src/HOL/Library/Topology_Euclidean_Space.thy
author huffman
Thu, 12 Mar 2009 08:57:03 -0700
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(* Title:      Topology
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   Author:     Amine Chaieb, University of Cambridge
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   Author:     Robert Himmelmann, TU Muenchen
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*)
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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header {* Elementary topology in Euclidean space. *}
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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theory Topology_Euclidean_Space
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  imports SEQ Euclidean_Space
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begin
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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declare fstcart_pastecart[simp] sndcart_pastecart[simp]
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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subsection{* General notion of a topology *}
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definition "istopology L \<longleftrightarrow> {} \<in> L \<and> (\<forall>S \<in>L. \<forall>T \<in>L. S \<inter> T \<in> L) \<and> (\<forall>K. K \<subseteq>L \<longrightarrow> \<Union> K \<in> L)"
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typedef (open) 'a topology = "{L::('a set) set. istopology L}"
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  morphisms "openin" "topology"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  unfolding istopology_def by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma istopology_open_in[intro]: "istopology(openin U)"
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  using openin[of U] by blast
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  using topology_inverse[unfolded mem_def Collect_def] .
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
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  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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proof-
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  {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
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  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    hence "openin T1 = openin T2" by (metis mem_def set_ext)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    hence "topology (openin T1) = topology (openin T2)" by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    hence "T1 = T2" unfolding openin_inverse .}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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text{* Infer the "universe" from union of all sets in the topology. *}
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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definition "topspace T =  \<Union>{S. openin T S}"
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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subsection{* Main properties of open sets *}
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_clauses:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  fixes U :: "'a topology"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  shows "openin U {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  using openin[of U] unfolding istopology_def Collect_def mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  by (metis mem_def subset_eq)+
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  unfolding topspace_def by blast
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lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  by (simp add: openin_clauses)
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  using openin_Union[of "{S,T}" U] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  {assume ?lhs then have ?rhs by auto }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    73
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  {assume H: ?rhs
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    then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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      unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
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    78
    have "\<Union> t`S = S" using t by auto
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    with openin_Union[OF th0] have "openin U S" by simp }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    80
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    82
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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subsection{* Closed sets *}
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    86
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    87
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
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    89
lemma closedin_topspace[intro,simp]:
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  "closedin U (topspace U)" by (simp add: closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    91
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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parents:
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    92
  by (auto simp add: Diff_Un closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    93
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    94
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    95
lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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    96
  shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    97
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    98
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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    99
  using closedin_Inter[of "{S,T}" U] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   100
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   101
lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   102
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   103
  apply (auto simp add: closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   104
  apply (metis openin_subset subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   105
  apply (auto simp add: Diff_Diff_Int)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   106
  apply (subgoal_tac "topspace U \<inter> S = S")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   107
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   108
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   109
lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   110
  by (simp add: openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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parents:
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   111
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   112
lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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parents:
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   113
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   114
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   115
    by (auto simp add: topspace_def openin_subset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   116
  then show ?thesis using oS cT by (auto simp add: closedin_def)
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huffman
parents: 30268
diff changeset
   117
qed
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   118
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   119
lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   120
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   121
  have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   122
    by (auto simp add: topspace_def )
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   123
  then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   124
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   125
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   126
subsection{* Subspace topology. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   127
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   128
definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   129
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   130
lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   131
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   132
  have "{} \<in> ?L" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   133
  {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   134
    from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   135
    have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   136
    then have "A \<inter> B \<in> ?L" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   137
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   138
  {fix K assume K: "K \<subseteq> ?L"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   139
    have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   140
      apply (rule set_ext)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   141
      apply (simp add: Ball_def image_iff)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   142
      by (metis mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   143
    from K[unfolded th0 subset_image_iff]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   144
    obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   145
    have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   146
    moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   147
    ultimately have "\<Union>K \<in> ?L" by blast}
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   148
  ultimately show ?thesis unfolding istopology_def by blast
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   149
qed
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   150
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   151
lemma openin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   152
  "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   153
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   154
  by (auto simp add: Collect_def)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   155
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   156
lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   157
  by (auto simp add: topspace_def openin_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   158
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   159
lemma closedin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   160
  "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   161
  unfolding closedin_def topspace_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   162
  apply (simp add: openin_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   163
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   164
  apply clarify
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   165
  apply (rule_tac x="topspace U - T" in exI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   166
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   167
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   168
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   169
  unfolding openin_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   170
  apply (rule iffI, clarify)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   171
  apply (frule openin_subset[of U])  apply blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   172
  apply (rule exI[where x="topspace U"])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   173
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   174
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   175
lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   176
  shows "subtopology U V = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   177
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   178
  {fix S
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   179
    {fix T assume T: "openin U T" "S = T \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   180
      from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   181
      have "openin U S" unfolding eq using T by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   182
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   183
    {assume S: "openin U S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   184
      hence "\<exists>T. openin U T \<and> S = T \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   185
	using openin_subset[OF S] UV by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   186
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   187
  then show ?thesis unfolding topology_eq openin_subtopology by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   188
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   189
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   190
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   191
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   192
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   193
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   194
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   195
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   196
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   197
subsection{* The universal Euclidean versions are what we use most of the time *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   198
definition "open S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>e >0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   199
definition "closed S \<longleftrightarrow> open(UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   200
definition "euclidean = topology open"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   201
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   202
lemma open_empty[intro,simp]: "open {}" by (simp add: open_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   203
lemma open_UNIV[intro,simp]:  "open UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   204
  by (simp add: open_def, rule exI[where x="1"], auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   205
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   206
lemma open_inter[intro]: assumes S: "open S" and T: "open T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   207
  shows "open (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   208
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   209
  note thS = S[unfolded open_def, rule_format]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   210
  note thT = T[unfolded open_def, rule_format]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   211
  {fix x assume x: "x \<in> S\<inter>T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   212
    hence xS: "x \<in> S" and xT: "x \<in> T" by simp_all
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   213
    from thS[OF xS] obtain eS where eS: "eS > 0" "\<forall>x'. dist x' x < eS \<longrightarrow> x' \<in> S" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   214
    from thT[OF xT] obtain eT where eT: "eT > 0" "\<forall>x'. dist x' x < eT \<longrightarrow> x' \<in> T" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   215
    from real_lbound_gt_zero[OF eS(1) eT(1)] obtain e where e: "e > 0" "e < eS" "e < eT" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   216
    { fix x' assume d: "dist x' x < e"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   217
      hence dS: "dist x' x < eS" and dT: "dist x' x < eT" using e by arith+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   218
      from eS(2)[rule_format, OF dS] eT(2)[rule_format, OF dT] have "x' \<in> S\<inter>T" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   219
    hence "\<exists>e >0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> (S\<inter>T)" using e by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   220
  then show ?thesis unfolding open_def by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   221
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   222
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   223
lemma open_Union[intro]: "(\<forall>S\<in>K. open S) \<Longrightarrow> open (\<Union> K)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   224
  by (simp add: open_def) metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   225
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   226
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   227
  unfolding euclidean_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   228
  apply (rule cong[where x=S and y=S])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   229
  apply (rule topology_inverse[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   230
  apply (auto simp add: istopology_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   231
  by (auto simp add: mem_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   232
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   233
lemma topspace_euclidean: "topspace euclidean = UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   234
  apply (simp add: topspace_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   235
  apply (rule set_ext)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   236
  by (auto simp add: open_openin[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   237
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   238
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   239
  by (simp add: topspace_euclidean topspace_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   240
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   241
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   242
  by (simp add: closed_def closedin_def topspace_euclidean open_openin)
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lemma open_Un[intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S\<union>T)"
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  by (auto simp add: open_openin)
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lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
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  by (simp add: open_openin openin_subopen[symmetric])
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lemma closed_empty[intro, simp]: "closed {}" by (simp add: closed_closedin)
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lemma closed_UNIV[simp,intro]: "closed UNIV"
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  by (simp add: closed_closedin topspace_euclidean[symmetric])
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lemma closed_Un[intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S\<union>T)"
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  by (auto simp add: closed_closedin)
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lemma closed_Int[intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S\<inter>T)"
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  by (auto simp add: closed_closedin)
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lemma closed_Inter[intro]: assumes H: "\<forall>S \<in>K. closed S" shows "closed (\<Inter>K)"
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  using H
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  unfolding closed_closedin
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  apply (cases "K = {}")
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  apply (simp add: closed_closedin[symmetric])
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  apply (rule closedin_Inter, auto)
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  done
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lemma open_closed: "open S \<longleftrightarrow> closed (UNIV - S)"
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  by (simp add: open_openin closed_closedin topspace_euclidean openin_closedin_eq)
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lemma closed_open: "closed S \<longleftrightarrow> open(UNIV - S)"
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  by (simp add: open_openin closed_closedin topspace_euclidean closedin_def)
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lemma open_diff[intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
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  by (auto simp add: open_openin closed_closedin)
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lemma closed_diff[intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed(S-T)"
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  by (auto simp add: open_openin closed_closedin)
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lemma open_Inter[intro]: assumes fS: "finite S" and h: "\<forall>T\<in>S. open T" shows "open (\<Inter>S)"
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  using h by (induct rule: finite_induct[OF fS], auto)
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lemma closed_Union[intro]: assumes fS: "finite S" and h: "\<forall>T\<in>S. closed T" shows "closed (\<Union>S)"
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  using h by (induct rule: finite_induct[OF fS], auto)
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subsection{* Open and closed balls. *}
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definition "ball x e = {y. dist x y < e}"
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definition "cball x e = {y. dist x y \<le> e}"
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lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
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lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
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lemma mem_ball_0[simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e" by (simp add: dist_def)
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lemma mem_cball_0[simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e" by (simp add: dist_def)
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lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
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lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
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lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
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lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
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lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
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  by (simp add: expand_set_eq) arith
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lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
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  by (simp add: expand_set_eq)
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subsection{* Topological properties of open balls *}
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lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
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  "(a::real) - b < 0 \<longleftrightarrow> a < b"
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  "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
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lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
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  "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
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lemma open_ball[intro, simp]: "open (ball x e)"
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  unfolding open_def ball_def Collect_def Ball_def mem_def
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  unfolding dist_sym
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  apply clarify
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  apply (rule_tac x="e - dist xa x" in exI)
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  using dist_triangle_alt[where z=x]
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  apply (clarsimp simp add: diff_less_iff)
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  apply atomize
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  apply (erule_tac x="x'" in allE)
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  apply (erule_tac x="xa" in allE)
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  by arith
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lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_refl)
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lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  unfolding open_def subset_eq mem_ball Ball_def dist_sym ..
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lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  by (metis open_contains_ball subset_eq centre_in_ball)
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lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  unfolding mem_ball expand_set_eq
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  apply (simp add: not_less)
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   336
  by (metis dist_pos_le order_trans dist_refl)
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
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subsection{* Basic "localization" results are handy for connectedness. *}
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   343
  by (auto simp add: openin_subtopology open_openin[symmetric])
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lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
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  by (auto simp add: openin_open)
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lemma open_openin_trans[trans]:
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 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  by (metis Int_absorb1  openin_open_Int)
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   351
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   353
  by (auto simp add: openin_open)
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   354
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lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   356
  by (simp add: closedin_subtopology closed_closedin Int_ac)
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   357
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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   359
  by (metis closedin_closed)
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   360
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   361
lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   362
  apply (subgoal_tac "S \<inter> T = T" )
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   363
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   364
  apply (frule closedin_closed_Int[of T S])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   365
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   366
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   367
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   368
  by (auto simp add: closedin_closed)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   369
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   370
lemma openin_euclidean_subtopology_iff: "openin (subtopology euclidean U) S
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   371
  \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   372
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   373
  {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   374
      by (simp add: open_def) blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   375
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   376
  {assume SU: "S \<subseteq> U" and H: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   377
    from H obtain d where d: "\<And>x . x\<in> S \<Longrightarrow> d x > 0 \<and> (\<forall>x' \<in> U. dist x' x < d x \<longrightarrow> x' \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   378
      by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   379
    let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   380
    have oT: "open ?T" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   381
    { fix x assume "x\<in>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   382
      hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   383
	apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   384
	unfolding dist_refl using d[of x] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   385
      hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   386
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   387
    { fix y assume "y\<in>?T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   388
      then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   389
      then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   390
      assume "y\<in>U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   391
      hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_sym) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   392
    ultimately have "S = ?T \<inter> U" by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   393
    with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   394
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   395
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   396
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   397
text{* These "transitivity" results are handy too. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   398
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   399
lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   400
  \<Longrightarrow> openin (subtopology euclidean U) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   401
  unfolding open_openin openin_open by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   402
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   403
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   404
  by (auto simp add: openin_open intro: openin_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   405
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   406
lemma closedin_trans[trans]:
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   407
 "closedin (subtopology euclidean T) S \<Longrightarrow>
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   408
           closedin (subtopology euclidean U) T
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   409
           ==> closedin (subtopology euclidean U) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   410
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   411
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   412
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   413
  by (auto simp add: closedin_closed intro: closedin_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   414
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   415
subsection{* Connectedness *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   416
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   417
definition "connected S \<longleftrightarrow>
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   418
  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   419
  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   420
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   421
lemma connected_local:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   422
 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   423
                 openin (subtopology euclidean S) e1 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   424
                 openin (subtopology euclidean S) e2 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   425
                 S \<subseteq> e1 \<union> e2 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   426
                 e1 \<inter> e2 = {} \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   427
                 ~(e1 = {}) \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   428
                 ~(e2 = {}))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   429
unfolding connected_def openin_open by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   430
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   431
lemma exists_diff: "(\<exists>S. P(UNIV - S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   432
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   433
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   434
  {assume "?lhs" hence ?rhs by blast }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   435
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   436
  {fix S assume H: "P S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   437
    have "S = UNIV - (UNIV - S)" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   438
    with H have "P (UNIV - (UNIV - S))" by metis }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   439
  ultimately show ?thesis by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   440
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   441
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   442
lemma connected_clopen: "connected S \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   443
        (\<forall>T. openin (subtopology euclidean S) T \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   444
            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   445
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   446
  have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (UNIV - e2) \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   447
    unfolding connected_def openin_open closedin_closed
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   448
    apply (subst exists_diff) by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   449
  hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   450
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   451
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   452
  have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   453
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   454
    unfolding connected_def openin_open closedin_closed by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   455
  {fix e2
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   456
    {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   457
	by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   458
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   459
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   460
  then show ?thesis unfolding th0 th1 by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   461
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   462
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   463
lemma connected_empty[simp, intro]: "connected {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   464
  by (simp add: connected_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   465
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   466
subsection{* Hausdorff and other separation properties *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   467
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   468
lemma hausdorff:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   469
  assumes xy: "x \<noteq> y"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   470
  shows "\<exists>U V. open U \<and> open V \<and> x\<in> U \<and> y \<in> V \<and> (U \<inter> V = {})" (is "\<exists>U V. ?P U V")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   471
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   472
  let ?U = "ball x (dist x y / 2)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   473
  let ?V = "ball y (dist x y / 2)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   474
  have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   475
               ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   476
  have "?P ?U ?V" using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_sym]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   477
    by (auto simp add: dist_refl expand_set_eq Arith_Tools.less_divide_eq_number_of1)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   478
  then show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   479
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   480
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   481
lemma separation_t2: "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   482
  using hausdorff[of x y] by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   483
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   484
lemma separation_t1: "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in>U \<and> y\<notin> U \<and> x\<notin>V \<and> y\<in>V)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   485
  using separation_t2[of x y] by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   486
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   487
lemma separation_t0: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))" by(metis separation_t1)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   488
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   489
subsection{* Limit points *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   490
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   491
definition islimpt:: "real ^'n \<Rightarrow> (real^'n) set \<Rightarrow> bool" (infixr "islimpt" 60) where
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   492
  islimpt_def: "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   493
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   494
  (* FIXME: Sure this form is OK????*)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   495
lemma islimptE: assumes "x islimpt S" and "x \<in> T" and "open T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   496
  obtains "(\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   497
  using assms unfolding islimpt_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   498
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   499
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   500
lemma islimpt_approachable: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   501
  unfolding islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   502
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   503
  apply(erule_tac x="ball x e" in allE)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   504
  apply (auto simp add: dist_refl)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   505
  apply(rule_tac x=y in bexI) apply (auto simp add: dist_sym)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   506
  by (metis open_def dist_sym open_ball centre_in_ball mem_ball)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   507
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   508
lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   509
  unfolding islimpt_approachable
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   510
  using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   511
  by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   512
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   513
lemma islimpt_UNIV[simp, intro]: "(x:: real ^'n) islimpt UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   514
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   515
  {
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   516
    fix e::real assume ep: "e>0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   517
    from vector_choose_size[of "e/2"] ep have "\<exists>(c:: real ^'n). norm c = e/2" by auto
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   518
    then obtain c ::"real^'n" where c: "norm c = e/2" by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   519
    let ?x = "x + c"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   520
    have "?x \<noteq> x" using c ep by (auto simp add: norm_eq_0_imp)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   521
    moreover have "dist ?x x < e" using c ep apply simp by norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   522
    ultimately have "\<exists>x'. x' \<noteq> x\<and> dist x' x < e" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   523
  then show ?thesis unfolding islimpt_approachable by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   524
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   525
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   526
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   527
  unfolding closed_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   528
  apply (subst open_subopen)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   529
  apply (simp add: islimpt_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   530
  by (metis DiffE DiffI UNIV_I insertCI insert_absorb mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   531
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   532
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   533
  unfolding islimpt_approachable apply auto by ferrack
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   534
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   535
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i\<in>{1.. dimindex(UNIV:: 'n set)}. 0 \<le>x$i}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   536
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   537
  let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   538
  let ?O = "{x::real^'n. \<forall>i\<in>?U. x$i\<ge>0}"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   539
  {fix x:: "real^'n" and i::nat assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e" and i: "i \<in> ?U"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   540
    and xi: "x$i < 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   541
    from xi have th0: "-x$i > 0" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   542
    from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   543
      have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   544
      have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   545
      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using i x'(1) xi
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   546
	apply (simp only: vector_component)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   547
	by (rule th') auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   548
      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[OF i, of "x'-x"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   549
	apply (simp add: dist_def) by norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   550
      from th[OF th1 th2] x'(3) have False by (simp add: dist_sym dist_pos_le) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   551
  then show ?thesis unfolding closed_limpt islimpt_approachable
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   552
    unfolding not_le[symmetric] by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   553
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   554
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   555
lemma finite_set_avoid: assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   556
proof(induct rule: finite_induct[OF fS])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   557
  case 1 thus ?case apply auto by ferrack
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   558
next
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   559
  case (2 x F)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   560
  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   561
  {assume "x = a" hence ?case using d by auto  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   562
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   563
  {assume xa: "x\<noteq>a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   564
    let ?d = "min d (dist a x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   565
    have dp: "?d > 0" using xa d(1) using dist_nz by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   566
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   567
    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   568
  ultimately show ?case by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   569
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   570
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   571
lemma islimpt_finite: assumes fS: "finite S" shows "\<not> a islimpt S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   572
  unfolding islimpt_approachable
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   573
  using finite_set_avoid[OF fS, of a] by (metis dist_sym  not_le)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   574
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   575
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   576
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   577
  defer
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   578
  apply (metis Un_upper1 Un_upper2 islimpt_subset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   579
  unfolding islimpt_approachable
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   580
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   581
  apply (erule_tac x="min e ea" in allE)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   582
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   583
  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   584
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   585
lemma discrete_imp_closed:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   586
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. norm(y - x) < e \<longrightarrow> y = x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   587
  shows "closed S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   588
proof-
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   589
  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   590
    from e have e2: "e/2 > 0" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   591
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   592
    let ?m = "min (e/2) (dist x y) "
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   593
    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   594
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   595
    have th: "norm (z - y) < e" using z y by norm
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   596
    from d[rule_format, OF y(1) z(1) th] y z
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   597
    have False by (auto simp add: dist_sym)}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   598
  then show ?thesis by (metis islimpt_approachable closed_limpt)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   599
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   600
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   601
subsection{* Interior of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   602
definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   603
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   604
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   605
  apply (simp add: expand_set_eq interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   606
  apply (subst (2) open_subopen) by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   607
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   608
lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   609
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   610
lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   611
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   612
lemma open_interior[simp, intro]: "open(interior S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   613
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   614
  apply (subst open_subopen) by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   615
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   616
lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   617
lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   618
lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   619
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   620
lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T  \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   621
  by (metis equalityI interior_maximal interior_subset open_interior)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   622
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   623
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   624
  by (metis open_contains_ball centre_in_ball open_ball subset_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   625
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   626
lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   627
  by (metis interior_maximal interior_subset subset_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   628
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   629
lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   630
  apply (rule equalityI, simp)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   631
  apply (metis Int_lower1 Int_lower2 subset_interior)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   632
  by (metis Int_mono interior_subset open_inter open_interior open_subset_interior)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   633
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   634
lemma interior_limit_point[intro]: assumes x: "x \<in> interior S" shows "x islimpt S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   635
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   636
  from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   637
    unfolding mem_interior subset_eq Ball_def mem_ball by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   638
  {fix d::real assume d: "d>0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   639
    let ?m = "min d e / 2"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   640
    have mde2: "?m \<ge> 0" using e(1) d(1) by arith
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   641
    from vector_choose_dist[OF mde2, of x]
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   642
    obtain y where y: "dist x y = ?m" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   643
    have th: "dist x y < e" "dist x y < d" unfolding y using e(1) d(1) by arith+
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   644
    have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   645
      apply (rule bexI[where x=y])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   646
      using e th y by (auto simp add: dist_sym)}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   647
  then show ?thesis unfolding islimpt_approachable by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   648
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   649
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   650
lemma interior_closed_Un_empty_interior:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   651
  assumes cS: "closed S" and iT: "interior T = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   652
  shows "interior(S \<union> T) = interior S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   653
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   654
  have "interior S \<subseteq> interior (S\<union>T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   655
    by (rule subset_interior, blast)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   656
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   657
  {fix x e assume e: "e > 0" "\<forall>x' \<in> ball x e. x'\<in>(S\<union>T)"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   658
    {fix y assume y: "y \<in> ball x e"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   659
      {fix d::real assume d: "d > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   660
	let ?k = "min d (e - dist x y)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   661
	have kp: "?k > 0" using d e(1) y[unfolded mem_ball] by norm
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   662
	have "?k/2 \<ge> 0" using kp by simp
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   663
	then obtain w where w: "dist y w = ?k/ 2" by (metis vector_choose_dist)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   664
	from iT[unfolded expand_set_eq mem_interior]
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   665
	have "\<not> ball w (?k/4) \<subseteq> T" using kp by (auto simp add: Arith_Tools.less_divide_eq_number_of1)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   666
	then obtain z where z: "dist w z < ?k/4" "z \<notin> T" by (auto simp add: subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   667
	have "z \<notin> T \<and> z\<noteq> y \<and> dist z y < d \<and> dist x z < e" using z apply simp
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   668
	  using w e(1) d apply (auto simp only: dist_sym)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   669
	  apply (auto simp add: min_def cong del: if_weak_cong)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   670
	  apply (cases "d \<le> e - dist x y", auto simp add: ring_simps cong del: if_weak_cong)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   671
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   672
	  apply (cases "d \<le> e - dist x y", auto simp add: ring_simps not_le not_less cong del: if_weak_cong)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   673
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   674
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   675
	  apply (cases "d \<le> e - dist x y", auto simp add: ring_simps not_le not_less cong del: if_weak_cong)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   676
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   677
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   678
	  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   679
	then have "\<exists>z. z \<notin> T \<and> z\<noteq> y \<and> dist z y < d \<and> dist x z < e" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   680
	then have "\<exists>x' \<in>S. x'\<noteq>y \<and> dist x' y < d" using e by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   681
      then have "y\<in>S" by (metis islimpt_approachable cS closed_limpt) }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   682
    then have "x \<in> interior S" unfolding mem_interior using e(1) by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   683
  hence "interior (S\<union>T) \<subseteq> interior S" unfolding mem_interior Ball_def subset_eq by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   684
  ultimately show ?thesis by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   685
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   686
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   687
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   688
subsection{* Closure of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   689
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   690
definition "closure S = S \<union> {x | x. x islimpt S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   691
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   692
lemma closure_interior: "closure S = UNIV - interior (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   693
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   694
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   695
    have "x\<in>UNIV - interior (UNIV - S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   696
    proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   697
      let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   698
      assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   699
      hence *:"\<not> ?exT x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   700
	unfolding interior_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   701
	by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   702
      { assume "\<not> ?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   703
	hence False using *
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   704
	  unfolding closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   705
	  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   706
      }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   707
      thus "?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   708
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   709
    next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   710
      assume "?rhs" thus "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   711
	unfolding closure_def interior_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   712
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   713
    qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   714
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   715
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   716
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   717
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   718
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   719
lemma interior_closure: "interior S = UNIV - (closure (UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   720
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   721
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   722
    have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   723
      unfolding interior_def closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   724
      by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   725
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   726
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   727
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   728
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   729
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   730
lemma closed_closure[simp, intro]: "closed (closure S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   731
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   732
  have "closed (UNIV - interior (UNIV -S))" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   733
  thus ?thesis using closure_interior[of S] by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   734
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   735
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   736
lemma closure_hull: "closure S = closed hull S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   737
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   738
  have "S \<subseteq> closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   739
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   740
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   741
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   742
  have "closed (closure S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   743
    using closed_closure[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   744
    by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   745
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   746
  { fix t
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   747
    assume *:"S \<subseteq> t" "closed t"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   748
    { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   749
      assume "x islimpt S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   750
      hence "x islimpt t" using *(1)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   751
	using islimpt_subset[of x, of S, of t]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   752
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   753
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   754
    with * have "closure S \<subseteq> t"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   755
      unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   756
      using closed_limpt[of t]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   757
      by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   758
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   759
  ultimately show ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   760
    using hull_unique[of S, of "closure S", of closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   761
    unfolding mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   762
    by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   763
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   764
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   765
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   766
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   767
  using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   768
  by (metis mem_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   769
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   770
lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   771
  using closure_eq[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   772
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   773
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   774
lemma closure_closure[simp]: "closure (closure S) = closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   775
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   776
  using hull_hull[of closed S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   777
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   778
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   779
lemma closure_subset: "S \<subseteq> closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   780
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   781
  using hull_subset[of S closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   782
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   783
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   784
lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   785
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   786
  using hull_mono[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   787
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   788
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   789
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   790
  using hull_minimal[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   791
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   792
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   793
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   794
lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   795
  using hull_unique[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   796
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   797
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   798
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   799
lemma closure_empty[simp]: "closure {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   800
  using closed_empty closure_closed[of "{}"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   801
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   802
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   803
lemma closure_univ[simp]: "closure UNIV = UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   804
  using closure_closed[of UNIV]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   805
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   806
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   807
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   808
  using closure_empty closure_subset[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   809
  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   810
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   811
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   812
  using closure_eq[of S] closure_subset[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   813
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   814
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   815
lemma open_inter_closure_eq_empty:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   816
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   817
  using open_subset_interior[of S "UNIV - T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   818
  using interior_subset[of "UNIV - T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   819
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   820
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   821
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   822
lemma open_inter_closure_subset: "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   823
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   824
  fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   825
  assume as: "open S" "x \<in> S \<inter> closure T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   826
  { assume *:"x islimpt T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   827
    { fix e::real
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   828
      assume "e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   829
      from as `open S` obtain e' where "e' > 0" and e':"\<forall>x'. dist x' x < e' \<longrightarrow> x' \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   830
	unfolding open_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   831
	by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   832
      let ?e = "min e e'"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   833
      from `e>0` `e'>0` have "?e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   834
	by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   835
      then obtain y where y:"y\<in>T" "y \<noteq> x \<and> dist y x < ?e"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   836
	using islimpt_approachable[of x T] using *
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   837
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   838
      hence "\<exists>x'\<in>S \<inter> T. x' \<noteq> x \<and> dist x' x < e" using e'
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   839
	using y
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   840
	by(rule_tac x=y in bexI, simp+)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   841
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   842
    hence "x islimpt S \<inter> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   843
      using islimpt_approachable[of x "S \<inter> T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   844
      by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   845
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   846
  then show "x \<in> closure (S \<inter> T)" using as
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   847
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   848
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   849
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   850
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   851
lemma closure_complement: "closure(UNIV - S) = UNIV - interior(S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   852
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   853
  have "S = UNIV - (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   854
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   855
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   856
    unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   857
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   858
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   859
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   860
lemma interior_complement: "interior(UNIV - S) = UNIV - closure(S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   861
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   862
  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   863
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   864
subsection{* Frontier (aka boundary) *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   865
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   866
definition "frontier S = closure S - interior S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   867
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   868
lemma frontier_closed: "closed(frontier S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   869
  by (simp add: frontier_def closed_diff closed_closure)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   870
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   871
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   872
  by (auto simp add: frontier_def interior_closure)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   873
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   874
lemma frontier_straddle: "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   875
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   876
  assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   877
  { fix e::real
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   878
    assume "e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   879
    let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   880
    { assume "a\<in>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   881
      have "\<exists>x\<in>S. dist a x < e" using dist_refl[of a] `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   882
      moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   883
	unfolding frontier_closures closure_def islimpt_def using dist_refl[of a] `e>0`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   884
	by (auto, erule_tac x="ball a e" in allE, auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   885
      ultimately have ?rhse by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   886
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   887
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   888
    { assume "a\<notin>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   889
      hence ?rhse using `?lhs`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   890
	unfolding frontier_closures closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   891
	using open_ball[of a e] dist_refl[of a] `e > 0`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   892
	by (auto, erule_tac x = "ball a e" in allE, auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   893
    }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   894
    ultimately have ?rhse by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   895
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   896
  thus ?rhs by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   897
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   898
  assume ?rhs
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   899
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   900
  { fix T assume "a\<notin>S" and
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   901
    as:"\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)" "a \<notin> S" "a \<in> T" "open T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   902
    from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   903
    then obtain e where "e>0" "ball a e \<subseteq> T" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   904
    then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   905
    have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   906
      using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   907
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   908
  hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   909
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   910
  { fix T assume "a \<in> T"  "open T" "a\<in>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   911
    then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   912
    obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   913
    hence "\<exists>y\<in>UNIV - S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   914
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   915
  hence "a islimpt (UNIV - S) \<or> a\<notin>S" unfolding islimpt_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   916
  ultimately show ?lhs unfolding frontier_closures using closure_def[of "UNIV - S"] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   917
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   918
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   919
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   920
  by (metis frontier_def closure_closed Diff_subset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   921
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   922
lemma frontier_empty: "frontier {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   923
  by (simp add: frontier_def closure_empty)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   924
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   925
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   926
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   927
  { assume "frontier S \<subseteq> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   928
    hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   929
    hence "closed S" using closure_subset_eq by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   930
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   931
  thus ?thesis using frontier_subset_closed[of S] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   932
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   933
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   934
lemma frontier_complement: "frontier(UNIV - S) = frontier S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   935
  by (auto simp add: frontier_def closure_complement interior_complement)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   936
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   937
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   938
  using frontier_complement frontier_subset_eq[of "UNIV - S"]
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   939
  unfolding open_closed by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   940
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   941
subsection{* A variant of nets (Slightly non-standard but good for our purposes). *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   942
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   943
typedef (open) 'a net =
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   944
  "{g :: 'a \<Rightarrow> 'a \<Rightarrow> bool. \<forall>x y. (\<forall>z. g z x \<longrightarrow> g z y) \<or> (\<forall>z. g z y \<longrightarrow> g z x)}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   945
  morphisms "netord" "mknet" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   946
lemma net: "(\<forall>z. netord n z x \<longrightarrow> netord n z y) \<or> (\<forall>z. netord n z y \<longrightarrow> netord n z x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   947
  using netord[of n] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   948
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   949
lemma oldnet: "netord n x x \<Longrightarrow> netord n y y \<Longrightarrow>
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   950
  \<exists>z. netord n z z \<and> (\<forall>w. netord n w z \<longrightarrow> netord n w x \<and> netord n w y)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   951
  by (metis net)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   952
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   953
lemma net_dilemma:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   954
 "\<exists>a. (\<exists>x. netord net x a) \<and> (\<forall>x. netord net x a \<longrightarrow> P x) \<Longrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   955
         \<exists>b. (\<exists>x. netord net x b) \<and> (\<forall>x. netord net x b \<longrightarrow> Q x)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   956
         \<Longrightarrow> \<exists>c. (\<exists>x. netord net x c) \<and> (\<forall>x. netord net x c \<longrightarrow> P x \<and> Q x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   957
  by (metis net)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   958
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   959
subsection{* Common nets and The "within" modifier for nets. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   960
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   961
definition "at a = mknet(\<lambda>x y. 0 < dist x a \<and> dist x a <= dist y a)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   962
definition "at_infinity = mknet(\<lambda>x y. norm x \<ge> norm y)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   963
definition "sequentially = mknet(\<lambda>(m::nat) n. m >= n)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   964
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   965
definition within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   966
  within_def: "net within S = mknet (\<lambda>x y. netord net x y \<and> x \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   967
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   968
definition indirection :: "real ^'n \<Rightarrow> real ^'n \<Rightarrow> (real ^'n) net" (infixr "indirection" 70) where
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   969
  indirection_def: "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = c*s v}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   970
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   971
text{* Prove That They are all nets. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   972
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   973
lemma mknet_inverse': "netord (mknet r) = r \<longleftrightarrow> (\<forall>x y. (\<forall>z. r z x \<longrightarrow> r z y) \<or> (\<forall>z. r z y \<longrightarrow> r z x))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   974
  using mknet_inverse[of r] apply (auto simp add: netord_inverse) by (metis net)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   975
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   976
method_setup net = {*
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   977
 let
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   978
  val ss1 = HOL_basic_ss addsimps [@{thm expand_fun_eq} RS sym]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   979
  val ss2 = HOL_basic_ss addsimps [@{thm mknet_inverse'}]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   980
  fun tac ths = ObjectLogic.full_atomize_tac THEN' Simplifier.simp_tac (ss1 addsimps ths) THEN' Simplifier.asm_full_simp_tac ss2
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   981
  in Method.thms_args (Method.SIMPLE_METHOD' o tac) end
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   982
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   983
*} "Reduces goals about net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   984
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   985
lemma at: "\<And>x y. netord (at a) x y \<longleftrightarrow> 0 < dist x a \<and> dist x a <= dist y a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   986
  apply (net at_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   987
  by (metis dist_sym real_le_linear real_le_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   988
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   989
lemma at_infinity:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   990
 "\<And>x y. netord at_infinity x y \<longleftrightarrow> norm x >= norm y"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   991
  apply (net at_infinity_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   992
  apply (metis real_le_linear real_le_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   993
  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   994
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   995
lemma sequentially: "\<And>m n. netord sequentially m n \<longleftrightarrow> m >= n"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   996
  apply (net sequentially_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   997
  apply (metis linorder_linear min_max.le_supI2 min_max.sup_absorb1)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   998
  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   999
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1000
lemma within: "netord (n within S) x y \<longleftrightarrow> netord n x y \<and> x \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1001
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1002
  have "\<forall>x y. (\<forall>z. netord n z x \<and> z \<in> S \<longrightarrow> netord n z y) \<or> (\<forall>z. netord n z y \<and> z \<in> S \<longrightarrow> netord n z x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1003
    by (metis net)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1004
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1005
    unfolding within_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1006
    using mknet_inverse[of "\<lambda>x y. netord n x y \<and> x \<in> S"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1007
    by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1008
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1009
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1010
lemma in_direction: "netord (a indirection v) x y \<longleftrightarrow> 0 < dist x a \<and> dist x a \<le> dist y a \<and> (\<exists>c \<ge> 0. x - a = c *s v)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1011
  by (simp add: within at indirection_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1012
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1013
lemma within_UNIV: "at x within UNIV = at x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1014
  by (simp add: within_def at_def netord_inverse)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1015
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1016
subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1017
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1018
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1019
definition "trivial_limit (net:: 'a net) \<longleftrightarrow>
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1020
  (\<forall>(a::'a) b. a = b) \<or> (\<exists>(a::'a) b. a \<noteq> b \<and> (\<forall>x. ~(netord (net) x a) \<and> ~(netord(net) x b)))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1021
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1022
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1023
lemma trivial_limit_within: "trivial_limit (at (a::real^'n) within S) \<longleftrightarrow> ~(a islimpt S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1024
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1025
  {assume "\<forall>(a::real^'n) b. a = b" hence "\<not> a islimpt S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1026
      apply (simp add: islimpt_approachable_le)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1027
      by (rule exI[where x=1], auto)}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1028
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1029
  {fix b c assume bc: "b \<noteq> c" "\<forall>x. \<not> netord (at a within S) x b \<and> \<not> netord (at a within S) x c"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1030
    have "dist a b > 0 \<or> dist a c > 0" using bc by (auto simp add: within at dist_nz[THEN sym])
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1031
    then have "\<not> a islimpt S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1032
      using bc
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1033
      unfolding within at dist_nz islimpt_approachable_le
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1034
      by(auto simp add: dist_triangle dist_sym dist_eq_0[THEN sym]) }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1035
  moreover
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1036
  {assume "\<not> a islimpt S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1037
    then obtain e where e: "e > 0" "\<forall>x' \<in> S. x' \<noteq> a \<longrightarrow> dist x' a > e"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1038
      unfolding islimpt_approachable_le by (auto simp add: not_le)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1039
    from e vector_choose_dist[of e a] obtain b where b: "dist a b = e" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1040
    from b e(1) have "a \<noteq> b" by (simp add: dist_nz)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1041
    moreover have "\<forall>x. \<not> ((0 < dist x a \<and> dist x a \<le> dist a a) \<and> x \<in> S) \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1042
                 \<not> ((0 < dist x a \<and> dist x a \<le> dist b a) \<and> x \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1043
      using e(2) b by (auto simp add: dist_refl dist_sym)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1044
    ultimately have "trivial_limit (at a within S)"  unfolding trivial_limit_def within at
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1045
      by blast}
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1046
  ultimately show ?thesis unfolding trivial_limit_def by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1047
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1048
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1049
lemma trivial_limit_at: "~(trivial_limit (at a))"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1050
  apply (subst within_UNIV[symmetric])
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1051
  by (simp add: trivial_limit_within islimpt_UNIV)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1052
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1053
lemma trivial_limit_at_infinity: "~(trivial_limit (at_infinity :: ('a::{norm,zero_neq_one}) net))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1054
  apply (simp add: trivial_limit_def at_infinity)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1055
  by (metis order_refl zero_neq_one)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1056
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1057
lemma trivial_limit_sequentially:  "~(trivial_limit sequentially)"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1058
  by (auto simp add: trivial_limit_def sequentially)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1059
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1060
subsection{* Some property holds "sufficiently close" to the limit point. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1061
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1062
definition "eventually P net \<longleftrightarrow> trivial_limit net \<or> (\<exists>y. (\<exists>x. netord net x y) \<and> (\<forall>x. netord net x y \<longrightarrow> P x))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1063
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1064
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1065
  by (metis eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1066
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1067
lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1068
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1069
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1070
  assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1071
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1072
  { assume "\<not> a islimpt S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1073
    then obtain e where "e>0" and e:"\<forall>x'\<in>S. \<not> (x' \<noteq> a \<and> dist x' a \<le> e)" unfolding islimpt_approachable_le by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1074
    hence  "?rhs" apply auto apply (rule_tac x=e in exI) by auto  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1075
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1076
  { assume "\<exists>y. (\<exists>x. netord (at a within S) x y) \<and> (\<forall>x. netord (at a within S) x y \<longrightarrow> P x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1077
    then obtain x y where xy:"netord (at a within S) x y \<and> (\<forall>x. netord (at a within S) x y \<longrightarrow> P x)" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1078
    hence "?rhs" unfolding within at by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1079
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1080
  ultimately show "?rhs" unfolding eventually_def trivial_limit_within by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1081
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1082
  assume "?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1083
  then obtain d where "d>0" and d:"\<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> P x" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1084
  thus "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1085
    unfolding eventually_def trivial_limit_within islimpt_approachable_le within at unfolding dist_nz[THEN sym] by (clarsimp, rule_tac x=d in exI, auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1086
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1087
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1088
lemma eventually_within:  " eventually P (at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1089
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1090
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1091
  { fix d
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1092
    assume "d>0" "\<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1093
    hence "\<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> (d/2) \<longrightarrow> P x" using order_less_imp_le by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1094
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1095
  thus ?thesis unfolding eventually_within_le using approachable_lt_le
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1096
    by (auto, rule_tac x="d/2" in exI, auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1097
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1098
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1099
lemma eventually_at: "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1100
  apply (subst within_UNIV[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1101
  by (simp add: eventually_within)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1102
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1103
lemma eventually_sequentially: "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1104
  apply (simp add: eventually_def sequentially trivial_limit_sequentially)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1105
apply (metis dlo_simps(7) dlo_simps(9) le_maxI2 min_max.le_iff_sup min_max.sup_absorb1 order_antisym_conv) done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1106
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1107
(* FIXME Declare this with P::'a::some_type \<Rightarrow> bool *)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1108
lemma eventually_at_infinity: "eventually (P::(real^'n \<Rightarrow> bool)) at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)" (is "?lhs = ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1109
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1110
  assume "?lhs" thus "?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1111
    unfolding eventually_def at_infinity
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1112
    by (auto simp add: trivial_limit_at_infinity)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1113
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1114
  assume "?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1115
  then obtain b where b:"\<forall>x. b \<le> norm x \<longrightarrow> P x" and "b\<ge>0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1116
    by (metis norm_ge_zero real_le_linear real_le_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1117
  obtain y::"real^'n" where y:"norm y = b" using `b\<ge>0`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1118
    using vector_choose_size[of b] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1119
  thus "?lhs" unfolding eventually_def at_infinity using b y by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1120
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1121
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1122
lemma always_eventually: "(\<forall>(x::'a::zero_neq_one). P x) ==> eventually P net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1123
  apply (auto simp add: eventually_def trivial_limit_def )
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1124
  by (rule exI[where x=0], rule exI[where x=1], rule zero_neq_one)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1125
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1126
text{* Combining theorems for "eventually" *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1127
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1128
lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1129
  apply (simp add: eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1130
  apply (cases "trivial_limit net")
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1131
  using net_dilemma[of net P Q] by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1132
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1133
lemma eventually_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net  \<Longrightarrow> eventually Q net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1134
  by (metis eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1135
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1136
lemma eventually_mp: "eventually (\<lambda>x. P x \<longrightarrow> Q x) net \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1137
  apply (atomize(full))
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1138
  unfolding imp_conjL[symmetric] eventually_and[symmetric]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1139
  by (auto simp add: eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1140
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1141
lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1142
  by (auto simp add: eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1143
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1144
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually P net)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1145
  by (auto simp add: eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1146
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1147
subsection{* Limits, defined as vacuously true when the limit is trivial. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1148
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1149
definition tendsto:: "('a \<Rightarrow> real ^'n) \<Rightarrow> real ^'n \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "--->" 55) where
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1150
  tendsto_def: "(f ---> l) net  \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1151
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1152
lemma tendstoD: "(f ---> l) net \<Longrightarrow> e>0 \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1153
  unfolding tendsto_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1154
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1155
  text{* Notation Lim to avoid collition with lim defined in analysis *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1156
definition "Lim net f = (THE l. (f ---> l) net)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1157
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1158
lemma Lim:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1159
 "(f ---> l) net \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1160
        trivial_limit net \<or>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1161
        (\<forall>e>0. \<exists>y. (\<exists>x. netord net x y) \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1162
                           (\<forall>x. netord(net) x y \<longrightarrow> dist (f x) l < e))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1163
  by (auto simp add: tendsto_def eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1164
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1165
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1166
text{* Show that they yield usual definitions in the various cases. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1167
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1168
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1169
           (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1170
  by (auto simp add: tendsto_def eventually_within_le)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1171
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1172
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1173
        (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1174
  by (auto simp add: tendsto_def eventually_within)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1175
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1176
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1177
        (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1178
  by (auto simp add: tendsto_def eventually_at)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1179
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1180
lemma Lim_at_infinity:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1181
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x::real^'n. norm x >= b \<longrightarrow> dist (f x) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1182
  by (auto simp add: tendsto_def eventually_at_infinity)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1183
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1184
lemma Lim_sequentially:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1185
 "(S ---> l) sequentially \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1186
          (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1187
  by (auto simp add: tendsto_def eventually_sequentially)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1188
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1189
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1190
  by (auto simp add: eventually_def Lim dist_refl)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1191
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1192
text{* The expected monotonicity property. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1193
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1194
lemma Lim_within_empty:  "(f ---> l) (at x within {})"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1195
  by (simp add: Lim_within_le)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1196
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1197
lemma Lim_within_subset: "(f ---> l) (at a within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at a within T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1198
  apply (auto simp add: Lim_within_le)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1199
  by (metis subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1200
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1201
lemma Lim_Un: assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1202
  shows "(f ---> l) (at x within (S \<union> T))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1203
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1204
  { fix e::real assume "e>0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1205
    obtain d1 where d1:"d1>0" "\<forall>xa\<in>T. 0 < dist xa x \<and> dist xa x < d1 \<longrightarrow> dist (f xa) l < e" using assms unfolding Lim_within using `e>0` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1206
    obtain d2 where d2:"d2>0" "\<forall>xa\<in>S. 0 < dist xa x \<and> dist xa x < d2 \<longrightarrow> dist (f xa) l < e" using assms unfolding Lim_within using `e>0` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1207
    have "\<exists>d>0. \<forall>xa\<in>S \<union> T. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) l < e" using d1 d2
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1208
      by (rule_tac x="min d1 d2" in exI)auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1209
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1210
  thus ?thesis unfolding Lim_within by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1211
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1212
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1213
lemma Lim_Un_univ:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1214
 "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>  S \<union> T = (UNIV::(real^'n) set)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1215
        ==> (f ---> l) (at x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1216
  by (metis Lim_Un within_UNIV)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1217
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1218
text{* Interrelations between restricted and unrestricted limits. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1219
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1220
lemma Lim_at_within: "(f ---> l)(at a) ==> (f ---> l)(at a within S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1221
  apply (simp add: Lim_at Lim_within)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1222
  by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1223
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1224
lemma Lim_within_open:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1225
  assumes"a \<in> S" "open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1226
  shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1227
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1228
  assume ?lhs
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1229
  { fix e::real assume "e>0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1230
    obtain d  where d:  "d >0" "\<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e" using `?lhs` `e>0` unfolding Lim_within by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1231
    obtain d' where d': "d'>0" "\<forall>x. dist x a < d' \<longrightarrow> x \<in> S" using assms  unfolding open_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1232
    from d d' have "\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e" by (rule_tac x= "min d d'" in exI)auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1233
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1234
  thus ?rhs unfolding Lim_at by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1235
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1236
  assume ?rhs
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1237
  { fix e::real assume "e>0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1238
    then obtain d where "d>0" and d:"\<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e" using `?rhs` unfolding Lim_at by auto
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1239
    hence "\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e" using `d>0` by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1240
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1241
  thus ?lhs using Lim_at_within[of f l a S] by (auto simp add: Lim_at)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1242
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1243
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1244
text{* Another limit point characterization. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1245
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1246
lemma islimpt_sequential:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1247
 "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)" (is "?lhs = ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents: