Library Coq.setoid_ring.BinList

Require Import BinPos.
Require Export List.
Set Implicit Arguments.
Local Open Scope positive_scope.

Section MakeBinList.
 Variable A : Type.
 Variable default : A.

 Fixpoint jump (p:positive) (l:list A) {struct p} : list A :=
  match p with
  | xH => tl l
  | xO p => jump p (jump p l)
  | xI p => jump p (jump p (tl l))
  end.

 Fixpoint nth (p:positive) (l:list A) {struct p} : A:=
  match p with
  | xH => hd default l
  | xO p => nth p (jump p l)
  | xI p => nth p (jump p (tl l))
  end.

 Lemma jump_tl : forall j l, tl (jump j l) = jump j (tl l).

 Lemma jump_succ : forall j l,
  jump (Pos.succ j) l = jump 1 (jump j l).

 Lemma jump_add : forall i j l,
  jump (i + j) l = jump i (jump j l).

 Lemma jump_pred_double : forall i l,
  jump (Pos.pred_double i) (tl l) = jump i (jump i l).

 Lemma nth_jump : forall p l, nth p (tl l) = hd default (jump p l).

 Lemma nth_pred_double :
  forall p l, nth (Pos.pred_double p) (tl l) = nth p (jump p l).

End MakeBinList.