--- a/src/HOL/Orderings.thy Sat Jul 16 21:53:50 2011 +0200
+++ b/src/HOL/Orderings.thy Sat Jul 16 22:04:02 2011 +0200
@@ -1084,35 +1084,54 @@
subsection {* (Unique) top and bottom elements *}
class bot = order +
- fixes bot :: 'a
- assumes bot_least [simp]: "bot \<le> x"
+ fixes bot :: 'a ("\<bottom>")
+ assumes bot_least [simp]: "\<bottom> \<le> a"
begin
+lemma le_bot:
+ "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
+ by (auto intro: antisym)
+
lemma bot_unique:
- "a \<le> bot \<longleftrightarrow> a = bot"
- by (auto simp add: intro: antisym)
+ "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
+ by (auto intro: antisym)
+
+lemma not_less_bot [simp]:
+ "\<not> (a < \<bottom>)"
+ using bot_least [of a] by (auto simp: le_less)
lemma bot_less:
- "a \<noteq> bot \<longleftrightarrow> bot < a"
+ "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
by (auto simp add: less_le_not_le intro!: antisym)
end
class top = order +
- fixes top :: 'a
- assumes top_greatest [simp]: "x \<le> top"
+ fixes top :: 'a ("\<top>")
+ assumes top_greatest [simp]: "a \<le> \<top>"
begin
+lemma top_le:
+ "\<top> \<le> a \<Longrightarrow> a = \<top>"
+ by (rule antisym) auto
+
lemma top_unique:
- "top \<le> a \<longleftrightarrow> a = top"
- by (auto simp add: intro: antisym)
+ "\<top> \<le> a \<longleftrightarrow> a = \<top>"
+ by (auto intro: antisym)
+
+lemma not_top_less [simp]: "\<not> (\<top> < a)"
+ using top_greatest [of a] by (auto simp: le_less)
lemma less_top:
- "a \<noteq> top \<longleftrightarrow> a < top"
+ "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
by (auto simp add: less_le_not_le intro!: antisym)
end
+no_notation
+ bot ("\<bottom>") and
+ top ("\<top>")
+
subsection {* Dense orders *}