src/HOL/Library/Product_Vector.thy
 author huffman Sun Jun 07 17:59:54 2009 -0700 (2009-06-07) changeset 31492 5400beeddb55 parent 31491 f7310185481d child 31562 10d0fb526643 permissions -rw-r--r--
replace 'topo' with 'open'; add extra type constraint for 'open'
```     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 :: ('a \<times> 'b) set) \<longleftrightarrow>
```
```    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)"
```
```    51
```
```    52 instance proof
```
```    53   show "open (UNIV :: ('a \<times> 'b) set)"
```
```    54     unfolding open_prod_def by auto
```
```    55 next
```
```    56   fix S T :: "('a \<times> 'b) set"
```
```    57   assume "open S" "open T" thus "open (S \<inter> T)"
```
```    58     unfolding open_prod_def
```
```    59     apply clarify
```
```    60     apply (drule (1) bspec)+
```
```    61     apply (clarify, rename_tac Sa Ta Sb Tb)
```
```    62     apply (rule_tac x="Sa \<inter> Ta" in exI)
```
```    63     apply (rule_tac x="Sb \<inter> Tb" in exI)
```
```    64     apply (simp add: open_Int)
```
```    65     apply fast
```
```    66     done
```
```    67 next
```
```    68   fix K :: "('a \<times> 'b) set set"
```
```    69   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```    70     unfolding open_prod_def by fast
```
```    71 qed
```
```    72
```
```    73 end
```
```    74
```
```    75 subsection {* Product is a metric space *}
```
```    76
```
```    77 instantiation
```
```    78   "*" :: (metric_space, metric_space) metric_space
```
```    79 begin
```
```    80
```
```    81 definition dist_prod_def:
```
```    82   "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
```
```    83
```
```    84 lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
```
```    85   unfolding dist_prod_def by simp
```
```    86
```
```    87 instance proof
```
```    88   fix x y :: "'a \<times> 'b"
```
```    89   show "dist x y = 0 \<longleftrightarrow> x = y"
```
```    90     unfolding dist_prod_def
```
```    91     by (simp add: expand_prod_eq)
```
```    92 next
```
```    93   fix x y z :: "'a \<times> 'b"
```
```    94   show "dist x y \<le> dist x z + dist y z"
```
```    95     unfolding dist_prod_def
```
```    96     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
```
```    97     apply (rule real_sqrt_le_mono)
```
```    98     apply (rule order_trans [OF add_mono])
```
```    99     apply (rule power_mono [OF dist_triangle2 [of _ _ "fst z"] zero_le_dist])
```
```   100     apply (rule power_mono [OF dist_triangle2 [of _ _ "snd z"] zero_le_dist])
```
```   101     apply (simp only: real_sum_squared_expand)
```
```   102     done
```
```   103 next
```
```   104   (* FIXME: long proof! *)
```
```   105   (* Maybe it would be easier to define topological spaces *)
```
```   106   (* in terms of neighborhoods instead of open sets? *)
```
```   107   fix S :: "('a \<times> 'b) set"
```
```   108   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   109     unfolding open_prod_def open_dist
```
```   110     apply safe
```
```   111     apply (drule (1) bspec)
```
```   112     apply clarify
```
```   113     apply (drule (1) bspec)+
```
```   114     apply (clarify, rename_tac r s)
```
```   115     apply (rule_tac x="min r s" in exI, simp)
```
```   116     apply (clarify, rename_tac c d)
```
```   117     apply (erule subsetD)
```
```   118     apply (simp add: dist_Pair_Pair)
```
```   119     apply (rule conjI)
```
```   120     apply (drule spec, erule mp)
```
```   121     apply (erule le_less_trans [OF real_sqrt_sum_squares_ge1])
```
```   122     apply (drule spec, erule mp)
```
```   123     apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2])
```
```   124
```
```   125     apply (drule (1) bspec)
```
```   126     apply clarify
```
```   127     apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
```
```   128     apply clarify
```
```   129     apply (rule_tac x="{y. dist y a < r}" in exI)
```
```   130     apply (rule_tac x="{y. dist y b < s}" in exI)
```
```   131     apply (rule conjI)
```
```   132     apply clarify
```
```   133     apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
```
```   134     apply clarify
```
```   135     apply (rule le_less_trans [OF dist_triangle])
```
```   136     apply (erule less_le_trans [OF add_strict_right_mono], simp)
```
```   137     apply (rule conjI)
```
```   138     apply clarify
```
```   139     apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
```
```   140     apply clarify
```
```   141     apply (rule le_less_trans [OF dist_triangle])
```
```   142     apply (erule less_le_trans [OF add_strict_right_mono], simp)
```
```   143     apply (rule conjI)
```
```   144     apply simp
```
```   145     apply (clarify, rename_tac c d)
```
```   146     apply (drule spec, erule mp)
```
```   147     apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
```
```   148     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
```
```   149     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
```
```   150     apply (simp add: power_divide)
```
```   151     done
```
```   152 qed
```
```   153
```
```   154 end
```
```   155
```
```   156 subsection {* Continuity of operations *}
```
```   157
```
```   158 lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
```
```   159 unfolding dist_prod_def by simp
```
```   160
```
```   161 lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
```
```   162 unfolding dist_prod_def by simp
```
```   163
```
```   164 lemma tendsto_fst:
```
```   165   assumes "(f ---> a) net"
```
```   166   shows "((\<lambda>x. fst (f x)) ---> fst a) net"
```
```   167 proof (rule topological_tendstoI)
```
```   168   fix S assume "open S" "fst a \<in> S"
```
```   169   then have "open (fst -` S)" "a \<in> fst -` S"
```
```   170     unfolding open_prod_def
```
```   171     apply simp_all
```
```   172     apply clarify
```
```   173     apply (rule exI, erule conjI)
```
```   174     apply (rule exI, rule conjI [OF open_UNIV])
```
```   175     apply auto
```
```   176     done
```
```   177   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
```
```   178     by (rule topological_tendstoD)
```
```   179   then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
```
```   180     by simp
```
```   181 qed
```
```   182
```
```   183 lemma tendsto_snd:
```
```   184   assumes "(f ---> a) net"
```
```   185   shows "((\<lambda>x. snd (f x)) ---> snd a) net"
```
```   186 proof (rule topological_tendstoI)
```
```   187   fix S assume "open S" "snd a \<in> S"
```
```   188   then have "open (snd -` S)" "a \<in> snd -` S"
```
```   189     unfolding open_prod_def
```
```   190     apply simp_all
```
```   191     apply clarify
```
```   192     apply (rule exI, rule conjI [OF open_UNIV])
```
```   193     apply (rule exI, erule conjI)
```
```   194     apply auto
```
```   195     done
```
```   196   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
```
```   197     by (rule topological_tendstoD)
```
```   198   then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
```
```   199     by simp
```
```   200 qed
```
```   201
```
```   202 lemma tendsto_Pair:
```
```   203   assumes "(f ---> a) net" and "(g ---> b) net"
```
```   204   shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
```
```   205 proof (rule topological_tendstoI)
```
```   206   fix S assume "open S" "(a, b) \<in> S"
```
```   207   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
```
```   208     unfolding open_prod_def by auto
```
```   209   have "eventually (\<lambda>x. f x \<in> A) net"
```
```   210     using `(f ---> a) net` `open A` `a \<in> A`
```
```   211     by (rule topological_tendstoD)
```
```   212   moreover
```
```   213   have "eventually (\<lambda>x. g x \<in> B) net"
```
```   214     using `(g ---> b) net` `open B` `b \<in> B`
```
```   215     by (rule topological_tendstoD)
```
```   216   ultimately
```
```   217   show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
```
```   218     by (rule eventually_elim2)
```
```   219        (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
```
```   220 qed
```
```   221
```
```   222 lemma LIMSEQ_fst: "(X ----> a) \<Longrightarrow> (\<lambda>n. fst (X n)) ----> fst a"
```
```   223 unfolding LIMSEQ_conv_tendsto by (rule tendsto_fst)
```
```   224
```
```   225 lemma LIMSEQ_snd: "(X ----> a) \<Longrightarrow> (\<lambda>n. snd (X n)) ----> snd a"
```
```   226 unfolding LIMSEQ_conv_tendsto by (rule tendsto_snd)
```
```   227
```
```   228 lemma LIMSEQ_Pair:
```
```   229   assumes "X ----> a" and "Y ----> b"
```
```   230   shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
```
```   231 using assms unfolding LIMSEQ_conv_tendsto
```
```   232 by (rule tendsto_Pair)
```
```   233
```
```   234 lemma LIM_fst: "f -- x --> a \<Longrightarrow> (\<lambda>x. fst (f x)) -- x --> fst a"
```
```   235 unfolding LIM_conv_tendsto by (rule tendsto_fst)
```
```   236
```
```   237 lemma LIM_snd: "f -- x --> a \<Longrightarrow> (\<lambda>x. snd (f x)) -- x --> snd a"
```
```   238 unfolding LIM_conv_tendsto by (rule tendsto_snd)
```
```   239
```
```   240 lemma LIM_Pair:
```
```   241   assumes "f -- x --> a" and "g -- x --> b"
```
```   242   shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
```
```   243 using assms unfolding LIM_conv_tendsto
```
```   244 by (rule tendsto_Pair)
```
```   245
```
```   246 lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
```
```   247 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
```
```   248
```
```   249 lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
```
```   250 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
```
```   251
```
```   252 lemma Cauchy_Pair:
```
```   253   assumes "Cauchy X" and "Cauchy Y"
```
```   254   shows "Cauchy (\<lambda>n. (X n, Y n))"
```
```   255 proof (rule metric_CauchyI)
```
```   256   fix r :: real assume "0 < r"
```
```   257   then have "0 < r / sqrt 2" (is "0 < ?s")
```
```   258     by (simp add: divide_pos_pos)
```
```   259   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
```
```   260     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
```
```   261   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
```
```   262     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
```
```   263   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
```
```   264     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
```
```   265   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
```
```   266 qed
```
```   267
```
```   268 lemma isCont_Pair [simp]:
```
```   269   "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
```
```   270   unfolding isCont_def by (rule LIM_Pair)
```
```   271
```
```   272 subsection {* Product is a complete metric space *}
```
```   273
```
```   274 instance "*" :: (complete_space, complete_space) complete_space
```
```   275 proof
```
```   276   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
```
```   277   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
```
```   278     using Cauchy_fst [OF `Cauchy X`]
```
```   279     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   280   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
```
```   281     using Cauchy_snd [OF `Cauchy X`]
```
```   282     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   283   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
```
```   284     using LIMSEQ_Pair [OF 1 2] by simp
```
```   285   then show "convergent X"
```
```   286     by (rule convergentI)
```
```   287 qed
```
```   288
```
```   289 subsection {* Product is a normed vector space *}
```
```   290
```
```   291 instantiation
```
```   292   "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
```
```   293 begin
```
```   294
```
```   295 definition norm_prod_def:
```
```   296   "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
```
```   297
```
```   298 definition sgn_prod_def:
```
```   299   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
```
```   300
```
```   301 lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
```
```   302   unfolding norm_prod_def by simp
```
```   303
```
```   304 instance proof
```
```   305   fix r :: real and x y :: "'a \<times> 'b"
```
```   306   show "0 \<le> norm x"
```
```   307     unfolding norm_prod_def by simp
```
```   308   show "norm x = 0 \<longleftrightarrow> x = 0"
```
```   309     unfolding norm_prod_def
```
```   310     by (simp add: expand_prod_eq)
```
```   311   show "norm (x + y) \<le> norm x + norm y"
```
```   312     unfolding norm_prod_def
```
```   313     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
```
```   314     apply (simp add: add_mono power_mono norm_triangle_ineq)
```
```   315     done
```
```   316   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
```
```   317     unfolding norm_prod_def
```
```   318     apply (simp add: norm_scaleR power_mult_distrib)
```
```   319     apply (simp add: right_distrib [symmetric])
```
```   320     apply (simp add: real_sqrt_mult_distrib)
```
```   321     done
```
```   322   show "sgn x = scaleR (inverse (norm x)) x"
```
```   323     by (rule sgn_prod_def)
```
```   324   show "dist x y = norm (x - y)"
```
```   325     unfolding dist_prod_def norm_prod_def
```
```   326     by (simp add: dist_norm)
```
```   327 qed
```
```   328
```
```   329 end
```
```   330
```
```   331 instance "*" :: (banach, banach) banach ..
```
```   332
```
```   333 subsection {* Product is an inner product space *}
```
```   334
```
```   335 instantiation "*" :: (real_inner, real_inner) real_inner
```
```   336 begin
```
```   337
```
```   338 definition inner_prod_def:
```
```   339   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
```
```   340
```
```   341 lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
```
```   342   unfolding inner_prod_def by simp
```
```   343
```
```   344 instance proof
```
```   345   fix r :: real
```
```   346   fix x y z :: "'a::real_inner * 'b::real_inner"
```
```   347   show "inner x y = inner y x"
```
```   348     unfolding inner_prod_def
```
```   349     by (simp add: inner_commute)
```
```   350   show "inner (x + y) z = inner x z + inner y z"
```
```   351     unfolding inner_prod_def
```
```   352     by (simp add: inner_left_distrib)
```
```   353   show "inner (scaleR r x) y = r * inner x y"
```
```   354     unfolding inner_prod_def
```
```   355     by (simp add: inner_scaleR_left right_distrib)
```
```   356   show "0 \<le> inner x x"
```
```   357     unfolding inner_prod_def
```
```   358     by (intro add_nonneg_nonneg inner_ge_zero)
```
```   359   show "inner x x = 0 \<longleftrightarrow> x = 0"
```
```   360     unfolding inner_prod_def expand_prod_eq
```
```   361     by (simp add: add_nonneg_eq_0_iff)
```
```   362   show "norm x = sqrt (inner x x)"
```
```   363     unfolding norm_prod_def inner_prod_def
```
```   364     by (simp add: power2_norm_eq_inner)
```
```   365 qed
```
```   366
```
```   367 end
```
```   368
```
```   369 subsection {* Pair operations are linear *}
```
```   370
```
```   371 interpretation fst: bounded_linear fst
```
```   372   apply (unfold_locales)
```
```   373   apply (rule fst_add)
```
```   374   apply (rule fst_scaleR)
```
```   375   apply (rule_tac x="1" in exI, simp add: norm_Pair)
```
```   376   done
```
```   377
```
```   378 interpretation snd: bounded_linear snd
```
```   379   apply (unfold_locales)
```
```   380   apply (rule snd_add)
```
```   381   apply (rule snd_scaleR)
```
```   382   apply (rule_tac x="1" in exI, simp add: norm_Pair)
```
```   383   done
```
```   384
```
```   385 text {* TODO: move to NthRoot *}
```
```   386 lemma sqrt_add_le_add_sqrt:
```
```   387   assumes x: "0 \<le> x" and y: "0 \<le> y"
```
```   388   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
```
```   389 apply (rule power2_le_imp_le)
```
```   390 apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
```
```   391 apply (simp add: mult_nonneg_nonneg x y)
```
```   392 apply (simp add: add_nonneg_nonneg x y)
```
```   393 done
```
```   394
```
```   395 lemma bounded_linear_Pair:
```
```   396   assumes f: "bounded_linear f"
```
```   397   assumes g: "bounded_linear g"
```
```   398   shows "bounded_linear (\<lambda>x. (f x, g x))"
```
```   399 proof
```
```   400   interpret f: bounded_linear f by fact
```
```   401   interpret g: bounded_linear g by fact
```
```   402   fix x y and r :: real
```
```   403   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
```
```   404     by (simp add: f.add g.add)
```
```   405   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
```
```   406     by (simp add: f.scaleR g.scaleR)
```
```   407   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
```
```   408     using f.pos_bounded by fast
```
```   409   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
```
```   410     using g.pos_bounded by fast
```
```   411   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
```
```   412     apply (rule allI)
```
```   413     apply (simp add: norm_Pair)
```
```   414     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
```
```   415     apply (simp add: right_distrib)
```
```   416     apply (rule add_mono [OF norm_f norm_g])
```
```   417     done
```
```   418   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
```
```   419 qed
```
```   420
```
```   421 subsection {* Frechet derivatives involving pairs *}
```
```   422
```
```   423 lemma FDERIV_Pair:
```
```   424   assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
```
```   425   shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
```
```   426 apply (rule FDERIV_I)
```
```   427 apply (rule bounded_linear_Pair)
```
```   428 apply (rule FDERIV_bounded_linear [OF f])
```
```   429 apply (rule FDERIV_bounded_linear [OF g])
```
```   430 apply (simp add: norm_Pair)
```
```   431 apply (rule real_LIM_sandwich_zero)
```
```   432 apply (rule LIM_add_zero)
```
```   433 apply (rule FDERIV_D [OF f])
```
```   434 apply (rule FDERIV_D [OF g])
```
```   435 apply (rename_tac h)
```
```   436 apply (simp add: divide_nonneg_pos)
```
```   437 apply (rename_tac h)
```
```   438 apply (subst add_divide_distrib [symmetric])
```
```   439 apply (rule divide_right_mono [OF _ norm_ge_zero])
```
```   440 apply (rule order_trans [OF sqrt_add_le_add_sqrt])
```
```   441 apply simp
```
```   442 apply simp
```
```   443 apply simp
```
```   444 done
```
```   445
```
```   446 end
```