src/HOL/Library/Topology_Euclidean_Space.thy
author huffman
Tue, 02 Jun 2009 15:13:22 -0700
changeset 31390 1d0478b16613
parent 31348 738eb25e1dd8
child 31391 97a2a3d4088e
permissions -rw-r--r--
redefine nets as filter bases
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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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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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. *}
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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theory Topology_Euclidean_Space
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254478a8dd05 dropped theory Arith_Tools
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imports SEQ 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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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]
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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subsection{* General notion of a topology *}
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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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)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  using openin[of U] 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 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] .
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 topology_inverse_iff: "istopology U \<longleftrightarrow> 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[of U] istopology_open_in[of "topology 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 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-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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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
chaieb
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    36
    hence "topology (openin T1) = topology (openin T2)" by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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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
chaieb
parents:
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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. *}
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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definition "topspace T =  \<Union>{S. openin T S}"
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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subsection{* Main properties of open sets *}
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_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
chaieb
parents:
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  by (metis mem_def subset_eq)+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    54
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
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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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)
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_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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parents:
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    65
  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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    68
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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    70
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
chaieb
parents:
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    72
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
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    73
  {assume H: ?rhs
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    74
    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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parents:
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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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    76
    from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
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    77
    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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parents:
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    78
    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
chaieb
parents:
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    79
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    80
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    81
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    82
subsection{* Closed sets *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    83
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    84
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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    85
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
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    86
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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    87
lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
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    88
lemma closedin_topspace[intro,simp]:
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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    89
  "closedin U (topspace U)" by (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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    90
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
chaieb
parents:
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    91
  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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parents:
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    92
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
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    93
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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    94
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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    95
  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
chaieb
parents:
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    96
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    97
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:
diff changeset
    98
  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:
diff changeset
    99
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   100
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
chaieb
parents:
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   101
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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   102
  apply (auto simp add: closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   103
  apply (metis openin_subset subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   104
  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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   105
  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:
diff changeset
   106
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   107
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   108
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:
diff changeset
   109
  by (simp add: openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   110
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   111
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
chaieb
parents:
diff changeset
   112
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   113
  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
   114
    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
   115
  then show ?thesis using oS cT by (auto simp add: closedin_def)
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5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   116
qed
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   117
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   118
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
   119
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   120
  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
   121
    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
   122
  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
   123
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   124
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   125
subsection{* Subspace topology. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   126
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   127
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
   128
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   129
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
   130
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   131
  have "{} \<in> ?L" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   132
  {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
   133
    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
   134
    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
   135
    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
   136
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   137
  {fix K assume K: "K \<subseteq> ?L"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   138
    have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   139
      apply (rule set_ext)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   140
      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
   141
      by (metis mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   142
    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
   143
    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
   144
    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
   145
    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
   146
    ultimately have "\<Union>K \<in> ?L" by blast}
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   147
  ultimately show ?thesis unfolding istopology_def by blast
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   148
qed
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   149
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   150
lemma openin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   151
  "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
   152
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   153
  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
   154
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   155
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
   156
  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
   157
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   158
lemma closedin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   159
  "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
   160
  unfolding closedin_def topspace_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   161
  apply (simp add: openin_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   162
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   163
  apply clarify
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   164
  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
   165
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   166
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   167
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
   168
  unfolding openin_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   169
  apply (rule iffI, clarify)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   170
  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
   171
  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
   172
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   173
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   174
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
   175
  shows "subtopology U V = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   176
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   177
  {fix S
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   178
    {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
   179
      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
   180
      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
   181
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   182
    {assume S: "openin U S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   183
      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
   184
	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
   185
    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
   186
  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
   187
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   188
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
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
   191
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   192
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   193
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
   194
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   195
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   196
subsection{* The universal Euclidean versions are what we use most of the time *}
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   197
definition
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   198
  "open" :: "'a::metric_space set \<Rightarrow> bool" where
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   199
  "open S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>e >0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> S)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   200
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
   201
definition "euclidean = topology open"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   202
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   203
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
   204
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
   205
  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
   206
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   207
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
   208
  shows "open (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   209
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   210
  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
   211
  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
   212
  {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
   213
    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
   214
    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
   215
    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
   216
    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
   217
    { 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
   218
      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
   219
      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
   220
    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
   221
  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
   222
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   223
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   224
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
   225
  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
   226
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   227
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
   228
  unfolding euclidean_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   229
  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
   230
  apply (rule topology_inverse[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   231
  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
   232
  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
   233
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   234
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
   235
  apply (simp add: topspace_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   236
  apply (rule set_ext)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   237
  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
   238
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   239
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
   240
  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
   241
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
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  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
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  ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
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  "ball x e = {y. dist x y < e}"
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definition
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  cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
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  "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]:
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  fixes x :: "'a::real_normed_vector"
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  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
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  by (simp add: dist_norm)
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lemma mem_cball_0 [simp]:
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  fixes x :: "'a::real_normed_vector"
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  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
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  by (simp add: dist_norm)
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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_commute
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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_self)
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lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
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  unfolding open_def subset_eq mem_ball Ball_def dist_commute ..
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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)"
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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"
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  unfolding mem_ball expand_set_eq
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  apply (simp add: not_less)
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  by (metis zero_le_dist order_trans dist_self)
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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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lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
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  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"
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   365
  by (metis Int_absorb1  openin_open_Int)
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 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
chaieb
parents:
diff changeset
   368
  by (auto simp add: openin_open)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   369
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   370
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
chaieb
parents:
diff changeset
   371
  by (simp add: closedin_subtopology closed_closedin Int_ac)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   372
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   373
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
chaieb
parents:
diff changeset
   374
  by (metis closedin_closed)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   375
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   376
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
   377
  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
   378
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   379
  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
   380
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   381
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   382
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
   383
  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
   384
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   385
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
   386
  \<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
   387
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   388
  {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
   389
      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
   390
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   391
  {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
   392
    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
   393
      by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   394
    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
   395
    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
   396
    { 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
   397
      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
   398
	apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   399
        by (rule d [THEN conjunct1])
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   400
      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
   401
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   402
    { 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
   403
      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
   404
      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
   405
      assume "y\<in>U"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   406
      hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   407
    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
   408
    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
   409
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   410
qed
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
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
   413
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   414
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
   415
  \<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
   416
  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
   417
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   418
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
   419
  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
   420
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   421
lemma closedin_trans[trans]:
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   422
 "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
   423
           closedin (subtopology euclidean U) T
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   424
           ==> closedin (subtopology euclidean U) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   425
  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
   426
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   427
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
   428
  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
   429
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   430
subsection{* Connectedness *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   431
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   432
definition "connected S \<longleftrightarrow>
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   433
  ~(\<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
   434
  \<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
   435
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   436
lemma connected_local:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   437
 "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
   438
                 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
   439
                 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
   440
                 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
   441
                 e1 \<inter> e2 = {} \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   442
                 ~(e1 = {}) \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   443
                 ~(e2 = {}))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   444
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
   445
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   446
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
   447
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   448
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   449
  {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
   450
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   451
  {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
   452
    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
   453
    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
   454
  ultimately show ?thesis by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   455
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   456
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   457
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
   458
        (\<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
   459
            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
   460
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   461
  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
   462
    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
   463
    apply (subst exists_diff) by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   464
  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
   465
    (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
   466
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   467
  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
   468
    (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
   469
    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
   470
  {fix e2
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   471
    {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
   472
	by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   473
    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
   474
  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
   475
  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
   476
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   477
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   478
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
   479
  by (simp add: connected_def)
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
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
   482
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   483
lemma hausdorff:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   484
  assumes xy: "x \<noteq> y"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   485
  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
   486
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   487
  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
   488
  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
   489
  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
   490
               ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   491
  have "?P ?U ?V" using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   492
    by (auto simp add: expand_set_eq less_divide_eq_number_of1)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   493
  then show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   494
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   495
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   496
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
   497
  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
   498
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   499
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
   500
  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
   501
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   502
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
   503
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   504
subsection{* Limit points *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   505
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   506
definition
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   507
  islimpt:: "'a::metric_space \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   508
  "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   509
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   510
  (* 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
   511
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
   512
  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
   513
  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
   514
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   515
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
   516
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
   517
  unfolding islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   518
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   519
  apply(erule_tac x="ball x e" in allE)
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   520
  apply auto
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   521
  apply(rule_tac x=y in bexI)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   522
  apply (auto simp add: dist_commute)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   523
  apply (simp add: open_def, drule (1) bspec)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   524
  apply (clarify, drule spec, drule (1) mp, auto)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   525
  done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   526
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   527
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
   528
  unfolding islimpt_approachable
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   529
  using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   530
  by metis (* FIXME: VERY slow! *)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   531
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   532
axclass perfect_space \<subseteq> metric_space
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   533
  islimpt_UNIV [simp, intro]: "x islimpt UNIV"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   534
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   535
lemma perfect_choose_dist:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   536
  fixes x :: "'a::perfect_space"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   537
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   538
using islimpt_UNIV [of x]
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   539
by (simp add: islimpt_approachable)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   540
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   541
instance real :: perfect_space
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   542
apply default
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   543
apply (rule islimpt_approachable [THEN iffD2])
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   544
apply (clarify, rule_tac x="x + e/2" in bexI)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   545
apply (auto simp add: dist_norm)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   546
done
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   547
  
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   548
instance "^" :: (perfect_space, finite) perfect_space
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   549
proof
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   550
  fix x :: "'a ^ 'b"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   551
  {
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   552
    fix e :: real assume "0 < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   553
    def a \<equiv> "x $ arbitrary"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   554
    have "a islimpt UNIV" by (rule islimpt_UNIV)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   555
    with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   556
      unfolding islimpt_approachable by auto
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   557
    def y \<equiv> "Cart_lambda ((Cart_nth x)(arbitrary := b))"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   558
    from `b \<noteq> a` have "y \<noteq> x"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   559
      unfolding a_def y_def by (simp add: Cart_eq)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   560
    from `dist b a < e` have "dist y x < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   561
      unfolding dist_vector_def a_def y_def
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   562
      apply simp
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   563
      apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   564
      apply (subst setsum_diff1' [where a=arbitrary], simp, simp, simp)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   565
      done
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   566
    from `y \<noteq> x` and `dist y x < e`
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   567
    have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   568
  }
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   569
  then show "x islimpt UNIV" unfolding islimpt_approachable by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   570
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   571
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   572
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
   573
  unfolding closed_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   574
  apply (subst open_subopen)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   575
  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
   576
  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
   577
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   578
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
   579
  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
   580
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   581
lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x$i}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   582
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   583
  let ?U = "UNIV :: 'n set"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   584
  let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   585
  {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   586
    and xi: "x$i < 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   587
    from xi have th0: "-x$i > 0" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   588
    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
   589
      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
   590
      have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   591
      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   592
	apply (simp only: vector_component)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   593
	by (rule th') auto
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   594
      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[of "x'-x" i]
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   595
	apply (simp add: dist_norm) by norm
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   596
      from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   597
  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
   598
    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
   599
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   600
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   601
lemma finite_set_avoid:
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   602
  fixes a :: "'a::metric_space"
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   603
  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   604
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
   605
  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
   606
next
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   607
  case (2 x F)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   608
  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
   609
  {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
   610
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   611
  {assume xa: "x\<noteq>a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   612
    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
   613
    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
   614
    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
   615
    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
   616
  ultimately show ?case by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   617
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   618
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   619
lemma islimpt_finite: assumes fS: "finite S" shows "\<not> a islimpt S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   620
  unfolding islimpt_approachable
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   621
  using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   622
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   623
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
   624
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   625
  defer
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   626
  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
   627
  unfolding islimpt_approachable
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   628
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   629
  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
   630
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   631
  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   632
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   633
lemma discrete_imp_closed:
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   634
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   635
  shows "closed S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   636
proof-
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   637
  {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
   638
    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
   639
    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
   640
    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
   641
    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
   642
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   643
    have th: "dist z y < e" using z y
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   644
      by (intro dist_triangle_lt [where z=x], simp)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   645
    from d[rule_format, OF y(1) z(1) th] y z
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   646
    have False by (auto simp add: dist_commute)}
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   647
  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
   648
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   649
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   650
subsection{* Interior of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   651
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
   652
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   653
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
   654
  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
   655
  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
   656
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   657
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
   658
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   659
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
   660
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   661
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
   662
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   663
  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
   664
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   665
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
   666
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
   667
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
   668
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
   669
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
   670
  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
   671
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
   672
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   673
  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
   674
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   675
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
   676
  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
   677
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   678
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
   679
  apply (rule equalityI, simp)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   680
  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
   681
  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
   682
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   683
lemma interior_limit_point [intro]:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   684
  fixes x :: "'a::perfect_space"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   685
  assumes x: "x \<in> interior S" shows "x islimpt S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   686
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   687
  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
   688
    unfolding mem_interior subset_eq Ball_def mem_ball by blast
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   689
  {
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   690
    fix d::real assume d: "d>0"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   691
    let ?m = "min d e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   692
    have mde2: "0 < ?m" using e(1) d(1) by simp
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   693
    from perfect_choose_dist [OF mde2, of x]
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   694
    obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   695
    then have "dist y x < e" "dist y x < d" by simp_all
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   696
    from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   697
    have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   698
      using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   699
  }
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   700
  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
   701
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   702
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   703
lemma interior_closed_Un_empty_interior:
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   704
  fixes S T :: "(real ^ 'n::finite) set" (* FIXME: generalize to perfect_space *)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   705
  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
   706
  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
   707
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   708
  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
   709
    by (rule subset_interior, blast)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   710
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   711
  {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
   712
    {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
   713
      {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
   714
	let ?k = "min d (e - dist x y)"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   715
	have kp: "?k > 0" using d e(1) y[unfolded mem_ball] by simp
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   716
	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
   717
	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
   718
	from iT[unfolded expand_set_eq mem_interior]
30654
254478a8dd05 dropped theory Arith_Tools
haftmann
parents: 30582
diff changeset
   719
	have "\<not> ball w (?k/4) \<subseteq> T" using kp by (auto simp add: less_divide_eq_number_of1)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   720
	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
   721
	have "z \<notin> T \<and> z\<noteq> y \<and> dist z y < d \<and> dist x z < e" using z apply simp
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   722
	  using w e(1) d apply (auto simp only: dist_commute)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   723
	  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
   724
	  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
   725
	  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
   726
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   727
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   728
	  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
   729
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   730
	  apply norm
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   731
	  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   732
	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
   733
	then have "\<exists>x' \<in>S. x'\<noteq>y \<and> dist x' y < d" using e by auto}
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   734
      then have "y\<in>S" by (metis islimpt_approachable [where 'a="real^'n"] cS closed_limpt[where 'a="real^'n"]) }
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   735
    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
   736
  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
   737
  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
   738
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   739
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   740
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   741
subsection{* Closure of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   742
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   743
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
   744
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   745
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
   746
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   747
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   748
    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
   749
    proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   750
      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
   751
      assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   752
      hence *:"\<not> ?exT x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   753
	unfolding interior_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   754
	by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   755
      { assume "\<not> ?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   756
	hence False using *
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   757
	  unfolding closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   758
	  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   759
      }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   760
      thus "?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   761
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   762
    next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   763
      assume "?rhs" thus "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   764
	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
   765
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   766
    qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   767
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   768
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   769
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   770
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   771
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   772
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
   773
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   774
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   775
    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
   776
      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
   777
      by blast
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
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   780
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   781
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   782
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   783
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
   784
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   785
  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
   786
  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
   787
qed
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_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
   790
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   791
  have "S \<subseteq> closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   792
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   793
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   794
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   795
  have "closed (closure S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   796
    using closed_closure[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   797
    by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   798
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   799
  { fix t
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   800
    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
   801
    { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   802
      assume "x islimpt S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   803
      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
   804
	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
   805
	by blast
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
    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
   808
      unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   809
      using closed_limpt[of t]
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   810
      by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   811
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   812
  ultimately show ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   813
    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
   814
    unfolding mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   815
    by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   816
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   817
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   818
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
   819
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   820
  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
   821
  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
   822
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   823
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
   824
  using closure_eq[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   825
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   826
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   827
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
   828
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   829
  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
   830
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   831
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   832
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
   833
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   834
  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
   835
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   836
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   837
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
   838
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   839
  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
   840
  by assumption
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
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
   843
  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
   844
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   845
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   846
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   847
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
   848
  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
   849
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   850
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   851
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   852
lemma closure_empty[simp]: "closure {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   853
  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
   854
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   855
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   856
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
   857
  using closure_closed[of UNIV]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   858
  by simp
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 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
   861
  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
   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
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
   865
  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
   866
  by simp
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 open_inter_closure_eq_empty:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   869
  "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
   870
  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
   871
  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
   872
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   873
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   874
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   875
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
   876
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   877
  fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   878
  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
   879
  { assume *:"x islimpt T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   880
    { fix e::real
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   881
      assume "e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   882
      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
   883
	unfolding open_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   884
	by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   885
      let ?e = "min e e'"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   886
      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
   887
	by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   888
      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
   889
	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
   890
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   891
      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
   892
	using y
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   893
	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
   894
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   895
    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
   896
      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
   897
      by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   898
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   899
  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
   900
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   901
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   902
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   903
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   904
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
   905
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   906
  have "S = UNIV - (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   907
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   908
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   909
    unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   910
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   911
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   912
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   913
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
   914
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   915
  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   916
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   917
subsection{* Frontier (aka boundary) *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   918
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   919
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
   920
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   921
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
   922
  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
   923
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   924
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
   925
  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
   926
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   927
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
   928
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   929
  assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   930
  { fix e::real
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   931
    assume "e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   932
    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
   933
    { assume "a\<in>S"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   934
      have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   935
      moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   936
	unfolding frontier_closures closure_def islimpt_def using `e>0`
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   937
	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
   938
      ultimately have ?rhse by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   939
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   940
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   941
    { assume "a\<notin>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   942
      hence ?rhse using `?lhs`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   943
	unfolding frontier_closures closure_def islimpt_def
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   944
	using open_ball[of a e] `e > 0`
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   945
	by (auto, erule_tac x = "ball a e" in allE, auto) (* FIXME: VERY slow! *)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   946
    }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   947
    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
   948
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   949
  thus ?rhs by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   950
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   951
  assume ?rhs
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   952
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   953
  { 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
   954
    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
   955
    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
   956
    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
   957
    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
   958
    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
   959
      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
   960
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   961
  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
   962
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   963
  { 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
   964
    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
   965
    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
   966
    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
   967
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   968
  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
   969
  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
   970
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   971
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   972
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
   973
  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
   974
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   975
lemma frontier_empty: "frontier {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   976
  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
   977
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   978
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
   979
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   980
  { assume "frontier S \<subseteq> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   981
    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
   982
    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
   983
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   984
  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
   985
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   986
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   987
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
   988
  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
   989
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   990
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   991
  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
   992
  unfolding open_closed by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   993
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   994
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
   995
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   996
typedef (open) 'a net =
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   997
  "{net :: 'a set set. (\<exists>A. A \<in> net)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   998
    \<and> (\<forall>A\<in>net. \<forall>B\<in>net. \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B)}"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   999
proof
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1000
  show "UNIV \<in> ?net" by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1001
qed
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1002
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1003
lemma Rep_net_nonempty: "\<exists>A. A \<in> Rep_net net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1004
using Rep_net [of net] by simp
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1005
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1006
lemma Rep_net_directed:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1007
  "A \<in> Rep_net net \<Longrightarrow> B \<in> Rep_net net \<Longrightarrow> \<exists>C\<in>Rep_net net. C \<subseteq> A \<and> C \<subseteq> B"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1008
using Rep_net [of net] by simp
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1009
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1010
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1011
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
  1012
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1013
definition
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1014
  at :: "'a::perfect_space \<Rightarrow> 'a net" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1015
  "at a = Abs_net ((\<lambda>r. {x. 0 < dist x a \<and> dist x a < r}) ` {r. 0 < r})"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1016
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1017
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1018
  at_infinity :: "'a::real_normed_vector net" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1019
  "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1020
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1021
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1022
  sequentially :: "nat net" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1023
  "sequentially = Abs_net (range (\<lambda>n. {n..}))"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1024
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1025
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1026
  within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1027
  "net within S = Abs_net ((\<lambda>A. A \<inter> S) ` Rep_net net)"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1028
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1029
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1030
  indirection :: "real ^'n::finite \<Rightarrow> real ^'n \<Rightarrow> (real ^'n) net" (infixr "indirection" 70) where
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1031
  "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = c*s v}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1032
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1033
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
  1034
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1035
lemma Abs_net_inverse':
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1036
  assumes "\<exists>A. A \<in> net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1037
  assumes "\<And>A B. A \<in> net \<Longrightarrow> B \<in> net \<Longrightarrow> \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B" 
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1038
  shows "Rep_net (Abs_net net) = net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1039
using assms by (simp add: Abs_net_inverse)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1040
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1041
lemma image_nonempty: "\<exists>x. x \<in> A \<Longrightarrow> \<exists>x. x \<in> f ` A"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1042
by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1043
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1044
lemma Rep_net_at:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1045
  "Rep_net (at a) = ((\<lambda>r. {x. 0 < dist x a \<and> dist x a < r}) ` {r. 0 < r})"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1046
unfolding at_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1047
apply (rule Abs_net_inverse')
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1048
apply (rule image_nonempty, rule_tac x=1 in exI, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1049
apply (clarsimp, rename_tac r s)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1050
apply (rule_tac x="min r s" in exI, auto)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1051
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1052
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1053
lemma Rep_net_at_infinity:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1054
  "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1055
unfolding at_infinity_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1056
apply (rule Abs_net_inverse')
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1057
apply (rule image_nonempty, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1058
apply (clarsimp, rename_tac r s)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1059
apply (rule_tac x="max r s" in exI, auto)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1060
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1061
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1062
lemma Rep_net_sequentially:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1063
  "Rep_net sequentially = range (\<lambda>n. {n..})"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1064
unfolding sequentially_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1065
apply (rule Abs_net_inverse')
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1066
apply (rule image_nonempty, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1067
apply (clarsimp, rename_tac m n)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1068
apply (rule_tac x="max m n" in exI, auto)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1069
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1070
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1071
lemma Rep_net_within:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1072
  "Rep_net (net within S) = (\<lambda>A. A \<inter> S) ` Rep_net net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1073
unfolding within_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1074
apply (rule Abs_net_inverse')
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1075
apply (rule image_nonempty, rule Rep_net_nonempty)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1076
apply (clarsimp, rename_tac A B)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1077
apply (drule (1) Rep_net_directed)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1078
apply (clarify, rule_tac x=C in bexI, auto)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1079
done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1080
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1081
lemma within_UNIV: "net within UNIV = net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1082
  by (simp add: Rep_net_inject [symmetric] Rep_net_within)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1083
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1084
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
  1085
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1086
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1087
  trivial_limit :: "'a net \<Rightarrow> bool" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1088
  "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1089
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1090
lemma trivial_limit_within:
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1091
  fixes a :: "'a::perfect_space"
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1092
  shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1093
proof
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1094
  assume "trivial_limit (at a within S)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1095
  thus "\<not> a islimpt S"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1096
    unfolding trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1097
    unfolding Rep_net_within Rep_net_at
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1098
    unfolding islimpt_approachable_le dist_nz
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1099
    apply (clarsimp simp add: not_le expand_set_eq)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1100
    apply (rule_tac x="r/2" in exI, clarsimp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1101
    apply (drule_tac x=x' in spec, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1102
    done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1103
next
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1104
  assume "\<not> a islimpt S"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1105
  thus "trivial_limit (at a within S)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1106
    unfolding trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1107
    unfolding Rep_net_within Rep_net_at
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1108
    unfolding islimpt_approachable_le dist_nz
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1109
    apply (clarsimp simp add: image_image not_le)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1110
    apply (rule_tac x=e in image_eqI)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1111
    apply (auto simp add: expand_set_eq)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1112
    done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1113
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1114
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1115
lemma trivial_limit_at: "\<not> trivial_limit (at a)"
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1116
  using trivial_limit_within [of a UNIV]
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1117
  by (simp add: within_UNIV)
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1118
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1119
lemma trivial_limit_at_infinity:
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1120
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1121
  unfolding trivial_limit_def Rep_net_at_infinity
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1122
  apply (clarsimp simp add: expand_set_eq)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1123
  apply (drule_tac x="scaleR r (sgn 1)" in spec)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1124
  apply (simp add: norm_scaleR norm_sgn)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1125
  done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1126
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1127
lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1128
  by (auto simp add: trivial_limit_def Rep_net_sequentially)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1129
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1130
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
  1131
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1132
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1133
  eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1134
  "eventually P net \<longleftrightarrow> (\<exists>A\<in>Rep_net net. \<forall>x\<in>A. P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1135
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1136
lemma eventually_at:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1137
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1138
unfolding eventually_def Rep_net_at by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1139
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1140
lemma eventually_at_infinity:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1141
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1142
unfolding eventually_def Rep_net_at_infinity by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1143
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1144
lemma eventually_sequentially:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1145
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1146
unfolding eventually_def Rep_net_sequentially by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1147
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1148
lemma eventually_within_eq:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1149
  "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1150
unfolding eventually_def Rep_net_within by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1151
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1152
lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1153
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1154
unfolding eventually_within_eq eventually_at by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1155
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1156
lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1157
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1158
unfolding eventually_within
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1159
apply safe
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1160
apply (rule_tac x="d/2" in exI, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1161
apply (rule_tac x="d" in exI, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1162
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1163
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1164
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1165
  unfolding eventually_def trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1166
  using Rep_net_nonempty [of net] by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1167
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1168
lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1169
  unfolding eventually_def trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1170
  using Rep_net_nonempty [of net] by auto
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1171
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1172
lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1173
  by (simp add: always_eventually)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1174
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1175
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1176
  unfolding trivial_limit_def eventually_def by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1177
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1178
lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1179
  unfolding trivial_limit_def eventually_def by auto
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1180
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1181
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1182
  apply (safe elim!: trivial_limit_eventually)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1183
  apply (simp add: eventually_False [symmetric])
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1184
  done
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1185
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1186
text{* Combining theorems for "eventually" *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1187
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1188
lemma eventually_conjI:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1189
  "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1190
    \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1191
unfolding eventually_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1192
apply (clarify, rename_tac A B)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1193
apply (drule (1) Rep_net_directed, clarify)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1194
apply (rule_tac x=C in bexI, auto)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1195
done
30262
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 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
  1198
  by (metis eventually_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1199
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1200
lemma eventually_rev_mono:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1201
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1202
using eventually_mono [of P Q] by fast
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1203
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1204
lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1205
  by (auto intro!: eventually_conjI elim: eventually_rev_mono)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1206
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1207
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
  1208
  apply (atomize(full))
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1209
  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
  1210
  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
  1211
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1212
lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1213
  by (auto simp add: eventually_False)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1214
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1215
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1216
  by (simp add: eventually_False)
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1217
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1218
lemma eventually_rev_mp:
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1219
  assumes 1: "eventually (\<lambda>x. P x) net"
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1220
  assumes 2: "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1221
  shows "eventually (\<lambda>x. Q x) net"
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1222
using 2 1 by (rule eventually_mp)
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1223
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1224
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
  1225
31343
9983f648f9bb generalize tendsto and related constants to class metric_space
huffman
parents: 31342
diff changeset
  1226
definition tendsto:: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "--->" 55) where
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1227
  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
  1228
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset