--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Wed Dec 04 12:44:38 2019 +0000
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Wed Dec 04 12:44:47 2019 +0000
@@ -455,6 +455,11 @@
shows "closure S \<subseteq> affine hull S"
by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
+lemma closed_affine_hull [iff]:
+ fixes S :: "'n::euclidean_space set"
+ shows "closed (affine hull S)"
+ by (metis affine_affine_hull affine_closed)
+
lemma closure_same_affine_hull [simp]:
fixes S :: "'n::euclidean_space set"
shows "affine hull (closure S) = affine hull S"
--- a/src/HOL/Analysis/Starlike.thy Wed Dec 04 12:44:38 2019 +0000
+++ b/src/HOL/Analysis/Starlike.thy Wed Dec 04 12:44:47 2019 +0000
@@ -1296,11 +1296,6 @@
shows "rel_frontier((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
-lemma closed_affine_hull [iff]:
- fixes S :: "'n::euclidean_space set"
- shows "closed (affine hull S)"
- by (metis affine_affine_hull affine_closed)
-
lemma rel_frontier_nonempty_interior:
fixes S :: "'n::euclidean_space set"
shows "interior S \<noteq> {} \<Longrightarrow> rel_frontier S = frontier S"