structure ROOT =
struct
structure Nat =
struct
datatype nat = Zero_nat | Suc of nat;
end; (*struct Nat*)
structure Integer =
struct
fun nat_aux n i =
(if IntInf.<= (i, (0 : IntInf.int)) then n
else nat_aux (Nat.Suc n) (IntInf.- (i, (1 : IntInf.int))));
fun nat i = nat_aux Nat.Zero_nat i;
fun op_eq_bit false false = true
| op_eq_bit true true = true
| op_eq_bit false true = false
| op_eq_bit true false = false;
end; (*struct Integer*)
structure Classes =
struct
type 'a semigroup = {Classes__mult : 'a -> 'a -> 'a};
fun mult (A_:'a semigroup) = #Classes__mult A_;
type 'a monoidl =
{Classes__monoidl_semigroup : 'a semigroup, Classes__neutral : 'a};
fun monoidl_semigroup (A_:'a monoidl) = #Classes__monoidl_semigroup A_;
fun neutral (A_:'a monoidl) = #Classes__neutral A_;
type 'a group =
{Classes__group_monoidl : 'a monoidl, Classes__inverse : 'a -> 'a};
fun group_monoidl (A_:'a group) = #Classes__group_monoidl A_;
fun inverse (A_:'a group) = #Classes__inverse A_;
fun inverse_int i = IntInf.~ i;
val neutral_int : IntInf.int = (0 : IntInf.int);
fun mult_int i j = IntInf.+ (i, j);
val semigroup_int = {Classes__mult = mult_int} : IntInf.int semigroup;
val monoidl_int =
{Classes__monoidl_semigroup = semigroup_int,
Classes__neutral = neutral_int}
: IntInf.int monoidl;
val group_int =
{Classes__group_monoidl = monoidl_int, Classes__inverse = inverse_int} :
IntInf.int group;
fun pow_nat A_ (Nat.Suc n) x =
mult (monoidl_semigroup A_) x (pow_nat A_ n x)
| pow_nat A_ Nat.Zero_nat x = neutral A_;
fun pow_int A_ k x =
(if IntInf.<= ((0 : IntInf.int), k)
then pow_nat (group_monoidl A_) (Integer.nat k) x
else inverse A_
(pow_nat (group_monoidl A_) (Integer.nat (IntInf.~ k)) x));
val example : IntInf.int =
pow_int group_int (10 : IntInf.int) (~2 : IntInf.int);
end; (*struct Classes*)
end; (*struct ROOT*)