src/HOL/IMP/Complete_Lattice.thy

+theory Complete_Lattice
+imports Main
+begin
+
+locale Complete_Lattice =
+fixes L :: "'a::order set" and Glb :: "'a set \<Rightarrow> 'a"
+assumes Glb_lower: "A \<subseteq> L \<Longrightarrow> a \<in> A \<Longrightarrow> Glb A \<le> a"
+and Glb_greatest: "b : L \<Longrightarrow> \<forall>a\<in>A. b \<le> a \<Longrightarrow> b \<le> Glb A"
+and Glb_in_L: "A \<subseteq> L \<Longrightarrow> Glb A : L"
+begin
+
+definition lfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" where
+"lfp f = Glb {a : L. f a \<le> a}"
+
+lemma index_lfp: "lfp f : L"
+by(auto simp: lfp_def intro: Glb_in_L)
+
+lemma lfp_lowerbound:
+  "\<lbrakk> a : L;  f a \<le> a \<rbrakk> \<Longrightarrow> lfp f \<le> a"
+by (auto simp add: lfp_def intro: Glb_lower)
+
+lemma lfp_greatest:
+  "\<lbrakk> a : L;  \<And>u. \<lbrakk> u : L; f u \<le> u\<rbrakk> \<Longrightarrow> a \<le> u \<rbrakk> \<Longrightarrow> a \<le> lfp f"
+by (auto simp add: lfp_def intro: Glb_greatest)
+
+lemma lfp_unfold: assumes "\<And>x. f x : L \<longleftrightarrow> x : L"
+and mono: "mono f" shows "lfp f = f (lfp f)"
+proof-
+  note assms(1)[simp] index_lfp[simp]
+  have 1: "f (lfp f) \<le> lfp f"
+    apply(rule lfp_greatest)
+    apply simp
+    by (blast intro: lfp_lowerbound monoD[OF mono] order_trans)
+  have "lfp f \<le> f (lfp f)"
+    by (fastforce intro: 1 monoD[OF mono] lfp_lowerbound)
+  with 1 show ?thesis by(blast intro: order_antisym)
+qed
+
+end
+
+end
+