author immler Mon Jan 11 15:20:17 2016 +0100 (2016-01-11) changeset 62127 d8e7738bd2e9 parent 62126 2d187ace2827 child 62128 3201ddb00097
generalized proofs
```     1.1 --- a/src/HOL/Multivariate_Analysis/Bounded_Linear_Function.thy	Mon Jan 11 13:15:15 2016 +0100
1.2 +++ b/src/HOL/Multivariate_Analysis/Bounded_Linear_Function.thy	Mon Jan 11 15:20:17 2016 +0100
1.3 @@ -467,74 +467,15 @@
1.4    "(if P then (1::'a::comm_semiring_1) else 0) * q = (if P then q else 0)"
1.5    by auto
1.6
1.7 -text \<open>TODO: generalize this and @{thm compact_lemma}?!\<close>
1.8  lemma compact_blinfun_lemma:
1.9    fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
1.10    assumes "bounded (range f)"
1.11    shows "\<forall>d\<subseteq>Basis. \<exists>l::'a \<Rightarrow>\<^sub>L 'b. \<exists> r.
1.12      subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) i) (l i) < e) sequentially)"
1.13 -proof safe
1.14 -  fix d :: "'a set"
1.15 -  assume d: "d \<subseteq> Basis"
1.16 -  with finite_Basis have "finite d"
1.17 -    by (blast intro: finite_subset)
1.18 -  from this d show "\<exists>l::'a \<Rightarrow>\<^sub>L 'b. \<exists>r. subseq r \<and>
1.19 -    (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) i) (l i) < e) sequentially)"
1.20 -  proof (induct d)
1.21 -    case empty
1.22 -    then show ?case
1.23 -      unfolding subseq_def by auto
1.24 -  next
1.25 -    case (insert k d)
1.26 -    have k[intro]: "k \<in> Basis"
1.27 -      using insert by auto
1.28 -    have s': "bounded ((\<lambda>x. blinfun_apply x k) ` range f)"
1.29 -      using \<open>bounded (range f)\<close>
1.30 -      by (auto intro!: bounded_linear_image bounded_linear_intros)
1.31 -    obtain l1::"'a \<Rightarrow>\<^sub>L 'b" and r1 where r1: "subseq r1"
1.32 -      and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) i) (l1 i) < e) sequentially"
1.33 -      using insert(3) using insert(4) by auto
1.34 -    have f': "\<forall>n. f (r1 n) k \<in> (\<lambda>x. blinfun_apply x k) ` range f"
1.35 -      by simp
1.36 -    have "bounded (range (\<lambda>i. f (r1 i) k))"
1.37 -      by (metis (lifting) bounded_subset f' image_subsetI s')
1.38 -    then obtain l2 r2
1.39 -      where r2: "subseq r2"
1.40 -      and lr2: "((\<lambda>i. f (r1 (r2 i)) k) \<longlongrightarrow> l2) sequentially"
1.41 -      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) k"]
1.42 -      by (auto simp: o_def)
1.43 -    def r \<equiv> "r1 \<circ> r2"
1.44 -    have r:"subseq r"
1.45 -      using r1 and r2 unfolding r_def o_def subseq_def by auto
1.46 -    moreover
1.47 -    def l \<equiv> "blinfun_of_matrix (\<lambda>i j. if j = k then l2 \<bullet> i else l1 j \<bullet> i)::'a \<Rightarrow>\<^sub>L 'b"
1.48 -    {
1.49 -      fix e::real
1.50 -      assume "e > 0"
1.51 -      from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n)  i) (l1  i) < e) sequentially"
1.52 -        by blast
1.53 -      from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n))  k) l2 < e) sequentially"
1.54 -        by (rule tendstoD)
1.55 -      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n))  i) (l1  i) < e) sequentially"
1.56 -        by (rule eventually_subseq)
1.57 -      have l2: "l k = l2"
1.58 -        using insert.prems
1.59 -        by (auto simp: blinfun_of_matrix.rep_eq inner_Basis l_def mult_if_delta
1.60 -          scaleR_setsum_left[symmetric] setsum.delta' intro!: euclidean_eqI[where 'a='b])
1.61 -      {
1.62 -        fix i assume "i \<in> d"
1.63 -        with insert have "i \<in> Basis" "i \<noteq> k" by auto
1.64 -        hence l1: "l i = (l1 i)"
1.65 -          by (auto simp: blinfun_of_matrix.rep_eq inner_Basis l_def mult_if_delta
1.66 -            scaleR_setsum_left[symmetric] setsum.delta' intro!: euclidean_eqI[where 'a='b])
1.67 -      } note l1 = this
1.68 -      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n)  i) (l  i) < e) sequentially"
1.69 -        using N1' N2
1.70 -        by eventually_elim (insert insert.prems, auto simp: r_def o_def l1 l2)
1.71 -    }
1.72 -    ultimately show ?case by auto
1.73 -  qed
1.74 -qed
1.75 +  by (rule compact_lemma_general[where unproj = "\<lambda>e. blinfun_of_matrix (\<lambda>i j. e j \<bullet> i)"])
1.76 +   (auto intro!: euclidean_eqI[where 'a='b] bounded_linear_image assms
1.77 +    simp: blinfun_of_matrix_works blinfun_of_matrix_apply inner_Basis mult_if_delta setsum.delta'
1.78 +      scaleR_setsum_left[symmetric])
1.79
1.80  lemma blinfun_euclidean_eqI: "(\<And>i. i \<in> Basis \<Longrightarrow> blinfun_apply x i = blinfun_apply y i) \<Longrightarrow> x = y"
1.81    apply (auto intro!: blinfun_eqI)
```
```     2.1 --- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Mon Jan 11 13:15:15 2016 +0100
2.2 +++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Mon Jan 11 15:20:17 2016 +0100
2.3 @@ -820,47 +820,14 @@
2.4  lemma compact_lemma_cart:
2.5    fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
2.6    assumes f: "bounded (range f)"
2.7 -  shows "\<forall>d.
2.8 -        \<exists>l r. subseq r \<and>
2.9 +  shows "\<exists>l r. subseq r \<and>
2.10          (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
2.11 -proof
2.12 -  fix d :: "'n set"
2.13 -  have "finite d" by simp
2.14 -  thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
2.15 -      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
2.16 -  proof (induct d)
2.17 -    case empty
2.18 -    thus ?case unfolding subseq_def by auto
2.19 -  next
2.20 -    case (insert k d)
2.21 -    obtain l1::"'a^'n" and r1 where r1:"subseq r1"
2.22 -      and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially"
2.23 -      using insert(3) by auto
2.24 -    have s': "bounded ((\<lambda>x. x \$ k) ` range f)" using \<open>bounded (range f)\<close>
2.25 -      by (auto intro!: bounded_component_cart)
2.26 -    have f': "\<forall>n. f (r1 n) \$ k \<in> (\<lambda>x. x \$ k) ` range f" by simp
2.27 -    have "bounded (range (\<lambda>i. f (r1 i) \$ k))"
2.28 -      by (metis (lifting) bounded_subset image_subsetI f' s')
2.29 -    then obtain l2 r2 where r2: "subseq r2"
2.30 -      and lr2: "((\<lambda>i. f (r1 (r2 i)) \$ k) \<longlongrightarrow> l2) sequentially"
2.31 -      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \$ k"] by (auto simp: o_def)
2.32 -    def r \<equiv> "r1 \<circ> r2"
2.33 -    have r: "subseq r"
2.34 -      using r1 and r2 unfolding r_def o_def subseq_def by auto
2.35 -    moreover
2.36 -    def l \<equiv> "(\<chi> i. if i = k then l2 else l1\$i)::'a^'n"
2.37 -    { fix e :: real assume "e > 0"
2.38 -      from lr1 \<open>e>0\<close> have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially"
2.39 -        by blast
2.40 -      from lr2 \<open>e>0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \$ k) l2 < e) sequentially"
2.41 -        by (rule tendstoD)
2.42 -      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \$ i) (l1 \$ i) < e) sequentially"
2.43 -        by (rule eventually_subseq)
2.44 -      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \$ i) (l \$ i) < e) sequentially"
2.45 -        using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
2.46 -    }
2.47 -    ultimately show ?case by auto
2.48 -  qed
2.49 +    (is "?th d")
2.50 +proof -
2.51 +  have "\<forall>d' \<subseteq> d. ?th d'"
2.52 +    by (rule compact_lemma_general[where unproj=vec_lambda])
2.53 +      (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
2.54 +  then show "?th d" by simp
2.55  qed
2.56
2.57  instance vec :: (heine_borel, finite) heine_borel
```
```     3.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Jan 11 13:15:15 2016 +0100
3.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Jan 11 15:20:17 2016 +0100
3.3 @@ -4443,61 +4443,75 @@
3.4      using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
3.5  qed
3.6
3.7 -lemma compact_lemma:
3.8 -  fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
3.9 -  assumes "bounded (range f)"
3.10 -  shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.
3.11 -    subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
3.12 +lemma compact_lemma_general:
3.13 +  fixes f :: "nat \<Rightarrow> 'a"
3.14 +  fixes proj::"'a \<Rightarrow> 'b \<Rightarrow> 'c::heine_borel" (infixl "proj" 60)
3.15 +  fixes unproj:: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"
3.16 +  assumes finite_basis: "finite basis"
3.17 +  assumes bounded_proj: "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. x proj k) ` range f)"
3.18 +  assumes proj_unproj: "\<And>e k. k \<in> basis \<Longrightarrow> (unproj e) proj k = e k"
3.19 +  assumes unproj_proj: "\<And>x. unproj (\<lambda>k. x proj k) = x"
3.20 +  shows "\<forall>d\<subseteq>basis. \<exists>l::'a. \<exists> r.
3.21 +    subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
3.22  proof safe
3.23 -  fix d :: "'a set"
3.24 -  assume d: "d \<subseteq> Basis"
3.25 -  with finite_Basis have "finite d"
3.26 +  fix d :: "'b set"
3.27 +  assume d: "d \<subseteq> basis"
3.28 +  with finite_basis have "finite d"
3.29      by (blast intro: finite_subset)
3.30    from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>
3.31 -    (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
3.32 +    (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
3.33    proof (induct d)
3.34      case empty
3.35      then show ?case
3.36        unfolding subseq_def by auto
3.37    next
3.38      case (insert k d)
3.39 -    have k[intro]: "k \<in> Basis"
3.40 +    have k[intro]: "k \<in> basis"
3.41        using insert by auto
3.42 -    have s': "bounded ((\<lambda>x. x \<bullet> k) ` range f)"
3.43 -      using \<open>bounded (range f)\<close>
3.44 -      by (auto intro!: bounded_linear_image bounded_linear_inner_left)
3.45 +    have s': "bounded ((\<lambda>x. x proj k) ` range f)"
3.46 +      using k
3.47 +      by (rule bounded_proj)
3.48      obtain l1::"'a" and r1 where r1: "subseq r1"
3.49 -      and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
3.50 +      and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
3.51        using insert(3) using insert(4) by auto
3.52 -    have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k) ` range f"
3.53 +    have f': "\<forall>n. f (r1 n) proj k \<in> (\<lambda>x. x proj k) ` range f"
3.54        by simp
3.55 -    have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"
3.56 +    have "bounded (range (\<lambda>i. f (r1 i) proj k))"
3.57        by (metis (lifting) bounded_subset f' image_subsetI s')
3.58 -    then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) \<longlongrightarrow> l2) sequentially"
3.59 -      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]
3.60 +    then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) proj k) \<longlongrightarrow> l2) sequentially"
3.61 +      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) proj k"]
3.62        by (auto simp: o_def)
3.63      def r \<equiv> "r1 \<circ> r2"
3.64      have r:"subseq r"
3.65        using r1 and r2 unfolding r_def o_def subseq_def by auto
3.66      moreover
3.67 -    def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"
3.68 +    def l \<equiv> "unproj (\<lambda>i. if i = k then l2 else l1 proj i)::'a"
3.69      {
3.70        fix e::real
3.71        assume "e > 0"
3.72 -      from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
3.73 +      from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
3.74          by blast
3.75 -      from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"
3.76 +      from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) proj k) l2 < e) sequentially"
3.77          by (rule tendstoD)
3.78 -      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
3.79 +      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e) sequentially"
3.80          by (rule eventually_subseq)
3.81 -      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
3.82 +      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) proj i) (l proj i) < e) sequentially"
3.83          using N1' N2
3.84 -        by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)
3.85 +        by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def proj_unproj)
3.86      }
3.87      ultimately show ?case by auto
3.88    qed
3.89  qed
3.90
3.91 +lemma compact_lemma:
3.92 +  fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
3.93 +  assumes "bounded (range f)"
3.94 +  shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.
3.95 +    subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
3.96 +  by (rule compact_lemma_general[where unproj="\<lambda>e. \<Sum>i\<in>Basis. e i *\<^sub>R i"])
3.97 +     (auto intro!: assms bounded_linear_inner_left bounded_linear_image
3.98 +       simp: euclidean_representation)
3.99 +
3.100  instance euclidean_space \<subseteq> heine_borel
3.101  proof
3.102    fix f :: "nat \<Rightarrow> 'a"
```