src/HOL/Real/Hyperreal/Lim.thy

final tuning;

(*  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