(* Title : HyperPow.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Powers theory for hyperreals
*)
HyperPow = HRealAbs + HyperNat + RealPow +
(** First: hypnat is linearly ordered **)
instance hypnat :: order (hypnat_le_refl,hypnat_le_trans,hypnat_le_anti_sym,
hypnat_less_le)
instance hypnat :: linorder (hypnat_le_linear)
instance hypnat :: plus_ac0(hypnat_add_commute,hypnat_add_assoc,
hypnat_add_zero_left)
instance hypreal :: {power}
consts hpowr :: "[hypreal,nat] => hypreal"
primrec
hpowr_0 "r ^ 0 = 1hr"
hpowr_Suc "r ^ (Suc n) = (r::hypreal) * (r ^ n)"
consts
"pow" :: [hypreal,hypnat] => hypreal (infixr 80)
defs
(* hypernatural powers of hyperreals *)
hyperpow_def
"(R::hypreal) pow (N::hypnat)
== Abs_hypreal(UN X:Rep_hypreal(R). UN Y: Rep_hypnat(N).
hyprel```{%n::nat. (X n) ^ (Y n)})"
end