
diff -r c197b3c3e7fa -r ff570720ba1c src/HOL/IMP/Complete_Lattice_ix.thy
--- a/src/HOL/IMP/Complete_Lattice_ix.thy	Thu Aug 09 22:31:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,61 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-header "Abstract Interpretation"
-
-theory Complete_Lattice_ix
-imports Main
-begin
-
-subsection "Complete Lattice (indexed)"
-
-text{* A complete lattice is an ordered type where every set of elements has
-a greatest lower (and thus also a leats upper) bound. Sets are the
-prototypical complete lattice where the greatest lower bound is
-intersection. Sometimes that set of all elements of a type is not a complete
-lattice although all elements of the same shape form a complete lattice, for
-example lists of the same length, where the list elements come from a
-complete lattice. We will have exactly this situation with annotated
-commands. This theory introduces a slightly generalised version of complete
-lattices where elements have an ``index'' and only the set of elements with
-the same index form a complete lattice; the type as a whole is a disjoint
-union of complete lattices. Because sets are not types, this requires a
-special treatment. *}
-
-locale Complete_Lattice_ix =
-fixes L :: "'i \<Rightarrow> 'a::order set"
-and Glb :: "'i \<Rightarrow> 'a set \<Rightarrow> 'a"
-assumes Glb_lower: "A \<subseteq> L i \<Longrightarrow> a \<in> A \<Longrightarrow> (Glb i A) \<le> a"
-and Glb_greatest: "b : L i \<Longrightarrow> \<forall>a\<in>A. b \<le> a \<Longrightarrow> b \<le> (Glb i A)"
-and Glb_in_L: "A \<subseteq> L i \<Longrightarrow> Glb i A : L i"
-begin
-
-definition lfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'i \<Rightarrow> 'a" where
-"lfp f i = Glb i {a : L i. f a \<le> a}"
-
-lemma index_lfp: "lfp f i : L i"
-by(auto simp: lfp_def intro: Glb_in_L)
-
-lemma lfp_lowerbound:
-  "\<lbrakk> a : L i;  f a \<le> a \<rbrakk> \<Longrightarrow> lfp f i \<le> a"
-by (auto simp add: lfp_def intro: Glb_lower)
-
-lemma lfp_greatest:
-  "\<lbrakk> a : L i;  \<And>u. \<lbrakk> u : L i; f u \<le> u\<rbrakk> \<Longrightarrow> a \<le> u \<rbrakk> \<Longrightarrow> a \<le> lfp f i"
-by (auto simp add: lfp_def intro: Glb_greatest)
-
-lemma lfp_unfold: assumes "\<And>x i. f x : L i \<longleftrightarrow> x : L i"
-and mono: "mono f" shows "lfp f i = f (lfp f i)"
-proof-
-  note assms(1)[simp] index_lfp[simp]
-  have 1: "f (lfp f i) \<le> lfp f i"
-    apply(rule lfp_greatest)
-    apply simp
-    by (blast intro: lfp_lowerbound monoD[OF mono] order_trans)
-  have "lfp f i \<le> f (lfp f i)"
-    by (fastforce intro: 1 monoD[OF mono] lfp_lowerbound)
-  with 1 show ?thesis by(blast intro: order_antisym)
-qed
-
-end
-
-end




