src/HOL/Library/Glbs.thy
 changeset 29838 a562ca0c408d child 30267 171b3bd93c90
```--- /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```