--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Glbs.thy Mon Feb 09 16:19:46 2009 +0000
@@ -0,0 +1,85 @@
+(* Title: Glbs
+ ID: $Id:
+ Author: Amine Chaieb, University of Cambridge
+*)
+
+header{*Definitions of Lower Bounds and Greatest Lower Bounds, analogous to Lubs*}
+
+theory Glbs
+imports Lubs
+begin
+
+definition
+ greatestP :: "['a =>bool,'a::ord] => bool" where
+ "greatestP P x = (P x & Collect P *<= x)"
+
+definition
+ isLb :: "['a set, 'a set, 'a::ord] => bool" where
+ "isLb R S x = (x <=* S & x: R)"
+
+definition
+ isGlb :: "['a set, 'a set, 'a::ord] => bool" where
+ "isGlb R S x = greatestP (isLb R S) x"
+
+definition
+ lbs :: "['a set, 'a::ord set] => 'a set" where
+ "lbs R S = Collect (isLb R S)"
+
+subsection{*Rules about the Operators @{term greatestP}, @{term isLb}
+ and @{term isGlb}*}
+
+lemma leastPD1: "greatestP P x ==> P x"
+by (simp add: greatestP_def)
+
+lemma greatestPD2: "greatestP P x ==> Collect P *<= x"
+by (simp add: greatestP_def)
+
+lemma greatestPD3: "[| greatestP P x; y: Collect P |] ==> x >= y"
+by (blast dest!: greatestPD2 setleD)
+
+lemma isGlbD1: "isGlb R S x ==> x <=* S"
+by (simp add: isGlb_def isLb_def greatestP_def)
+
+lemma isGlbD1a: "isGlb R S x ==> x: R"
+by (simp add: isGlb_def isLb_def greatestP_def)
+
+lemma isGlb_isLb: "isGlb R S x ==> isLb R S x"
+apply (simp add: isLb_def)
+apply (blast dest: isGlbD1 isGlbD1a)
+done
+
+lemma isGlbD2: "[| isGlb R S x; y : S |] ==> y >= x"
+by (blast dest!: isGlbD1 setgeD)
+
+lemma isGlbD3: "isGlb R S x ==> greatestP(isLb R S) x"
+by (simp add: isGlb_def)
+
+lemma isGlbI1: "greatestP(isLb R S) x ==> isGlb R S x"
+by (simp add: isGlb_def)
+
+lemma isGlbI2: "[| isLb R S x; Collect (isLb R S) *<= x |] ==> isGlb R S x"
+by (simp add: isGlb_def greatestP_def)
+
+lemma isLbD: "[| isLb R S x; y : S |] ==> y >= x"
+by (simp add: isLb_def setge_def)
+
+lemma isLbD2: "isLb R S x ==> x <=* S "
+by (simp add: isLb_def)
+
+lemma isLbD2a: "isLb R S x ==> x: R"
+by (simp add: isLb_def)
+
+lemma isLbI: "[| x <=* S ; x: R |] ==> isLb R S x"
+by (simp add: isLb_def)
+
+lemma isGlb_le_isLb: "[| isGlb R S x; isLb R S y |] ==> x >= y"
+apply (simp add: isGlb_def)
+apply (blast intro!: greatestPD3)
+done
+
+lemma isGlb_ubs: "isGlb R S x ==> lbs R S *<= x"
+apply (simp add: lbs_def isGlb_def)
+apply (erule greatestPD2)
+done
+
+end