src/HOL/Real/Lubs.thy
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(*  Title       : Lubs.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Description : Upper bounds, lubs definitions
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                  suggested by James Margetson
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*) 
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Lubs = Set +
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consts
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    "*<=" :: ['a set, 'a] => bool     (infixl 70)
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    "<=*" :: ['a, 'a set] => bool     (infixl 70)
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constdefs
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    leastP      :: ['a =>bool,'a] => bool
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    "leastP P x == (P x & x <=* Collect P)"
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    isLub       :: ['a set, 'a set, 'a] => bool    
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    "isLub R S x  == leastP (isUb R S) x"
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    isUb        :: ['a set, 'a set, 'a] => bool     
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    "isUb R S x   == S *<= x & x: R"
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    ubs         :: ['a set, 'a set] => 'a set
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    "ubs R S      == Collect (isUb R S)"
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defs
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    setle_def
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    "S *<= x    == (ALL y: S. y <= x)"
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    setge_def
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    "x <=* S    == (ALL y: S. x <= y)"
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end                    
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