
diff -r cfb21e03fe2a -r d64a4ef26edb src/HOL/Lubs.thy
--- a/src/HOL/Lubs.thy	Thu Dec 05 17:52:12 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,103 +0,0 @@
-(*  Title:      HOL/Lubs.thy
-    Author:     Jacques D. Fleuriot, University of Cambridge
-*)
-
-header {* Definitions of Upper Bounds and Least Upper Bounds *}
-
-theory Lubs
-imports Main
-begin
-
-text {* Thanks to suggestions by James Margetson *}
-
-definition setle :: "'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"  (infixl "*<=" 70)
-  where "S *<= x = (ALL y: S. y \<le> x)"
-
-definition setge :: "'a::ord \<Rightarrow> 'a set \<Rightarrow> bool"  (infixl "<=*" 70)
-  where "x <=* S = (ALL y: S. x \<le> y)"
-
-definition leastP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
-  where "leastP P x = (P x \<and> x <=* Collect P)"
-
-definition isUb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
-  where "isUb R S x = (S *<= x \<and> x: R)"
-
-definition isLub :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
-  where "isLub R S x = leastP (isUb R S) x"
-
-definition ubs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
-  where "ubs R S = Collect (isUb R S)"
-
-
-subsection {* Rules for the Relations @{text "*<="} and @{text "<=*"} *}
-
-lemma setleI: "ALL y: S. y \<le> x \<Longrightarrow> S *<= x"
-  by (simp add: setle_def)
-
-lemma setleD: "S *<= x \<Longrightarrow> y: S \<Longrightarrow> y \<le> x"
-  by (simp add: setle_def)
-
-lemma setgeI: "ALL y: S. x \<le> y \<Longrightarrow> x <=* S"
-  by (simp add: setge_def)
-
-lemma setgeD: "x <=* S \<Longrightarrow> y: S \<Longrightarrow> x \<le> y"
-  by (simp add: setge_def)
-
-
-subsection {* Rules about the Operators @{term leastP}, @{term ub} and @{term lub} *}
-
-lemma leastPD1: "leastP P x \<Longrightarrow> P x"
-  by (simp add: leastP_def)
-
-lemma leastPD2: "leastP P x \<Longrightarrow> x <=* Collect P"
-  by (simp add: leastP_def)
-
-lemma leastPD3: "leastP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<le> y"
-  by (blast dest!: leastPD2 setgeD)
-
-lemma isLubD1: "isLub R S x \<Longrightarrow> S *<= x"
-  by (simp add: isLub_def isUb_def leastP_def)
-
-lemma isLubD1a: "isLub R S x \<Longrightarrow> x: R"
-  by (simp add: isLub_def isUb_def leastP_def)
-
-lemma isLub_isUb: "isLub R S x \<Longrightarrow> isUb R S x"
-  unfolding isUb_def by (blast dest: isLubD1 isLubD1a)
-
-lemma isLubD2: "isLub R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
-  by (blast dest!: isLubD1 setleD)
-
-lemma isLubD3: "isLub R S x \<Longrightarrow> leastP (isUb R S) x"
-  by (simp add: isLub_def)
-
-lemma isLubI1: "leastP(isUb R S) x \<Longrightarrow> isLub R S x"
-  by (simp add: isLub_def)
-
-lemma isLubI2: "isUb R S x \<Longrightarrow> x <=* Collect (isUb R S) \<Longrightarrow> isLub R S x"
-  by (simp add: isLub_def leastP_def)
-
-lemma isUbD: "isUb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
-  by (simp add: isUb_def setle_def)
-
-lemma isUbD2: "isUb R S x \<Longrightarrow> S *<= x"
-  by (simp add: isUb_def)
-
-lemma isUbD2a: "isUb R S x \<Longrightarrow> x: R"
-  by (simp add: isUb_def)
-
-lemma isUbI: "S *<= x \<Longrightarrow> x: R \<Longrightarrow> isUb R S x"
-  by (simp add: isUb_def)
-
-lemma isLub_le_isUb: "isLub R S x \<Longrightarrow> isUb R S y \<Longrightarrow> x \<le> y"
-  unfolding isLub_def by (blast intro!: leastPD3)
-
-lemma isLub_ubs: "isLub R S x \<Longrightarrow> x <=* ubs R S"
-  unfolding ubs_def isLub_def by (rule leastPD2)
-
-lemma isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::'a::linorder)"
-  apply (frule isLub_isUb)
-  apply (frule_tac x = y in isLub_isUb)
-  apply (blast intro!: order_antisym dest!: isLub_le_isUb)
-  done
-
-end




