src/HOL/Library/Product_Vector.thy
 author huffman Wed Jun 03 08:43:29 2009 -0700 (2009-06-03) changeset 31415 80686a815b59 parent 31405 1f72869f1a2e child 31417 c12b25b7f015 permissions -rw-r--r--
instance * :: topological_space
```     1 (*  Title:      HOL/Library/Product_Vector.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Cartesian Products as Vector Spaces *}
```
```     6
```
```     7 theory Product_Vector
```
```     8 imports Inner_Product Product_plus
```
```     9 begin
```
```    10
```
```    11 subsection {* Product is a real vector space *}
```
```    12
```
```    13 instantiation "*" :: (real_vector, real_vector) real_vector
```
```    14 begin
```
```    15
```
```    16 definition scaleR_prod_def:
```
```    17   "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
```
```    18
```
```    19 lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
```
```    20   unfolding scaleR_prod_def by simp
```
```    21
```
```    22 lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
```
```    23   unfolding scaleR_prod_def by simp
```
```    24
```
```    25 lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
```
```    26   unfolding scaleR_prod_def by simp
```
```    27
```
```    28 instance proof
```
```    29   fix a b :: real and x y :: "'a \<times> 'b"
```
```    30   show "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```    31     by (simp add: expand_prod_eq scaleR_right_distrib)
```
```    32   show "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```    33     by (simp add: expand_prod_eq scaleR_left_distrib)
```
```    34   show "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```    35     by (simp add: expand_prod_eq)
```
```    36   show "scaleR 1 x = x"
```
```    37     by (simp add: expand_prod_eq)
```
```    38 qed
```
```    39
```
```    40 end
```
```    41
```
```    42 subsection {* Product is a topological space *}
```
```    43
```
```    44 instantiation
```
```    45   "*" :: (topological_space, topological_space) topological_space
```
```    46 begin
```
```    47
```
```    48 definition open_prod_def:
```
```    49   "open S = (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
```
```    50
```
```    51 instance proof
```
```    52   show "open (UNIV :: ('a \<times> 'b) set)"
```
```    53     unfolding open_prod_def by (fast intro: open_UNIV)
```
```    54 next
```
```    55   fix S T :: "('a \<times> 'b) set"
```
```    56   assume "open S" "open T" thus "open (S \<inter> T)"
```
```    57     unfolding open_prod_def
```
```    58     apply clarify
```
```    59     apply (drule (1) bspec)+
```
```    60     apply (clarify, rename_tac Sa Ta Sb Tb)
```
```    61     apply (rule_tac x="Sa \<inter> Ta" in exI)
```
```    62     apply (rule_tac x="Sb \<inter> Tb" in exI)
```
```    63     apply (simp add: open_Int)
```
```    64     apply fast
```
```    65     done
```
```    66 next
```
```    67   fix T :: "('a \<times> 'b) set set"
```
```    68   assume "\<forall>A\<in>T. open A" thus "open (\<Union>T)"
```
```    69     unfolding open_prod_def by fast
```
```    70 qed
```
```    71
```
```    72 end
```
```    73
```
```    74 subsection {* Product is a metric space *}
```
```    75
```
```    76 instantiation
```
```    77   "*" :: (metric_space, metric_space) metric_space
```
```    78 begin
```
```    79
```
```    80 definition dist_prod_def:
```
```    81   "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
```
```    82
```
```    83 lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
```
```    84   unfolding dist_prod_def by simp
```
```    85
```
```    86 instance proof
```
```    87   fix x y :: "'a \<times> 'b"
```
```    88   show "dist x y = 0 \<longleftrightarrow> x = y"
```
```    89     unfolding dist_prod_def
```
```    90     by (simp add: expand_prod_eq)
```
```    91 next
```
```    92   fix x y z :: "'a \<times> 'b"
```
```    93   show "dist x y \<le> dist x z + dist y z"
```
```    94     unfolding dist_prod_def
```
```    95     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
```
```    96     apply (rule real_sqrt_le_mono)
```
```    97     apply (rule order_trans [OF add_mono])
```
```    98     apply (rule power_mono [OF dist_triangle2 [of _ _ "fst z"] zero_le_dist])
```
```    99     apply (rule power_mono [OF dist_triangle2 [of _ _ "snd z"] zero_le_dist])
```
```   100     apply (simp only: real_sum_squared_expand)
```
```   101     done
```
```   102 next
```
```   103   (* FIXME: long proof! *)
```
```   104   (* Maybe it would be easier to define topological spaces *)
```
```   105   (* in terms of neighborhoods instead of open sets? *)
```
```   106   fix S :: "('a \<times> 'b) set"
```
```   107   show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   108     unfolding open_prod_def open_dist
```
```   109     apply safe
```
```   110     apply (drule (1) bspec)
```
```   111     apply clarify
```
```   112     apply (drule (1) bspec)+
```
```   113     apply (clarify, rename_tac r s)
```
```   114     apply (rule_tac x="min r s" in exI, simp)
```
```   115     apply (clarify, rename_tac c d)
```
```   116     apply (erule subsetD)
```
```   117     apply (simp add: dist_Pair_Pair)
```
```   118     apply (rule conjI)
```
```   119     apply (drule spec, erule mp)
```
```   120     apply (erule le_less_trans [OF real_sqrt_sum_squares_ge1])
```
```   121     apply (drule spec, erule mp)
```
```   122     apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2])
```
```   123
```
```   124     apply (drule (1) bspec)
```
```   125     apply clarify
```
```   126     apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
```
```   127     apply clarify
```
```   128     apply (rule_tac x="{y. dist y a < r}" in exI)
```
```   129     apply (rule_tac x="{y. dist y b < s}" in exI)
```
```   130     apply (rule conjI)
```
```   131     apply clarify
```
```   132     apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
```
```   133     apply clarify
```
```   134     apply (rule le_less_trans [OF dist_triangle])
```
```   135     apply (erule less_le_trans [OF add_strict_right_mono], simp)
```
```   136     apply (rule conjI)
```
```   137     apply clarify
```
```   138     apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
```
```   139     apply clarify
```
```   140     apply (rule le_less_trans [OF dist_triangle])
```
```   141     apply (erule less_le_trans [OF add_strict_right_mono], simp)
```
```   142     apply (rule conjI)
```
```   143     apply simp
```
```   144     apply (clarify, rename_tac c d)
```
```   145     apply (drule spec, erule mp)
```
```   146     apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
```
```   147     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
```
```   148     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
```
```   149     apply (simp add: power_divide)
```
```   150     done
```
```   151 qed
```
```   152
```
```   153 end
```
```   154
```
```   155 subsection {* Continuity of operations *}
```
```   156
```
```   157 lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
```
```   158 unfolding dist_prod_def by simp
```
```   159
```
```   160 lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
```
```   161 unfolding dist_prod_def by simp
```
```   162
```
```   163 lemma tendsto_fst:
```
```   164   assumes "tendsto f a net"
```
```   165   shows "tendsto (\<lambda>x. fst (f x)) (fst a) net"
```
```   166 proof (rule tendstoI)
```
```   167   fix r :: real assume "0 < r"
```
```   168   have "eventually (\<lambda>x. dist (f x) a < r) net"
```
```   169     using `tendsto f a net` `0 < r` by (rule tendstoD)
```
```   170   thus "eventually (\<lambda>x. dist (fst (f x)) (fst a) < r) net"
```
```   171     by (rule eventually_elim1)
```
```   172        (rule le_less_trans [OF dist_fst_le])
```
```   173 qed
```
```   174
```
```   175 lemma tendsto_snd:
```
```   176   assumes "tendsto f a net"
```
```   177   shows "tendsto (\<lambda>x. snd (f x)) (snd a) net"
```
```   178 proof (rule tendstoI)
```
```   179   fix r :: real assume "0 < r"
```
```   180   have "eventually (\<lambda>x. dist (f x) a < r) net"
```
```   181     using `tendsto f a net` `0 < r` by (rule tendstoD)
```
```   182   thus "eventually (\<lambda>x. dist (snd (f x)) (snd a) < r) net"
```
```   183     by (rule eventually_elim1)
```
```   184        (rule le_less_trans [OF dist_snd_le])
```
```   185 qed
```
```   186
```
```   187 lemma tendsto_Pair:
```
```   188   assumes "tendsto X a net" and "tendsto Y b net"
```
```   189   shows "tendsto (\<lambda>i. (X i, Y i)) (a, b) net"
```
```   190 proof (rule tendstoI)
```
```   191   fix r :: real assume "0 < r"
```
```   192   then have "0 < r / sqrt 2" (is "0 < ?s")
```
```   193     by (simp add: divide_pos_pos)
```
```   194   have "eventually (\<lambda>i. dist (X i) a < ?s) net"
```
```   195     using `tendsto X a net` `0 < ?s` by (rule tendstoD)
```
```   196   moreover
```
```   197   have "eventually (\<lambda>i. dist (Y i) b < ?s) net"
```
```   198     using `tendsto Y b net` `0 < ?s` by (rule tendstoD)
```
```   199   ultimately
```
```   200   show "eventually (\<lambda>i. dist (X i, Y i) (a, b) < r) net"
```
```   201     by (rule eventually_elim2)
```
```   202        (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
```
```   203 qed
```
```   204
```
```   205 lemma LIMSEQ_fst: "(X ----> a) \<Longrightarrow> (\<lambda>n. fst (X n)) ----> fst a"
```
```   206 unfolding LIMSEQ_conv_tendsto by (rule tendsto_fst)
```
```   207
```
```   208 lemma LIMSEQ_snd: "(X ----> a) \<Longrightarrow> (\<lambda>n. snd (X n)) ----> snd a"
```
```   209 unfolding LIMSEQ_conv_tendsto by (rule tendsto_snd)
```
```   210
```
```   211 lemma LIMSEQ_Pair:
```
```   212   assumes "X ----> a" and "Y ----> b"
```
```   213   shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
```
```   214 using assms unfolding LIMSEQ_conv_tendsto
```
```   215 by (rule tendsto_Pair)
```
```   216
```
```   217 lemma LIM_fst: "f -- x --> a \<Longrightarrow> (\<lambda>x. fst (f x)) -- x --> fst a"
```
```   218 unfolding LIM_conv_tendsto by (rule tendsto_fst)
```
```   219
```
```   220 lemma LIM_snd: "f -- x --> a \<Longrightarrow> (\<lambda>x. snd (f x)) -- x --> snd a"
```
```   221 unfolding LIM_conv_tendsto by (rule tendsto_snd)
```
```   222
```
```   223 lemma LIM_Pair:
```
```   224   assumes "f -- x --> a" and "g -- x --> b"
```
```   225   shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
```
```   226 using assms unfolding LIM_conv_tendsto
```
```   227 by (rule tendsto_Pair)
```
```   228
```
```   229 lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
```
```   230 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
```
```   231
```
```   232 lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
```
```   233 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
```
```   234
```
```   235 lemma Cauchy_Pair:
```
```   236   assumes "Cauchy X" and "Cauchy Y"
```
```   237   shows "Cauchy (\<lambda>n. (X n, Y n))"
```
```   238 proof (rule metric_CauchyI)
```
```   239   fix r :: real assume "0 < r"
```
```   240   then have "0 < r / sqrt 2" (is "0 < ?s")
```
```   241     by (simp add: divide_pos_pos)
```
```   242   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
```
```   243     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
```
```   244   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
```
```   245     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
```
```   246   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
```
```   247     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
```
```   248   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
```
```   249 qed
```
```   250
```
```   251 lemma isCont_Pair [simp]:
```
```   252   "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
```
```   253   unfolding isCont_def by (rule LIM_Pair)
```
```   254
```
```   255 subsection {* Product is a complete metric space *}
```
```   256
```
```   257 instance "*" :: (complete_space, complete_space) complete_space
```
```   258 proof
```
```   259   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
```
```   260   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
```
```   261     using Cauchy_fst [OF `Cauchy X`]
```
```   262     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   263   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
```
```   264     using Cauchy_snd [OF `Cauchy X`]
```
```   265     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   266   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
```
```   267     using LIMSEQ_Pair [OF 1 2] by simp
```
```   268   then show "convergent X"
```
```   269     by (rule convergentI)
```
```   270 qed
```
```   271
```
```   272 subsection {* Product is a normed vector space *}
```
```   273
```
```   274 instantiation
```
```   275   "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
```
```   276 begin
```
```   277
```
```   278 definition norm_prod_def:
```
```   279   "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
```
```   280
```
```   281 definition sgn_prod_def:
```
```   282   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
```
```   283
```
```   284 lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
```
```   285   unfolding norm_prod_def by simp
```
```   286
```
```   287 instance proof
```
```   288   fix r :: real and x y :: "'a \<times> 'b"
```
```   289   show "0 \<le> norm x"
```
```   290     unfolding norm_prod_def by simp
```
```   291   show "norm x = 0 \<longleftrightarrow> x = 0"
```
```   292     unfolding norm_prod_def
```
```   293     by (simp add: expand_prod_eq)
```
```   294   show "norm (x + y) \<le> norm x + norm y"
```
```   295     unfolding norm_prod_def
```
```   296     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
```
```   297     apply (simp add: add_mono power_mono norm_triangle_ineq)
```
```   298     done
```
```   299   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
```
```   300     unfolding norm_prod_def
```
```   301     apply (simp add: norm_scaleR power_mult_distrib)
```
```   302     apply (simp add: right_distrib [symmetric])
```
```   303     apply (simp add: real_sqrt_mult_distrib)
```
```   304     done
```
```   305   show "sgn x = scaleR (inverse (norm x)) x"
```
```   306     by (rule sgn_prod_def)
```
```   307   show "dist x y = norm (x - y)"
```
```   308     unfolding dist_prod_def norm_prod_def
```
```   309     by (simp add: dist_norm)
```
```   310 qed
```
```   311
```
```   312 end
```
```   313
```
```   314 instance "*" :: (banach, banach) banach ..
```
```   315
```
```   316 subsection {* Product is an inner product space *}
```
```   317
```
```   318 instantiation "*" :: (real_inner, real_inner) real_inner
```
```   319 begin
```
```   320
```
```   321 definition inner_prod_def:
```
```   322   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
```
```   323
```
```   324 lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
```
```   325   unfolding inner_prod_def by simp
```
```   326
```
```   327 instance proof
```
```   328   fix r :: real
```
```   329   fix x y z :: "'a::real_inner * 'b::real_inner"
```
```   330   show "inner x y = inner y x"
```
```   331     unfolding inner_prod_def
```
```   332     by (simp add: inner_commute)
```
```   333   show "inner (x + y) z = inner x z + inner y z"
```
```   334     unfolding inner_prod_def
```
```   335     by (simp add: inner_left_distrib)
```
```   336   show "inner (scaleR r x) y = r * inner x y"
```
```   337     unfolding inner_prod_def
```
```   338     by (simp add: inner_scaleR_left right_distrib)
```
```   339   show "0 \<le> inner x x"
```
```   340     unfolding inner_prod_def
```
```   341     by (intro add_nonneg_nonneg inner_ge_zero)
```
```   342   show "inner x x = 0 \<longleftrightarrow> x = 0"
```
```   343     unfolding inner_prod_def expand_prod_eq
```
```   344     by (simp add: add_nonneg_eq_0_iff)
```
```   345   show "norm x = sqrt (inner x x)"
```
```   346     unfolding norm_prod_def inner_prod_def
```
```   347     by (simp add: power2_norm_eq_inner)
```
```   348 qed
```
```   349
```
```   350 end
```
```   351
```
```   352 subsection {* Pair operations are linear *}
```
```   353
```
```   354 interpretation fst: bounded_linear fst
```
```   355   apply (unfold_locales)
```
```   356   apply (rule fst_add)
```
```   357   apply (rule fst_scaleR)
```
```   358   apply (rule_tac x="1" in exI, simp add: norm_Pair)
```
```   359   done
```
```   360
```
```   361 interpretation snd: bounded_linear snd
```
```   362   apply (unfold_locales)
```
```   363   apply (rule snd_add)
```
```   364   apply (rule snd_scaleR)
```
```   365   apply (rule_tac x="1" in exI, simp add: norm_Pair)
```
```   366   done
```
```   367
```
```   368 text {* TODO: move to NthRoot *}
```
```   369 lemma sqrt_add_le_add_sqrt:
```
```   370   assumes x: "0 \<le> x" and y: "0 \<le> y"
```
```   371   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
```
```   372 apply (rule power2_le_imp_le)
```
```   373 apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
```
```   374 apply (simp add: mult_nonneg_nonneg x y)
```
```   375 apply (simp add: add_nonneg_nonneg x y)
```
```   376 done
```
```   377
```
```   378 lemma bounded_linear_Pair:
```
```   379   assumes f: "bounded_linear f"
```
```   380   assumes g: "bounded_linear g"
```
```   381   shows "bounded_linear (\<lambda>x. (f x, g x))"
```
```   382 proof
```
```   383   interpret f: bounded_linear f by fact
```
```   384   interpret g: bounded_linear g by fact
```
```   385   fix x y and r :: real
```
```   386   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
```
```   387     by (simp add: f.add g.add)
```
```   388   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
```
```   389     by (simp add: f.scaleR g.scaleR)
```
```   390   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
```
```   391     using f.pos_bounded by fast
```
```   392   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
```
```   393     using g.pos_bounded by fast
```
```   394   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
```
```   395     apply (rule allI)
```
```   396     apply (simp add: norm_Pair)
```
```   397     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
```
```   398     apply (simp add: right_distrib)
```
```   399     apply (rule add_mono [OF norm_f norm_g])
```
```   400     done
```
```   401   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
```
```   402 qed
```
```   403
```
```   404 subsection {* Frechet derivatives involving pairs *}
```
```   405
```
```   406 lemma FDERIV_Pair:
```
```   407   assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
```
```   408   shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
```
```   409 apply (rule FDERIV_I)
```
```   410 apply (rule bounded_linear_Pair)
```
```   411 apply (rule FDERIV_bounded_linear [OF f])
```
```   412 apply (rule FDERIV_bounded_linear [OF g])
```
```   413 apply (simp add: norm_Pair)
```
```   414 apply (rule real_LIM_sandwich_zero)
```
```   415 apply (rule LIM_add_zero)
```
```   416 apply (rule FDERIV_D [OF f])
```
```   417 apply (rule FDERIV_D [OF g])
```
```   418 apply (rename_tac h)
```
```   419 apply (simp add: divide_nonneg_pos)
```
```   420 apply (rename_tac h)
```
```   421 apply (subst add_divide_distrib [symmetric])
```
```   422 apply (rule divide_right_mono [OF _ norm_ge_zero])
```
```   423 apply (rule order_trans [OF sqrt_add_le_add_sqrt])
```
```   424 apply simp
```
```   425 apply simp
```
```   426 apply simp
```
```   427 done
```
```   428
```
```   429 end
```