src/HOL/Library/Product_Vector.thy
author huffman
Sun, 07 Jun 2009 17:59:54 -0700
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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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subsection {* Product is a topological space *}
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instantiation
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  "*" :: (topological_space, topological_space) topological_space
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begin
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definition open_prod_def:
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  "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
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    (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
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instance proof
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  show "open (UNIV :: ('a \<times> 'b) set)"
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    unfolding open_prod_def by auto
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next
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  fix S T :: "('a \<times> 'b) set"
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  assume "open S" "open T" thus "open (S \<inter> T)"
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    unfolding open_prod_def
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    apply clarify
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    apply (drule (1) bspec)+
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    apply (clarify, rename_tac Sa Ta Sb Tb)
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    apply (rule_tac x="Sa \<inter> Ta" in exI)
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    apply (rule_tac x="Sb \<inter> Tb" in exI)
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    apply (simp add: open_Int)
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    apply fast
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    done
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next
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  fix K :: "('a \<times> 'b) set set"
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  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
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    unfolding open_prod_def by fast
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qed
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end
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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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next
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  (* FIXME: long proof! *)
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  (* Maybe it would be easier to define topological spaces *)
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  (* in terms of neighborhoods instead of open sets? *)
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  fix S :: "('a \<times> 'b) set"
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  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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    unfolding open_prod_def open_dist
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    apply safe
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    apply (drule (1) bspec)
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    apply clarify
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    apply (drule (1) bspec)+
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    apply (clarify, rename_tac r s)
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    apply (rule_tac x="min r s" in exI, simp)
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    apply (clarify, rename_tac c d)
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    apply (erule subsetD)
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    apply (simp add: dist_Pair_Pair)
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    apply (rule conjI)
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    apply (drule spec, erule mp)
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    apply (erule le_less_trans [OF real_sqrt_sum_squares_ge1])
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    apply (drule spec, erule mp)
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    apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2])
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    apply (drule (1) bspec)
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    apply clarify
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    apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
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    apply clarify
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    apply (rule_tac x="{y. dist y a < r}" in exI)
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    apply (rule_tac x="{y. dist y b < s}" in exI)
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    apply (rule conjI)
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    apply clarify
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    apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
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    apply clarify
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    apply (rule le_less_trans [OF dist_triangle])
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    apply (erule less_le_trans [OF add_strict_right_mono], simp)
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    apply (rule conjI)
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    apply clarify
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    apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
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    apply clarify
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    apply (rule le_less_trans [OF dist_triangle])
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    apply (erule less_le_trans [OF add_strict_right_mono], simp)
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    apply (rule conjI)
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    apply simp
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    apply (clarify, rename_tac c d)
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    apply (drule spec, erule mp)
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   147
    apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
80686a815b59 instance * :: topological_space
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diff changeset
   148
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
80686a815b59 instance * :: topological_space
huffman
parents: 31405
diff changeset
   149
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
80686a815b59 instance * :: topological_space
huffman
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diff changeset
   150
    apply (simp add: power_divide)
80686a815b59 instance * :: topological_space
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diff changeset
   151
    done
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
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   152
qed
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
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   153
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
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   154
end
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
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parents: 31290
diff changeset
   155
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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   156
subsection {* Continuity of operations *}
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   157
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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   158
lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   159
unfolding dist_prod_def by simp
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   160
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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   161
lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   162
unfolding dist_prod_def by simp
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
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diff changeset
   163
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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   164
lemma tendsto_fst:
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   165
  assumes "(f ---> a) net"
f7310185481d generalize tendsto lemmas for products
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   166
  shows "((\<lambda>x. fst (f x)) ---> fst a) net"
f7310185481d generalize tendsto lemmas for products
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   167
proof (rule topological_tendstoI)
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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diff changeset
   168
  fix S assume "open S" "fst a \<in> S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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   169
  then have "open (fst -` S)" "a \<in> fst -` S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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   170
    unfolding open_prod_def
31491
f7310185481d generalize tendsto lemmas for products
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   171
    apply simp_all
f7310185481d generalize tendsto lemmas for products
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   172
    apply clarify
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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diff changeset
   173
    apply (rule exI, erule conjI)
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
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diff changeset
   174
    apply (rule exI, rule conjI [OF open_UNIV])
31491
f7310185481d generalize tendsto lemmas for products
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   175
    apply auto
f7310185481d generalize tendsto lemmas for products
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   176
    done
f7310185481d generalize tendsto lemmas for products
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   177
  with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
f7310185481d generalize tendsto lemmas for products
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   178
    by (rule topological_tendstoD)
f7310185481d generalize tendsto lemmas for products
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   179
  then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
f7310185481d generalize tendsto lemmas for products
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   180
    by simp
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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   181
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   182
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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   183
lemma tendsto_snd:
31491
f7310185481d generalize tendsto lemmas for products
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   184
  assumes "(f ---> a) net"
f7310185481d generalize tendsto lemmas for products
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   185
  shows "((\<lambda>x. snd (f x)) ---> snd a) net"
f7310185481d generalize tendsto lemmas for products
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diff changeset
   186
proof (rule topological_tendstoI)
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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   187
  fix S assume "open S" "snd a \<in> S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
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diff changeset
   188
  then have "open (snd -` S)" "a \<in> snd -` S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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diff changeset
   189
    unfolding open_prod_def
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f7310185481d generalize tendsto lemmas for products
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diff changeset
   190
    apply simp_all
f7310185481d generalize tendsto lemmas for products
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diff changeset
   191
    apply clarify
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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diff changeset
   192
    apply (rule exI, rule conjI [OF open_UNIV])
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
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diff changeset
   193
    apply (rule exI, erule conjI)
31491
f7310185481d generalize tendsto lemmas for products
huffman
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diff changeset
   194
    apply auto
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   195
    done
f7310185481d generalize tendsto lemmas for products
huffman
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diff changeset
   196
  with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   197
    by (rule topological_tendstoD)
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   198
  then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
f7310185481d generalize tendsto lemmas for products
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diff changeset
   199
    by simp
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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parents: 31388
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   200
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
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diff changeset
   201
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   202
lemma tendsto_Pair:
31491
f7310185481d generalize tendsto lemmas for products
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   203
  assumes "(f ---> a) net" and "(g ---> b) net"
f7310185481d generalize tendsto lemmas for products
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   204
  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   205
proof (rule topological_tendstoI)
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
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diff changeset
   206
  fix S assume "open S" "(a, b) \<in> S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   207
  then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
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diff changeset
   208
    unfolding open_prod_def by auto
31491
f7310185481d generalize tendsto lemmas for products
huffman
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diff changeset
   209
  have "eventually (\<lambda>x. f x \<in> A) net"
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   210
    using `(f ---> a) net` `open A` `a \<in> A`
31491
f7310185481d generalize tendsto lemmas for products
huffman
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diff changeset
   211
    by (rule topological_tendstoD)
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   212
  moreover
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   213
  have "eventually (\<lambda>x. g x \<in> B) net"
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   214
    using `(g ---> b) net` `open B` `b \<in> B`
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   215
    by (rule topological_tendstoD)
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
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diff changeset
   216
  ultimately
31491
f7310185481d generalize tendsto lemmas for products
huffman
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diff changeset
   217
  show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   218
    by (rule eventually_elim2)
31491
f7310185481d generalize tendsto lemmas for products
huffman
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diff changeset
   219
       (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   220
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   221
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
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diff changeset
   222
lemma LIMSEQ_fst: "(X ----> a) \<Longrightarrow> (\<lambda>n. fst (X n)) ----> fst a"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   223
unfolding LIMSEQ_conv_tendsto by (rule tendsto_fst)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   224
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   225
lemma LIMSEQ_snd: "(X ----> a) \<Longrightarrow> (\<lambda>n. snd (X n)) ----> snd a"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   226
unfolding LIMSEQ_conv_tendsto by (rule tendsto_snd)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   227
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   228
lemma LIMSEQ_Pair:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
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diff changeset
   229
  assumes "X ----> a" and "Y ----> b"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   230
  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
   231
using assms unfolding LIMSEQ_conv_tendsto
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   232
by (rule tendsto_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   233
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
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diff changeset
   234
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
   235
unfolding LIM_conv_tendsto by (rule tendsto_fst)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   236
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   237
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
   238
unfolding LIM_conv_tendsto by (rule tendsto_snd)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   239
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   240
lemma LIM_Pair:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   241
  assumes "f -- x --> a" and "g -- x --> b"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   242
  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
   243
using assms unfolding LIM_conv_tendsto
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   244
by (rule tendsto_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   245
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   246
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
   247
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
   248
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   249
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
   250
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
   251
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   252
lemma Cauchy_Pair:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   253
  assumes "Cauchy X" and "Cauchy Y"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   254
  shows "Cauchy (\<lambda>n. (X n, Y n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   255
proof (rule metric_CauchyI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   256
  fix r :: real assume "0 < r"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   257
  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
   258
    by (simp add: divide_pos_pos)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   259
  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
   260
    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   261
  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
   262
    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   263
  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
   264
    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
   265
  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
   266
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   267
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   268
lemma isCont_Pair [simp]:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   269
  "\<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
   270
  unfolding isCont_def by (rule LIM_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   271
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   272
subsection {* Product is a complete metric space *}
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   273
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   274
instance "*" :: (complete_space, complete_space) complete_space
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   275
proof
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   276
  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
   277
  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
   278
    using Cauchy_fst [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   279
    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
   280
  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
   281
    using Cauchy_snd [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   282
    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
   283
  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
   284
    using LIMSEQ_Pair [OF 1 2] by simp
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   285
  then show "convergent X"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   286
    by (rule convergentI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   287
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   288
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   289
subsection {* Product is a normed vector space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   290
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   291
instantiation
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   292
  "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   293
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   294
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   295
definition norm_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   296
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   297
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   298
definition sgn_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   299
  "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
   300
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   301
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
   302
  unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   303
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   304
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   305
  fix r :: real and x y :: "'a \<times> 'b"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   306
  show "0 \<le> norm x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   307
    unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   308
  show "norm x = 0 \<longleftrightarrow> x = 0"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   309
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   310
    by (simp add: expand_prod_eq)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   311
  show "norm (x + y) \<le> norm x + norm y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   312
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   313
    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
   314
    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
   315
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   316
  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
   317
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   318
    apply (simp add: norm_scaleR power_mult_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   319
    apply (simp add: right_distrib [symmetric])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   320
    apply (simp add: real_sqrt_mult_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   321
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   322
  show "sgn x = scaleR (inverse (norm x)) x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   323
    by (rule sgn_prod_def)
31290
f41c023d90bc define dist for products
huffman
parents: 30729
diff changeset
   324
  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
   325
    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
   326
    by (simp add: dist_norm)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   327
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   328
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   329
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   330
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   331
instance "*" :: (banach, banach) banach ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   332
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   333
subsection {* Product is an inner product space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   334
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   335
instantiation "*" :: (real_inner, real_inner) real_inner
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   336
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   337
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   338
definition inner_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   339
  "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
   340
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   341
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
   342
  unfolding inner_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   343
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   344
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   345
  fix r :: real
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   346
  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
   347
  show "inner x y = inner y x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   348
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   349
    by (simp add: inner_commute)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   350
  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
   351
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   352
    by (simp add: inner_left_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   353
  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
   354
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   355
    by (simp add: inner_scaleR_left right_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   356
  show "0 \<le> inner x x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   357
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   358
    by (intro add_nonneg_nonneg inner_ge_zero)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   359
  show "inner x x = 0 \<longleftrightarrow> x = 0"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   360
    unfolding inner_prod_def expand_prod_eq
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   361
    by (simp add: add_nonneg_eq_0_iff)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   362
  show "norm x = sqrt (inner x x)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   363
    unfolding norm_prod_def inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   364
    by (simp add: power2_norm_eq_inner)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   365
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   366
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   367
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   368
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   369
subsection {* Pair operations are linear *}
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   370
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30019
diff changeset
   371
interpretation fst: bounded_linear fst
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   372
  apply (unfold_locales)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   373
  apply (rule fst_add)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   374
  apply (rule fst_scaleR)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   375
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   376
  done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   377
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30019
diff changeset
   378
interpretation snd: bounded_linear snd
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   379
  apply (unfold_locales)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   380
  apply (rule snd_add)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   381
  apply (rule snd_scaleR)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   382
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   383
  done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   384
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   385
text {* TODO: move to NthRoot *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   386
lemma sqrt_add_le_add_sqrt:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   387
  assumes x: "0 \<le> x" and y: "0 \<le> y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   388
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   389
apply (rule power2_le_imp_le)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   390
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   391
apply (simp add: mult_nonneg_nonneg x y)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   392
apply (simp add: add_nonneg_nonneg x y)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   393
done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   394
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   395
lemma bounded_linear_Pair:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   396
  assumes f: "bounded_linear f"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   397
  assumes g: "bounded_linear g"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   398
  shows "bounded_linear (\<lambda>x. (f x, g x))"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   399
proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   400
  interpret f: bounded_linear f by fact
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   401
  interpret g: bounded_linear g by fact
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   402
  fix x y and r :: real
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   403
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   404
    by (simp add: f.add g.add)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   405
  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   406
    by (simp add: f.scaleR g.scaleR)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   407
  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   408
    using f.pos_bounded by fast
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   409
  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   410
    using g.pos_bounded by fast
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   411
  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   412
    apply (rule allI)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   413
    apply (simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   414
    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   415
    apply (simp add: right_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   416
    apply (rule add_mono [OF norm_f norm_g])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   417
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   418
  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   419
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   420
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   421
subsection {* Frechet derivatives involving pairs *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   422
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   423
lemma FDERIV_Pair:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   424
  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   425
  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   426
apply (rule FDERIV_I)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   427
apply (rule bounded_linear_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   428
apply (rule FDERIV_bounded_linear [OF f])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   429
apply (rule FDERIV_bounded_linear [OF g])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   430
apply (simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   431
apply (rule real_LIM_sandwich_zero)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   432
apply (rule LIM_add_zero)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   433
apply (rule FDERIV_D [OF f])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   434
apply (rule FDERIV_D [OF g])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   435
apply (rename_tac h)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   436
apply (simp add: divide_nonneg_pos)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   437
apply (rename_tac h)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   438
apply (subst add_divide_distrib [symmetric])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   439
apply (rule divide_right_mono [OF _ norm_ge_zero])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   440
apply (rule order_trans [OF sqrt_add_le_add_sqrt])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   441
apply simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   442
apply simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   443
apply simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   444
done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   445
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   446
end