(* Title : Lim.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Theory of limits, continuity and
differentiation of real=>real functions
*)
Lim = SEQ + RealAbs +
(*-----------------------------------------------------------------------
Limits, continuity and differentiation: standard and NS definitions
-----------------------------------------------------------------------*)
constdefs
LIM :: [real=>real,real,real] => bool ("((_)/ -- (_)/ --> (_))" 60)
"f -- a --> L == (ALL r. #0 < r -->
(EX s. #0 < s & (ALL x. (#0 < abs(x + -a) & (abs(x + -a) < s)
--> abs(f x + -L) < r))))"
NSLIM :: [real=>real,real,real] => bool ("((_)/ -- (_)/ --NS> (_))" 60)
"f -- a --NS> L == (ALL x. (x ~= hypreal_of_real a &
x @= hypreal_of_real a --> (*f* f) x @= hypreal_of_real L))"
isCont :: [real=>real,real] => bool
"isCont f a == (f -- a --> (f a))"
(* NS definition dispenses with limit notions *)
isNSCont :: [real=>real,real] => bool
"isNSCont f a == (ALL y. y @= hypreal_of_real a -->
(*f* f) y @= hypreal_of_real (f a))"
(* differentiation: D is derivative of function f at x *)
deriv:: [real=>real,real,real] => bool ("(DERIV (_)/ (_)/ :> (_))" 60)
"DERIV f x :> D == ((%h. (f(x + h) + -f(x))*rinv(h)) -- #0 --> D)"
nsderiv :: [real=>real,real,real] => bool ("(NSDERIV (_)/ (_)/ :> (_))" 60)
"NSDERIV f x :> D == (ALL h: Infinitesimal - {0}.
((*f* f)(hypreal_of_real x + h) +
-hypreal_of_real (f x))*hrinv(h) @= hypreal_of_real D)"
differentiable :: [real=>real,real] => bool (infixl 60)
"f differentiable x == (EX D. DERIV f x :> D)"
NSdifferentiable :: [real=>real,real] => bool (infixl 60)
"f NSdifferentiable x == (EX D. NSDERIV f x :> D)"
increment :: [real=>real,real,hypreal] => hypreal
"increment f x h == (@inc. f NSdifferentiable x &
inc = (*f* f)(hypreal_of_real x + h) + -hypreal_of_real (f x))"
isUCont :: (real=>real) => bool
"isUCont f == (ALL r. #0 < r -->
(EX s. #0 < s & (ALL x y. abs(x + -y) < s
--> abs(f x + -f y) < r)))"
isNSUCont :: (real=>real) => bool
"isNSUCont f == (ALL x y. x @= y --> (*f* f) x @= (*f* f) y)"
end