equal
deleted
inserted
replaced
5298 |
5298 |
5299 lemma compact_eq_seq_compact_metric: |
5299 lemma compact_eq_seq_compact_metric: |
5300 "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s" |
5300 "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s" |
5301 using compact_imp_seq_compact seq_compact_imp_heine_borel by blast |
5301 using compact_imp_seq_compact seq_compact_imp_heine_borel by blast |
5302 |
5302 |
5303 lemma compact_def: --\<open>this is the definition of compactness in HOL Light\<close> |
5303 lemma compact_def: \<comment>\<open>this is the definition of compactness in HOL Light\<close> |
5304 "compact (S :: 'a::metric_space set) \<longleftrightarrow> |
5304 "compact (S :: 'a::metric_space set) \<longleftrightarrow> |
5305 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))" |
5305 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))" |
5306 unfolding compact_eq_seq_compact_metric seq_compact_def by auto |
5306 unfolding compact_eq_seq_compact_metric seq_compact_def by auto |
5307 |
5307 |
5308 subsubsection \<open>Complete the chain of compactness variants\<close> |
5308 subsubsection \<open>Complete the chain of compactness variants\<close> |