(* 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 randomRBoolCorrect :
forall b b1 b2, implb b1 b2 ->
(implb b1 b && implb b b2 <->
exists seed, (fst (randomRBool (b1, b2) seed)) = b).