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 jump_simpl p l x :
jump p l x =
match p with
| xH => tail l x
| xO p => jump p (jump p l) x
| xI p  => jump p (jump p (tail l)) x
end.