Definition posFunToPathFun (f : positive -> RandomSeed) (p : SplitPath) 
: RandomSeed :=
  match pathToPos p with 
    | Some a => f a
    | None   => newRandomSeed
  end.

(* Specifications: *)
Theorem coarbComplete' : forall (max : positive) (f : positive -> RandomSeed) , exists seed, forall p, p <= max -> varySeed (posToPath p) seed = f p.