--- a/src/HOL/Library/More_List.thy Fri Oct 14 18:55:29 2011 +0200
+++ b/src/HOL/Library/More_List.thy Fri Oct 14 18:55:59 2011 +0200
@@ -295,4 +295,37 @@
"nth_map (Suc n) f (x # xs) = x # nth_map n f xs"
by (simp add: nth_map_def)
+text {* Enumeration of all sublists of a list *}
+
+primrec sublists :: "'a list \<Rightarrow> 'a list list" where
+ "sublists [] = [[]]"
+ | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
+
+lemma length_sublists:
+ "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
+ by (induct xs) (simp_all add: Let_def)
+
+lemma sublists_powset:
+ "set ` set (sublists xs) = Pow (set xs)"
+proof -
+ have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
+ by (auto simp add: image_def)
+ have "set (map set (sublists xs)) = Pow (set xs)"
+ by (induct xs)
+ (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
+ then show ?thesis by simp
+qed
+
+lemma distinct_set_sublists:
+ assumes "distinct xs"
+ shows "distinct (map set (sublists xs))"
+proof (rule card_distinct)
+ have "finite (set xs)" by rule
+ then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
+ with assms distinct_card [of xs]
+ have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
+ then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
+ by (simp add: sublists_powset length_sublists)
+qed
+
end