# Library Coq.Numbers.Cyclic.Abstract.DoubleType

Set Implicit Arguments.

Require Import BinInt.
Require Import CarryType.
Local Open Scope Z_scope.

Definition base digits := Z.pow 2 (Zpos digits).
Arguments base digits: simpl never.

Notation carry := carry (only parsing).
Notation C0 := C0 (only parsing).
Notation C1 := C1 (only parsing).

Definition interp_carry {A} (sign:Z)(B:Z)(interp:A -> Z) c :=
match c with
| C0 x => interp x
| C1 x => sign*B + interp x
end.

From a type znz representing a cyclic structure Z/nZ, we produce a representation of Z/2nZ by pairs of elements of znz (plus a special case for zero). High half of the new number comes first.
#[universes(template)]
Variant zn2z {znz : Type} :=
| W0 : zn2z
| WW : znz -> znz -> zn2z.
Arguments zn2z : clear implicits.

Definition zn2z_to_Z znz (wB:Z) (w_to_Z:znz->Z) (x:zn2z znz) :=
match x with
| W0 => 0
| WW xh xl => w_to_Z xh * wB + w_to_Z xl
end.

Arguments W0 {znz}.

From a cyclic representation w, we iterate the zn2z construct n times, gaining the type of binary trees of depth at most n, whose leafs are either W0 (if depth < n) or elements of w (if depth = n).

Fixpoint word (w:Set) (n:nat) : Set :=
match n with
| O => w
| S n => zn2z (word w n)
end.