src/HOL/Real/Lubs.thy
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(*  Title       : Lubs.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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*)
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header{*Definitions of Upper Bounds and Least Upper Bounds*}
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theory Lubs
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imports Main
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begin
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text{*Thanks to suggestions by James Margetson*}
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definition
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  setle :: "['a set, 'a::ord] => bool"  (infixl "*<=" 70) where
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  "S *<= x = (ALL y: S. y <= x)"
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definition
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  setge :: "['a::ord, 'a set] => bool"  (infixl "<=*" 70) where
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  "x <=* S = (ALL y: S. x <= y)"
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definition
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  leastP      :: "['a =>bool,'a::ord] => bool" where
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  "leastP P x = (P x & x <=* Collect P)"
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definition
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  isUb        :: "['a set, 'a set, 'a::ord] => bool" where
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  "isUb R S x = (S *<= x & x: R)"
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definition
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  isLub       :: "['a set, 'a set, 'a::ord] => bool" where
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  "isLub R S x = leastP (isUb R S) x"
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definition
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  ubs         :: "['a set, 'a::ord set] => 'a set" where
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  "ubs R S = Collect (isUb R S)"
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subsection{*Rules for the Relations @{text "*<="} and @{text "<=*"}*}
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lemma setleI: "ALL y: S. y <= x ==> S *<= x"
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by (simp add: setle_def)
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lemma setleD: "[| S *<= x; y: S |] ==> y <= x"
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by (simp add: setle_def)
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lemma setgeI: "ALL y: S. x<= y ==> x <=* S"
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by (simp add: setge_def)
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lemma setgeD: "[| x <=* S; y: S |] ==> x <= y"
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by (simp add: setge_def)
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subsection{*Rules about the Operators @{term leastP}, @{term ub}
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    and @{term lub}*}
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lemma leastPD1: "leastP P x ==> P x"
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lemma leastPD2: "leastP P x ==> x <=* Collect P"
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lemma leastPD3: "[| leastP P x; y: Collect P |] ==> x <= y"
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by (blast dest!: leastPD2 setgeD)
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lemma isLubD1: "isLub R S x ==> S *<= x"
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by (simp add: isLub_def isUb_def leastP_def)
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lemma isLubD1a: "isLub R S x ==> x: R"
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by (simp add: isLub_def isUb_def leastP_def)
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lemma isLub_isUb: "isLub R S x ==> isUb R S x"
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apply (simp add: isUb_def)
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apply (blast dest: isLubD1 isLubD1a)
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done
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lemma isLubD2: "[| isLub R S x; y : S |] ==> y <= x"
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by (blast dest!: isLubD1 setleD)
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lemma isLubD3: "isLub R S x ==> leastP(isUb R S) x"
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by (simp add: isLub_def)
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lemma isLubI1: "leastP(isUb R S) x ==> isLub R S x"
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by (simp add: isLub_def)
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lemma isLubI2: "[| isUb R S x; x <=* Collect (isUb R S) |] ==> isLub R S x"
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by (simp add: isLub_def leastP_def)
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lemma isUbD: "[| isUb R S x; y : S |] ==> y <= x"
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by (simp add: isUb_def setle_def)
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lemma isUbD2: "isUb R S x ==> S *<= x"
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lemma isUbD2a: "isUb R S x ==> x: R"
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lemma isUbI: "[| S *<= x; x: R |] ==> isUb R S x"
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lemma isLub_le_isUb: "[| isLub R S x; isUb R S y |] ==> x <= y"
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apply (simp add: isLub_def)
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apply (blast intro!: leastPD3)
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done
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lemma isLub_ubs: "isLub R S x ==> x <=* ubs R S"
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apply (simp add: ubs_def isLub_def)
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apply (erule leastPD2)
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done
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end