Section EquivalenceInversion.
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_head_var:
exists (h: nat) T s', s = AppR (var h) T /\ nth S h = Some s'.