--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Wellfounded_More.thy Wed Sep 12 05:29:21 2012 +0200
@@ -0,0 +1,50 @@
+(* Title: HOL/Cardinals/Wellfounded_More.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+More on well-founded relations.
+*)
+
+header {* More on Well-Founded Relations *}
+
+theory Wellfounded_More
+imports Wellfounded_More_Base Order_Relation_More
+begin
+
+
+subsection {* Well-founded recursion via genuine fixpoints *}
+
+(*2*)lemma adm_wf_unique_fixpoint:
+fixes r :: "('a * 'a) set" and
+ H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" and
+ f :: "'a \<Rightarrow> 'b" and g :: "'a \<Rightarrow> 'b"
+assumes WF: "wf r" and ADM: "adm_wf r H" and fFP: "f = H f" and gFP: "g = H g"
+shows "f = g"
+proof-
+ {fix x
+ have "f x = g x"
+ proof(rule wf_induct[of r "(\<lambda>x. f x = g x)"],
+ auto simp add: WF)
+ fix x assume "\<forall>y. (y, x) \<in> r \<longrightarrow> f y = g y"
+ hence "H f x = H g x" using ADM adm_wf_def[of r H] by auto
+ thus "f x = g x" using fFP and gFP by simp
+ qed
+ }
+ thus ?thesis by (simp add: ext)
+qed
+
+(*2*)lemma wfrec_unique_fixpoint:
+fixes r :: "('a * 'a) set" and
+ H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" and
+ f :: "'a \<Rightarrow> 'b"
+assumes WF: "wf r" and ADM: "adm_wf r H" and
+ fp: "f = H f"
+shows "f = wfrec r H"
+proof-
+ have "H (wfrec r H) = wfrec r H"
+ using assms wfrec_fixpoint[of r H] by simp
+ thus ?thesis
+ using assms adm_wf_unique_fixpoint[of r H "wfrec r H"] by simp
+qed
+
+end