Section BackwardDirection.
Variables  (s t: exp X) (x: nat) (sigma: fin -> exp X) (S: list (exp X)).
Hypothesis (H1: forall y, isVar (sigma y) /\ sigma y <> var x).
Hypothesis (N: normal s).
Hypothesis (EQ: S .+ sigma • s ≡ (var x) t).

Lemma end_is_var:
(forall x, x ∈ S -> isVar x) -> exists i e, i < |S| /\ s = var i e.