author | paulson |
Tue, 12 Dec 2000 12:01:19 +0100 | |
changeset 10648 | a8c647cfa31f |
parent 10607 | 352f6f209775 |
child 10677 | 36625483213f |
permissions | -rw-r--r-- |
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(* Title : Lim.thy |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : Theory of limits, continuity and |
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differentiation of real=>real functions |
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*) |
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Lim = SEQ + RealAbs + |
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(*----------------------------------------------------------------------- |
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Limits, continuity and differentiation: standard and NS definitions |
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-----------------------------------------------------------------------*) |
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constdefs |
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LIM :: [real=>real,real,real] => bool |
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("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) |
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"f -- a --> L == |
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ALL r. #0 < r --> |
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(EX s. #0 < s & (ALL x. (x ~= a & (abs(x + -a) < s) |
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--> abs(f x + -L) < r)))" |
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NSLIM :: [real=>real,real,real] => bool |
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("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60) |
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"f -- a --NS> L == (ALL x. (x ~= hypreal_of_real a & |
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x @= hypreal_of_real a --> (*f* f) x @= hypreal_of_real L))" |
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isCont :: [real=>real,real] => bool |
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"isCont f a == (f -- a --> (f a))" |
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(* NS definition dispenses with limit notions *) |
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isNSCont :: [real=>real,real] => bool |
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"isNSCont f a == (ALL y. y @= hypreal_of_real a --> |
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(*f* f) y @= hypreal_of_real (f a))" |
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(* differentiation: D is derivative of function f at x *) |
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deriv:: [real=>real,real,real] => bool |
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("(DERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60) |
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"DERIV f x :> D == ((%h. (f(x + h) + -f(x))*inverse(h)) -- #0 --> D)" |
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nsderiv :: [real=>real,real,real] => bool |
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("(NSDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60) |
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"NSDERIV f x :> D == (ALL h: Infinitesimal - {0}. |
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((*f* f)(hypreal_of_real x + h) + |
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-hypreal_of_real (f x))*inverse(h) @= hypreal_of_real D)" |
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differentiable :: [real=>real,real] => bool (infixl 60) |
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"f differentiable x == (EX D. DERIV f x :> D)" |
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NSdifferentiable :: [real=>real,real] => bool (infixl 60) |
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"f NSdifferentiable x == (EX D. NSDERIV f x :> D)" |
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increment :: [real=>real,real,hypreal] => hypreal |
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"increment f x h == (@inc. f NSdifferentiable x & |
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inc = (*f* f)(hypreal_of_real x + h) + -hypreal_of_real (f x))" |
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isUCont :: (real=>real) => bool |
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"isUCont f == (ALL r. #0 < r --> |
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(EX s. #0 < s & (ALL x y. abs(x + -y) < s |
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--> abs(f x + -f y) < r)))" |
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isNSUCont :: (real=>real) => bool |
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"isNSUCont f == (ALL x y. x @= y --> (*f* f) x @= (*f* f) y)" |
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end |
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