(* end SplitPathCompleteness *)
intros Pref.
pose proof (listToTree l f Pref) as ExSeedTree.
inversion ExSeedTree as [st Corr]; clear ExSeedTree.
inversion Corr as [Vary_In In_Vary Pref']; clear Corr.
pose proof (splitExpand st) as ExRst.
inversion ExRst as [rst Sub]; clear ExRst.
exists rst.
intros p InPL.
pose proof (pathAgreesOnSubTree st (mkSeedTree rst) p (f p)) as Hyp.
auto.
Qed.

Axiom randomRNCorrect:
forall n n1 n2,
N.leb n1 n2 ->
N.leb n1 n && N.leb n n2 <->
exists seed, fst (randomRN (n1, n2) seed) = n.