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(* Title : NSA.thy
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Author : Jacques D. Fleuriot
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Copyright : 1998 University of Cambridge
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Description : Infinite numbers, Infinitesimals,
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infinitely close relation etc.
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*)
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NSA = HRealAbs + RComplete +
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constdefs
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Infinitesimal :: "hypreal set"
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"Infinitesimal == {x. ALL r: SReal. 0 < r --> abs x < r}"
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HFinite :: "hypreal set"
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"HFinite == {x. EX r: SReal. abs x < r}"
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HInfinite :: "hypreal set"
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"HInfinite == {x. ALL r: SReal. r < abs x}"
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(* standard part map *)
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st :: hypreal => hypreal
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"st == (%x. @r. x : HFinite & r:SReal & r @= x)"
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monad :: hypreal => hypreal set
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"monad x == {y. x @= y}"
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galaxy :: hypreal => hypreal set
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"galaxy x == {y. (x + -y) : HFinite}"
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(* infinitely close *)
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inf_close :: "[hypreal, hypreal] => bool" (infixl "@=" 50)
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"x @= y == (x + -y) : Infinitesimal"
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defs
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(*standard real numbers as a subset of the hyperreals*)
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SReal_def "SReal == {x. EX r. x = hypreal_of_real r}"
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syntax (symbols)
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inf_close :: "[hypreal, hypreal] => bool" (infixl "\\<approx>" 50)
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end
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