Section S.
Variable D :Type.
Definition Env := positive -> D.
Definition jump (j:positive) (e:Env) := fun x => e (x+j).
Definition nth (n:positive) (e:Env) := e n.
Definition hd (e:Env) := nth 1 e.
Definition tail (e:Env) := jump 1 e.

Lemma nth_pred_double p l :
nth (Pos.pred_double p) (tail l) = nth p (jump p l).