(* Title : HSeries.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Finite summation and infinite series
for hyperreals
*)
HSeries = Series +
consts
sumhr :: "(hypnat * hypnat * (nat=>real)) => hypreal"
defs
sumhr_def
"sumhr p
== Abs_hypreal(UN X:Rep_hypnat(fst p).
UN Y: Rep_hypnat(fst(snd p)).
hyprel ^^{%n::nat. sumr (X n) (Y n) (snd(snd p))})"
constdefs
NSsums :: [nat=>real,real] => bool (infixr 80)
"f NSsums s == (%n. sumr 0 n f) ----NS> s"
NSsummable :: (nat=>real) => bool
"NSsummable f == (EX s. f NSsums s)"
NSsuminf :: (nat=>real) => real
"NSsuminf f == (@s. f NSsums s)"
end