--- a/src/HOL/Library/Extended_Real.thy Thu Apr 25 10:35:56 2013 +0200
+++ b/src/HOL/Library/Extended_Real.thy Thu Apr 25 11:59:21 2013 +0200
@@ -1151,6 +1151,12 @@
instance ereal :: complete_linorder ..
+instance ereal :: linear_continuum
+proof
+ show "\<exists>a b::ereal. a \<noteq> b"
+ using ereal_zero_one by blast
+qed
+
lemma ereal_Sup_uminus_image_eq: "Sup (uminus ` S::ereal set) = - Inf S"
by (auto intro!: Sup_eqI
simp: Ball_def[symmetric] ereal_uminus_le_reorder le_Inf_iff
@@ -1572,7 +1578,7 @@
subsubsection "Topological space"
-instantiation ereal :: connected_linorder_topology
+instantiation ereal :: linear_continuum_topology
begin
definition "open_ereal" :: "ereal set \<Rightarrow> bool" where
@@ -1697,31 +1703,6 @@
show thesis by auto
qed
-instance ereal :: perfect_space
-proof (default, rule)
- fix a :: ereal assume a: "open {a}"
- show False
- proof (cases a)
- case MInf
- then obtain y where "{..<ereal y} \<le> {a}" using a open_MInfty2[of "{a}"] by auto
- then have "ereal (y - 1) \<in> {a}" apply (subst subsetD[of "{..<ereal y}"]) by auto
- then show False using `a = -\<infinity>` by auto
- next
- case PInf
- then obtain y where "{ereal y<..} \<le> {a}" using a open_PInfty2[of "{a}"] by auto
- then have "ereal (y + 1) \<in> {a}" apply (subst subsetD[of "{ereal y<..}"]) by auto
- then show False using `a = \<infinity>` by auto
- next
- case (real r)
- then have fin: "\<bar>a\<bar> \<noteq> \<infinity>" by simp
- from ereal_open_cont_interval[OF a singletonI this] guess e . note e = this
- then obtain b where b_def: "a<b \<and> b<a+e"
- using fin ereal_between dense[of a "a+e"] by auto
- then have "b: {a-e <..< a+e}" using fin ereal_between[of a e] e by auto
- then show False using b_def e by auto
- qed
-qed
-
subsubsection {* Convergent sequences *}
lemma lim_ereal[simp]: