--- a/src/HOL/Library/Glbs.thy Tue Nov 05 09:45:00 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,79 +0,0 @@
-(* 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 \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
- where "greatestP P x = (P x \<and> Collect P *<= x)"
-
-definition isLb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
- where "isLb R S x = (x <=* S \<and> x: R)"
-
-definition isGlb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
- where "isGlb R S x = greatestP (isLb R S) x"
-
-definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> '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 \<Longrightarrow> P x"
- by (simp add: greatestP_def)
-
-lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x"
- by (simp add: greatestP_def)
-
-lemma greatestPD3: "greatestP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<ge> y"
- by (blast dest!: greatestPD2 setleD)
-
-lemma isGlbD1: "isGlb R S x \<Longrightarrow> x <=* S"
- by (simp add: isGlb_def isLb_def greatestP_def)
-
-lemma isGlbD1a: "isGlb R S x \<Longrightarrow> x: R"
- by (simp add: isGlb_def isLb_def greatestP_def)
-
-lemma isGlb_isLb: "isGlb R S x \<Longrightarrow> isLb R S x"
- unfolding isLb_def by (blast dest: isGlbD1 isGlbD1a)
-
-lemma isGlbD2: "isGlb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
- by (blast dest!: isGlbD1 setgeD)
-
-lemma isGlbD3: "isGlb R S x \<Longrightarrow> greatestP (isLb R S) x"
- by (simp add: isGlb_def)
-
-lemma isGlbI1: "greatestP (isLb R S) x \<Longrightarrow> isGlb R S x"
- by (simp add: isGlb_def)
-
-lemma isGlbI2: "isLb R S x \<Longrightarrow> Collect (isLb R S) *<= x \<Longrightarrow> isGlb R S x"
- by (simp add: isGlb_def greatestP_def)
-
-lemma isLbD: "isLb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
- by (simp add: isLb_def setge_def)
-
-lemma isLbD2: "isLb R S x \<Longrightarrow> x <=* S "
- by (simp add: isLb_def)
-
-lemma isLbD2a: "isLb R S x \<Longrightarrow> x: R"
- by (simp add: isLb_def)
-
-lemma isLbI: "x <=* S \<Longrightarrow> x: R \<Longrightarrow> isLb R S x"
- by (simp add: isLb_def)
-
-lemma isGlb_le_isLb: "isGlb R S x \<Longrightarrow> isLb R S y \<Longrightarrow> x \<ge> y"
- unfolding isGlb_def by (blast intro!: greatestPD3)
-
-lemma isGlb_ubs: "isGlb R S x \<Longrightarrow> lbs R S *<= x"
- unfolding lbs_def isGlb_def by (rule greatestPD2)
-
-lemma isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::'a::linorder)"
- apply (frule isGlb_isLb)
- apply (frule_tac x = y in isGlb_isLb)
- apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
- done
-
-end