x))"
huffman@30019
   465
proof
huffman@30019
   466
  interpret f: bounded_linear f by fact
huffman@30019
   467
  interpret g: bounded_linear g by fact
huffman@30019
   468
  fix x y and r :: real
huffman@30019
   469
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
huffman@30019
   470
    by (simp add: f.add g.add)
huffman@30019
   471
  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
huffman@30019
   472
    by (simp add: f.scaleR g.scaleR)
huffman@30019
   473
  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
huffman@30019
   474
    using f.pos_bounded by fast
huffman@30019
   475
  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
huffman@30019
   476
    using g.pos_bounded by fast
huffman@30019
   477
  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
huffman@30019
   478
    apply (rule allI)
huffman@30019
   479
    apply (simp add: norm_Pair)
huffman@30019
   480
    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
huffman@30019
   481
    apply (simp add: right_distrib)
huffman@30019
   482
    apply (rule add_mono [OF norm_f norm_g])
huffman@30019
   483
    done
huffman@30019
   484
  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
huffman@30019
   485
qed
huffman@30019
   486
huffman@30019
   487
subsection {* Frechet derivatives involving pairs *}
huffman@30019
   488
huffman@30019
   489
lemma FDERIV_Pair:
huffman@30019
   490
  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
huffman@30019
   491
  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
huffman@30019
   492
apply (rule FDERIV_I)
huffman@30019
   493
apply (rule bounded_linear_Pair)
huffman@30019
   494
apply (rule FDERIV_bounded_linear [OF f])
huffman@30019
   495
apply (rule FDERIV_bounded_linear [OF g])
huffman@30019
   496
apply (simp add: norm_Pair)
huffman@30019
   497
apply (rule real_LIM_sandwich_zero)
huffman@30019
   498
apply (rule LIM_add_zero)
huffman@30019
   499
apply (rule FDERIV_D [OF f])
huffman@30019
   500
apply (rule FDERIV_D [OF g])
huffman@30019
   501
apply (rename_tac h)
huffman@30019
   502
apply (simp add: divide_nonneg_pos)
huffman@30019
   503
apply (rename_tac h)
huffman@30019
   504
apply (subst add_divide_distrib [symmetric])
huffman@30019
   505
apply (rule divide_right_mono [OF _ norm_ge_zero])
huffman@30019
   506
apply (rule order_trans [OF sqrt_add_le_add_sqrt])
huffman@30019
   507
apply simp
huffman@30019
   508
apply simp
huffman@30019
   509
apply simp
huffman@30019
   510
done
huffman@30019
   511
huffman@30019
   512
end