Instance coqArbNat : CoArbitrary nat.Definition pathToPos p := pathToPosAux p (fun x => x).

Lemma evenPathAux : forall l l' l'' lApp f n p, length l = (2 * n)%nat -> pathToPosAux (l ++ l' ++ l'') f = Some p -> exists f', pathToPosAux (l ++ l') f = pathToPosAux l' f' /\ pathToPosAux (l ++ l' ++ lApp) f = pathToPosAux (l' ++ lApp) f'.