src/HOL/Library/Product_Vector.thy
author huffman
Tue, 02 Jun 2009 23:35:52 -0700
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child 31415 80686a815b59
permissions -rw-r--r--
instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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(*  Title:      HOL/Library/Product_Vector.thy
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    Author:     Brian Huffman
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*)
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header {* Cartesian Products as Vector Spaces *}
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theory Product_Vector
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imports Inner_Product Product_plus
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begin
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subsection {* Product is a real vector space *}
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instantiation "*" :: (real_vector, real_vector) real_vector
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begin
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definition scaleR_prod_def:
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  "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
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lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
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  unfolding scaleR_prod_def by simp
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lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
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  unfolding scaleR_prod_def by simp
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lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
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  unfolding scaleR_prod_def by simp
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instance proof
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  fix a b :: real and x y :: "'a \<times> 'b"
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  show "scaleR a (x + y) = scaleR a x + scaleR a y"
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    by (simp add: expand_prod_eq scaleR_right_distrib)
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  show "scaleR (a + b) x = scaleR a x + scaleR b x"
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    by (simp add: expand_prod_eq scaleR_left_distrib)
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  show "scaleR a (scaleR b x) = scaleR (a * b) x"
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    by (simp add: expand_prod_eq)
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  show "scaleR 1 x = x"
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    by (simp add: expand_prod_eq)
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qed
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end
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31339
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subsection {* Product is a metric space *}
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instantiation
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  "*" :: (metric_space, metric_space) metric_space
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begin
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definition dist_prod_def:
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  "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
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lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
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  unfolding dist_prod_def by simp
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instance proof
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  fix x y :: "'a \<times> 'b"
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  show "dist x y = 0 \<longleftrightarrow> x = y"
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    unfolding dist_prod_def
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    by (simp add: expand_prod_eq)
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next
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  fix x y z :: "'a \<times> 'b"
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  show "dist x y \<le> dist x z + dist y z"
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    unfolding dist_prod_def
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    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
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    apply (rule real_sqrt_le_mono)
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    apply (rule order_trans [OF add_mono])
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    apply (rule power_mono [OF dist_triangle2 [of _ _ "fst z"] zero_le_dist])
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    apply (rule power_mono [OF dist_triangle2 [of _ _ "snd z"] zero_le_dist])
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    apply (simp only: real_sum_squared_expand)
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    done
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qed
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end
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subsection {* Continuity of operations *}
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lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
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unfolding dist_prod_def by simp
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lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
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unfolding dist_prod_def by simp
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lemma tendsto_fst:
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  assumes "tendsto f a net"
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  shows "tendsto (\<lambda>x. fst (f x)) (fst a) net"
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proof (rule tendstoI)
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  fix r :: real assume "0 < r"
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  have "eventually (\<lambda>x. dist (f x) a < r) net"
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    using `tendsto f a net` `0 < r` by (rule tendstoD)
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  thus "eventually (\<lambda>x. dist (fst (f x)) (fst a) < r) net"
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    by (rule eventually_elim1)
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       (rule le_less_trans [OF dist_fst_le])
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qed
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lemma tendsto_snd:
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  assumes "tendsto f a net"
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  shows "tendsto (\<lambda>x. snd (f x)) (snd a) net"
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proof (rule tendstoI)
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  fix r :: real assume "0 < r"
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  have "eventually (\<lambda>x. dist (f x) a < r) net"
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    using `tendsto f a net` `0 < r` by (rule tendstoD)
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  thus "eventually (\<lambda>x. dist (snd (f x)) (snd a) < r) net"
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    by (rule eventually_elim1)
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       (rule le_less_trans [OF dist_snd_le])
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qed
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lemma tendsto_Pair:
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  assumes "tendsto X a net" and "tendsto Y b net"
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  shows "tendsto (\<lambda>i. (X i, Y i)) (a, b) net"
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proof (rule tendstoI)
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  fix r :: real assume "0 < r"
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  then have "0 < r / sqrt 2" (is "0 < ?s")
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    by (simp add: divide_pos_pos)
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  have "eventually (\<lambda>i. dist (X i) a < ?s) net"
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    using `tendsto X a net` `0 < ?s` by (rule tendstoD)
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  moreover
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  have "eventually (\<lambda>i. dist (Y i) b < ?s) net"
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    using `tendsto Y b net` `0 < ?s` by (rule tendstoD)
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  ultimately
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  show "eventually (\<lambda>i. dist (X i, Y i) (a, b) < r) net"
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    by (rule eventually_elim2)
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       (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
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qed
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lemma LIMSEQ_fst: "(X ----> a) \<Longrightarrow> (\<lambda>n. fst (X n)) ----> fst a"
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unfolding LIMSEQ_conv_tendsto by (rule tendsto_fst)
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lemma LIMSEQ_snd: "(X ----> a) \<Longrightarrow> (\<lambda>n. snd (X n)) ----> snd a"
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unfolding LIMSEQ_conv_tendsto by (rule tendsto_snd)
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lemma LIMSEQ_Pair:
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  assumes "X ----> a" and "Y ----> b"
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   132
  shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   133
using assms unfolding LIMSEQ_conv_tendsto
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   134
by (rule tendsto_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   135
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   136
lemma LIM_fst: "f -- x --> a \<Longrightarrow> (\<lambda>x. fst (f x)) -- x --> fst a"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   137
unfolding LIM_conv_tendsto by (rule tendsto_fst)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   138
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   139
lemma LIM_snd: "f -- x --> a \<Longrightarrow> (\<lambda>x. snd (f x)) -- x --> snd a"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   140
unfolding LIM_conv_tendsto by (rule tendsto_snd)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   141
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   142
lemma LIM_Pair:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
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   143
  assumes "f -- x --> a" and "g -- x --> b"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   144
  shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   145
using assms unfolding LIM_conv_tendsto
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   146
by (rule tendsto_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   147
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   148
lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   149
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   150
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   151
lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   152
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   153
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   154
lemma Cauchy_Pair:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   155
  assumes "Cauchy X" and "Cauchy Y"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   156
  shows "Cauchy (\<lambda>n. (X n, Y n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   157
proof (rule metric_CauchyI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   158
  fix r :: real assume "0 < r"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   159
  then have "0 < r / sqrt 2" (is "0 < ?s")
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   160
    by (simp add: divide_pos_pos)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   161
  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   162
    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   163
  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   164
    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   165
  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   166
    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   167
  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   168
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   169
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   170
lemma isCont_Pair [simp]:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   171
  "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   172
  unfolding isCont_def by (rule LIM_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   173
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   174
subsection {* Product is a complete metric space *}
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   175
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   176
instance "*" :: (complete_space, complete_space) complete_space
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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parents: 31388
diff changeset
   177
proof
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   178
  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   179
  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   180
    using Cauchy_fst [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   181
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   182
  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   183
    using Cauchy_snd [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   184
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   185
  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   186
    using LIMSEQ_Pair [OF 1 2] by simp
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   187
  then show "convergent X"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   188
    by (rule convergentI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   189
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   190
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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diff changeset
   191
subsection {* Product is a normed vector space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
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diff changeset
   192
a2f19e0a28b2 add theory of products as real vector spaces to Library
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   193
instantiation
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
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   194
  "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
diff changeset
   195
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
diff changeset
   196
a2f19e0a28b2 add theory of products as real vector spaces to Library
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diff changeset
   197
definition norm_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
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   198
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
diff changeset
   199
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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diff changeset
   200
definition sgn_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   201
  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   202
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   203
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   204
  unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   205
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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   206
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   207
  fix r :: real and x y :: "'a \<times> 'b"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   208
  show "0 \<le> norm x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   209
    unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   210
  show "norm x = 0 \<longleftrightarrow> x = 0"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   211
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   212
    by (simp add: expand_prod_eq)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   213
  show "norm (x + y) \<le> norm x + norm y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   214
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   215
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   216
    apply (simp add: add_mono power_mono norm_triangle_ineq)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   217
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   218
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   219
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   220
    apply (simp add: norm_scaleR power_mult_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   221
    apply (simp add: right_distrib [symmetric])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   222
    apply (simp add: real_sqrt_mult_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   223
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   224
  show "sgn x = scaleR (inverse (norm x)) x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   225
    by (rule sgn_prod_def)
31290
f41c023d90bc define dist for products
huffman
parents: 30729
diff changeset
   226
  show "dist x y = norm (x - y)"
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   227
    unfolding dist_prod_def norm_prod_def
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   228
    by (simp add: dist_norm)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
diff changeset
   229
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
diff changeset
   230
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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diff changeset
   231
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
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diff changeset
   232
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
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   233
instance "*" :: (banach, banach) banach ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   234
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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diff changeset
   235
subsection {* Product is an inner product space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
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diff changeset
   236
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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diff changeset
   237
instantiation "*" :: (real_inner, real_inner) real_inner
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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diff changeset
   238
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   239
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   240
definition inner_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
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   241
  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   242
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   243
lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   244
  unfolding inner_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   245
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
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diff changeset
   246
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   247
  fix r :: real
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   248
  fix x y z :: "'a::real_inner * 'b::real_inner"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   249
  show "inner x y = inner y x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   250
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   251
    by (simp add: inner_commute)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   252
  show "inner (x + y) z = inner x z + inner y z"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   253
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   254
    by (simp add: inner_left_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   255
  show "inner (scaleR r x) y = r * inner x y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   256
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   257
    by (simp add: inner_scaleR_left right_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   258
  show "0 \<le> inner x x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   259
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   260
    by (intro add_nonneg_nonneg inner_ge_zero)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   261
  show "inner x x = 0 \<longleftrightarrow> x = 0"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   262
    unfolding inner_prod_def expand_prod_eq
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   263
    by (simp add: add_nonneg_eq_0_iff)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   264
  show "norm x = sqrt (inner x x)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   265
    unfolding norm_prod_def inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   266
    by (simp add: power2_norm_eq_inner)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   267
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
diff changeset
   268
a2f19e0a28b2 add theory of products as real vector spaces to Library
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diff changeset
   269
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
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parents:
diff changeset
   270
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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subsection {* Pair operations are linear *}
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interpretation fst: bounded_linear fst
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  apply (unfold_locales)
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  apply (rule fst_add)
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  apply (rule fst_scaleR)
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  apply (rule_tac x="1" in exI, simp add: norm_Pair)
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  done
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interpretation snd: bounded_linear snd
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  apply (unfold_locales)
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  apply (rule snd_add)
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  apply (rule snd_scaleR)
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  apply (rule_tac x="1" in exI, simp add: norm_Pair)
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  done
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text {* TODO: move to NthRoot *}
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lemma sqrt_add_le_add_sqrt:
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  assumes x: "0 \<le> x" and y: "0 \<le> y"
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  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
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apply (rule power2_le_imp_le)
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apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
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apply (simp add: mult_nonneg_nonneg x y)
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apply (simp add: add_nonneg_nonneg x y)
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done
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lemma bounded_linear_Pair:
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  assumes f: "bounded_linear f"
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  assumes g: "bounded_linear g"
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  shows "bounded_linear (\<lambda>x. (f x, g x))"
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proof
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  interpret f: bounded_linear f by fact
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  interpret g: bounded_linear g by fact
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  fix x y and r :: real
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  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
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    by (simp add: f.add g.add)
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  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
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    by (simp add: f.scaleR g.scaleR)
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  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
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    using f.pos_bounded by fast
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  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
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    using g.pos_bounded by fast
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  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
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    apply (rule allI)
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    apply (simp add: norm_Pair)
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    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
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    apply (simp add: right_distrib)
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    apply (rule add_mono [OF norm_f norm_g])
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    done
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  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
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qed
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subsection {* Frechet derivatives involving pairs *}
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lemma FDERIV_Pair:
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  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
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  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
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apply (rule FDERIV_I)
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apply (rule bounded_linear_Pair)
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apply (rule FDERIV_bounded_linear [OF f])
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apply (rule FDERIV_bounded_linear [OF g])
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apply (simp add: norm_Pair)
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apply (rule real_LIM_sandwich_zero)
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apply (rule LIM_add_zero)
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apply (rule FDERIV_D [OF f])
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apply (rule FDERIV_D [OF g])
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apply (rename_tac h)
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apply (simp add: divide_nonneg_pos)
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apply (rename_tac h)
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apply (subst add_divide_distrib [symmetric])
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apply (rule divide_right_mono [OF _ norm_ge_zero])
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apply (rule order_trans [OF sqrt_add_le_add_sqrt])
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apply simp
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apply simp
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apply simp
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done
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end