--- a/src/HOL/Library/Infinite_Set.thy Thu Jul 30 14:02:19 2015 +0200
+++ b/src/HOL/Library/Infinite_Set.thy Thu Jul 30 21:56:19 2015 +0200
@@ -62,6 +62,16 @@
with S show False by simp
qed
+lemma infinite_coinduct [consumes 1, case_names infinite]:
+ assumes "X A"
+ and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
+ shows "infinite A"
+proof
+ assume "finite A"
+ then show False using \<open>X A\<close>
+ by(induction rule: finite_psubset_induct)(meson Diff_subset card_Diff1_less card_psubset finite_Diff step)
+qed
+
text \<open>
As a concrete example, we prove that the set of natural numbers is
infinite.