(* Title: HOL/Nat.thy
ID: $Id$
Author: Tobias Nipkow and Lawrence C Paulson
Type "nat" is a linear order, and a datatype; arithmetic operators + -
and * (for div, mod and dvd, see theory Divides).
*)
Nat = NatDef + Inductive +
(* type "nat" is a linear order, and a datatype *)
rep_datatype nat
distinct Suc_not_Zero, Zero_not_Suc
inject Suc_Suc_eq
induct nat_induct
instance nat :: order (le_refl,le_trans,le_anti_sym,nat_less_le)
instance nat :: linorder (nat_le_linear)
axclass power < term
consts
power :: ['a::power, nat] => 'a (infixr "^" 80)
(* arithmetic operators + - and * *)
instance
nat :: {plus, minus, times, power}
(* size of a datatype value; overloaded *)
consts size :: 'a => nat
primrec
add_0 "0 + n = n"
add_Suc "Suc m + n = Suc(m + n)"
primrec
diff_0 "m - 0 = m"
diff_Suc "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
primrec
mult_0 "0 * n = 0"
mult_Suc "Suc m * n = n + (m * n)"
end