--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Nov 05 09:44:59 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Nov 05 09:44:59 2013 +0100
@@ -1972,22 +1972,25 @@
subsection {* Infimum Distance *}
-definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"
-
-lemma bdd_below_infdist[intro, simp]: "bdd_below {dist x a|a. a \<in> A}"
+definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"
+
+lemma bdd_below_infdist[intro, simp]: "bdd_below (dist x`A)"
by (auto intro!: zero_le_dist)
-lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"
+lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"
by (simp add: infdist_def)
lemma infdist_nonneg: "0 \<le> infdist x A"
- by (auto simp add: infdist_def intro: cInf_greatest)
-
-lemma infdist_le: "a \<in> A \<Longrightarrow> d = dist x a \<Longrightarrow> infdist x A \<le> d"
- using assms by (auto intro: cInf_lower simp add: infdist_def)
+ by (auto simp add: infdist_def intro: cINF_greatest)
+
+lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"
+ by (auto intro: cINF_lower simp add: infdist_def)
+
+lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"
+ by (auto intro!: cINF_lower2 simp add: infdist_def)
lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"
- by (auto intro!: antisym infdist_nonneg infdist_le)
+ by (auto intro!: antisym infdist_nonneg infdist_le2)
lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
proof (cases "A = {}")
@@ -2006,9 +2009,8 @@
by auto
show "infdist x A \<le> d"
unfolding infdist_notempty[OF `A \<noteq> {}`]
- proof (rule cInf_lower2)
- show "dist x a \<in> {dist x a |a. a \<in> A}"
- using `a \<in> A` by auto
+ proof (rule cINF_lower2)
+ show "a \<in> A" by fact
show "dist x a \<le> d"
unfolding d by (rule dist_triangle)
qed simp
@@ -2024,7 +2026,7 @@
assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
then have "i - dist x y \<le> infdist y A"
unfolding infdist_notempty[OF `A \<noteq> {}`] using `a \<in> A`
- by (intro cInf_greatest) (auto simp: field_simps)
+ by (intro cINF_greatest) (auto simp: field_simps)
then show "i \<le> dist x y + infdist y A"
by simp
qed
@@ -2063,7 +2065,7 @@
assume "\<not> (\<exists>y\<in>A. dist y x < e)"
then have "infdist x A \<ge> e" using `a \<in> A`
unfolding infdist_def
- by (force simp: dist_commute intro: cInf_greatest)
+ by (force simp: dist_commute intro: cINF_greatest)
with x `e > 0` show False by auto
qed
qed
@@ -5701,28 +5703,27 @@
text {* We can state this in terms of diameter of a set. *}
-definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"
+definition diameter :: "'a::metric_space set \<Rightarrow> real" where
+ "diameter S = (if S = {} then 0 else SUP (x,y):S\<times>S. dist x y)"
lemma diameter_bounded_bound:
fixes s :: "'a :: metric_space set"
assumes s: "bounded s" "x \<in> s" "y \<in> s"
shows "dist x y \<le> diameter s"
proof -
- let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
unfolding bounded_def by auto
- have "dist x y \<le> Sup ?D"
- proof (rule cSup_upper)
- show "bdd_above ?D"
- unfolding bdd_above_def
- proof (safe intro!: exI)
- fix a b
- assume "a \<in> s" "b \<in> s"
- with z[of a] z[of b] dist_triangle[of a b z]
- show "dist a b \<le> 2 * d"
- by (simp add: dist_commute)
- qed
- qed (insert s, auto)
+ have "bdd_above (split dist ` (s\<times>s))"
+ proof (intro bdd_aboveI, safe)
+ fix a b
+ assume "a \<in> s" "b \<in> s"
+ with z[of a] z[of b] dist_triangle[of a b z]
+ show "dist a b \<le> 2 * d"
+ by (simp add: dist_commute)
+ qed
+ moreover have "(x,y) \<in> s\<times>s" using s by auto
+ ultimately have "dist x y \<le> (SUP (x,y):s\<times>s. dist x y)"
+ by (rule cSUP_upper2) simp
with `x \<in> s` show ?thesis
by (auto simp add: diameter_def)
qed
@@ -5733,16 +5734,12 @@
and d: "0 < d" "d < diameter s"
shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"
proof (rule ccontr)
- let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
assume contr: "\<not> ?thesis"
- moreover
- from d have "s \<noteq> {}"
- by (auto simp: diameter_def)
- then have "?D \<noteq> {}" by auto
- ultimately have "Sup ?D \<le> d"
- by (intro cSup_least) (auto simp: not_less)
- with `d < diameter s` `s \<noteq> {}` show False
- by (auto simp: diameter_def)
+ moreover have "s \<noteq> {}"
+ using d by (auto simp add: diameter_def)
+ ultimately have "diameter s \<le> d"
+ by (auto simp: not_less diameter_def intro!: cSUP_least)
+ with `d < diameter s` show False by auto
qed
lemma diameter_bounded:
@@ -5765,7 +5762,7 @@
then have "diameter s \<le> dist x y"
unfolding diameter_def
apply clarsimp
- apply (rule cSup_least)
+ apply (rule cSUP_least)
apply fast+
done
then show ?thesis