author huffman Thu Jan 17 08:31:16 2013 -0800 (2013-01-17) changeset 50955 ada575c605e1 parent 50954 7bc58677860e child 50963 23ed79fc2b4d
simplify proof of compact_imp_bounded
```     1.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Thu Jan 17 19:20:56 2013 +0100
1.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Thu Jan 17 08:31:16 2013 -0800
1.3 @@ -2205,6 +2205,9 @@
1.4  lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1.5    by (induct rule: finite_induct[of F], auto)
1.6
1.7 +lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
1.8 +  by (induct set: finite, auto)
1.9 +
1.10  lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
1.11  proof -
1.12    have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp
1.13 @@ -2583,21 +2586,10 @@
1.14    have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto
1.15    then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
1.16      by (elim compactE_image)
1.17 -  def d \<equiv> "SOME d. d \<in> D"
1.18 -  show "bounded U"
1.19 -    unfolding bounded_def
1.20 -  proof (intro exI, safe)
1.21 -    fix x assume "x \<in> U"
1.22 -    with D obtain d' where "d' \<in> D" "x \<in> ball d' 1" by auto
1.23 -    moreover have "dist d x \<le> dist d d' + dist d' x"
1.24 -      using dist_triangle[of d x d'] by (simp add: dist_commute)
1.25 -    moreover
1.26 -    from `x\<in>U` D have "d \<in> D"
1.27 -      unfolding d_def by (rule_tac someI_ex) auto
1.28 -    ultimately
1.29 -    show "dist d x \<le> Max ((\<lambda>d'. dist d d' + 1) ` D)"
1.30 -      using D by (subst Max_ge_iff) (auto intro!: bexI[of _ d'])
1.31 -  qed
1.32 +  from `finite D` have "bounded (\<Union>x\<in>D. ball x 1)"
1.33 +    by (simp add: bounded_UN)
1.34 +  thus "bounded U" using `U \<subseteq> (\<Union>x\<in>D. ball x 1)`
1.35 +    by (rule bounded_subset)
1.36  qed
1.37
1.38  text{* In particular, some common special cases. *}
```