(* Title : Lubs.thy
Author : Jacques D. Fleuriot
Copyright : 1998 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, 'a::ord] => bool" (infixl "*<=" 70) where
"S *<= x = (ALL y: S. y <= x)"
definition
setge :: "['a::ord, 'a set] => bool" (infixl "<=*" 70) where
"x <=* S = (ALL y: S. x <= y)"
definition
leastP :: "['a =>bool,'a::ord] => bool" where
"leastP P x = (P x & x <=* Collect P)"
definition
isUb :: "['a set, 'a set, 'a::ord] => bool" where
"isUb R S x = (S *<= x & x: R)"
definition
isLub :: "['a set, 'a set, 'a::ord] => bool" where
"isLub R S x = leastP (isUb R S) x"
definition
ubs :: "['a set, 'a::ord set] => 'a set" where
"ubs R S = Collect (isUb R S)"
subsection{*Rules for the Relations @{text "*<="} and @{text "<=*"}*}
lemma setleI: "ALL y: S. y <= x ==> S *<= x"
by (simp add: setle_def)
lemma setleD: "[| S *<= x; y: S |] ==> y <= x"
by (simp add: setle_def)
lemma setgeI: "ALL y: S. x<= y ==> x <=* S"
by (simp add: setge_def)
lemma setgeD: "[| x <=* S; y: S |] ==> x <= y"
by (simp add: setge_def)
subsection{*Rules about the Operators @{term leastP}, @{term ub}
and @{term lub}*}
lemma leastPD1: "leastP P x ==> P x"
by (simp add: leastP_def)
lemma leastPD2: "leastP P x ==> x <=* Collect P"
by (simp add: leastP_def)
lemma leastPD3: "[| leastP P x; y: Collect P |] ==> x <= y"
by (blast dest!: leastPD2 setgeD)
lemma isLubD1: "isLub R S x ==> S *<= x"
by (simp add: isLub_def isUb_def leastP_def)
lemma isLubD1a: "isLub R S x ==> x: R"
by (simp add: isLub_def isUb_def leastP_def)
lemma isLub_isUb: "isLub R S x ==> isUb R S x"
apply (simp add: isUb_def)
apply (blast dest: isLubD1 isLubD1a)
done
lemma isLubD2: "[| isLub R S x; y : S |] ==> y <= x"
by (blast dest!: isLubD1 setleD)
lemma isLubD3: "isLub R S x ==> leastP(isUb R S) x"
by (simp add: isLub_def)
lemma isLubI1: "leastP(isUb R S) x ==> isLub R S x"
by (simp add: isLub_def)
lemma isLubI2: "[| isUb R S x; x <=* Collect (isUb R S) |] ==> isLub R S x"
by (simp add: isLub_def leastP_def)
lemma isUbD: "[| isUb R S x; y : S |] ==> y <= x"
by (simp add: isUb_def setle_def)
lemma isUbD2: "isUb R S x ==> S *<= x"
by (simp add: isUb_def)
lemma isUbD2a: "isUb R S x ==> x: R"
by (simp add: isUb_def)
lemma isUbI: "[| S *<= x; x: R |] ==> isUb R S x"
by (simp add: isUb_def)
lemma isLub_le_isUb: "[| isLub R S x; isUb R S y |] ==> x <= y"
apply (simp add: isLub_def)
apply (blast intro!: leastPD3)
done
lemma isLub_ubs: "isLub R S x ==> x <=* ubs R S"
apply (simp add: ubs_def isLub_def)
apply (erule leastPD2)
done
end