# Library Coq.Numbers.Cyclic.DoubleCyclic.DoubleType

Require Import ZArith.
Local Open Scope Z_scope.
Definition base digits := Zpower 2 (Zpos digits).
Section Carry.
Variable A : Type.
Inductive carry :=
| C0 : A -> carry
| C1 : A -> carry.
Definition interp_carry (sign:Z)(B:Z)(interp:A -> Z) c :=
match c with
| C0 x => interp x
| C1 x => sign * B + interp x
end.
End Carry.
Section Zn2Z.
Variable znz : Type.
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.
Inductive zn2z :=
| W0 : zn2z
| WW : znz -> znz -> zn2z.
Definition zn2z_to_Z (wB:Z) (w_to_Z:znz->Z) (x:zn2z) :=
match x with
| W0 => 0
| WW xh xl => w_to_Z xh * wB + w_to_Z xl
end.
End Zn2Z.
Implicit 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:Type) (n:nat) : Type :=
match n with
| O => w
| S n => zn2z (word w n)
end.