(* Title : PNat.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : The positive naturals
*)
PNat = Arith +
(** type pnat **)
(* type definition *)
typedef
pnat = "lfp(%X. {1} Un (Suc``X))" (lfp_def)
instance
pnat :: {ord, plus, times}
consts
pSuc :: pnat => pnat
"1p" :: pnat ("1p")
constdefs
pnat_nat :: nat => pnat ("*# _" [80] 80)
"*# n == Abs_pnat(n + 1)"
defs
pnat_one_def "1p == Abs_pnat(1)"
pnat_Suc_def "pSuc == (%n. Abs_pnat(Suc(Rep_pnat(n))))"
pnat_add_def
"x + y == Abs_pnat(Rep_pnat(x) + Rep_pnat(y))"
pnat_mult_def
"x * y == Abs_pnat(Rep_pnat(x) * Rep_pnat(y))"
pnat_less_def
"x < (y::pnat) == Rep_pnat(x) < Rep_pnat(y)"
pnat_le_def
"x <= (y::pnat) == ~(y < x)"
end