Section Reduction.
Variables (M : SMN) (R R': list Symbol) (A: Symbol) (X Y: State).
Variable confluent_M : confluent M.
Variable length_preserving_M : length_preserving M.
Variable derivable : reachable M ([A], R, X) ([A], R', Y).
Variable non_nil : 1 <= length R.
Variable XY_neq : X <> Y.
Variable unique_step_l : forall l r y, step M (l, r, X) y -> l = [A] ++ skipn 1 l.
Variable unique_step_l' : forall l' r' y, step M y (l', r', X) -> l' = [A] ++ skipn 1 l'.
Definition M' : SMN := [(([], R, X), ([], R', Y))] ++ M.

Lemma inverse_simulation {x y} : reachable M x y -> reachable M' x y.