intros s l f Corr.
inversion Corr as [Vary_In In_Vary Pref]; clear Corr.
pose proof (Vary_In [] s) as Hyp; clear Vary_In.
simpl in Hyp.
assert (InNilL : In [] l) by (apply Hyp; auto); clear Hyp.
split.
+ destruct l eqn:L.
- inversion InNilL.
- destruct s0 eqn : S0.
* apply PrefixFreeWithNil in Pref; subst; auto.
* pose proof (In_Vary (s1 :: s2)) as Hyp; clear In_Vary.
+ inversion Pref.
fireInLeft Hyp'. simpl in Hyp'. inversion Hyp'.
+ pose proof In_Vary [] as Hyp; clear In_Vary.
apply Hyp in InNilL; clear Hyp.
simpl in *.
congruence.
Qed.

Lemma corrNodeNonEmpty : forall st1 st2 l p f,
correspondingSeedTree l f (SeedTreeNode st1 st2) ->
In p l -> p <> [].