induction l.
- unfold refineList; constructor.
- destruct a eqn:A; subst.
* apply prefixFreeEmpty in H. subst. unfold refineList. simpl. constructor.
* destruct s eqn:S; destruct d; subst.
+ unfold refineList; simpl.
eapply FreeCons.
- apply IHl. inversion H; auto.
- intros. inversion H; subst; clear H.
apply in_map_iff in H0.
inversion H0; subst; clear H0.
inversion H; subst; clear H.
apply filter_In in H2. inversion H2; subst; clear H2.
destruct x eqn:X; simpl in *. inversion H0.
destruct s eqn:S; simpl in *.
pose proof H5 (Left :: l0).
eapply H2 in H; auto.
simpl. instantiate (1 := p2). instantiate (1:= p1). rewrite H1. auto.
inversion H0.
+ unfold refineList; simpl. apply IHl.
inversion H. auto.
+ unfold refineList; simpl. apply IHl.
inversion H. auto.
+ unfold refineList; simpl.
eapply FreeCons.
- apply IHl. inversion H; auto.
- intros. inversion H; subst; clear H.
apply in_map_iff in H0.
inversion H0; subst; clear H0.
inversion H; subst; clear H.
apply filter_In in H2. inversion H2; subst; clear H2.
destruct x eqn:X; simpl in *. inversion H0.
destruct s eqn:S; simpl in *.
inversion H0.
pose proof H5 (Right :: l0).
eapply H2 in H; auto.
simpl. instantiate (1 := p2). instantiate (1:= p1). rewrite H1. auto.
Qed.
Definition refineFunction (f : SplitPath -> RandomSeed) (d : SplitDirection) (arg : SplitPath) :
RandomSeed :=
f (d :: arg).

Lemma refineFunCorrect : forall f d p, f (d :: p) = (refineFunction f d) p.
auto.