Section Backward.
Definition decode_subst (sigma: fin -> exp ag) (N: forall x, normal (sigma x)) (x: fin) :=
match dec_enc (N (F x)) with
| inl  (exist _ n _) => n
| inr _ => 0
end.

Lemma decode_subst_encodes (sigma: fin -> exp ag) N x n:
encodes (sigma (F x)) n -> decode_subst sigma N x = n.