Library Coq.Numbers.Cyclic.DoubleCyclic.DoubleType

Set Implicit Arguments.

Require Import ZArith.
Local Open Scope Z_scope.

Definition base digits := Z.pow 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.

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.