Definition demorgan_or (P:list bool -> Prop) l l' := ~ (P l /\ P l') -> ~ P l \/ ~ P l'.
Definition demorgan_inductively_barred_at P :=
forall n l, demorgan_or (inductively_barred_at P n) (true::l) (false::l).

(* Specifications: *)
Lemma inductively_barred_at_imp_is_path_from :
forall P, demorgan_inductively_barred_at P -> forall n l,
~ inductively_barred_at P n l -> is_path_from P n l.
Lemma is_path_from_imp_inductively_barred_at : forall P n l,
is_path_from P n l -> inductively_barred_at P n l -> False.
Lemma find_left_path : forall P l n,
is_path_from P (S n) l -> inductively_barred_at P n (false :: l) -> is_path_from P n (true :: l).
Lemma Y_unique : forall P, demorgan_inductively_barred_at P ->
forall l1 l2, length l1 = length l2 -> Y P l1 -> Y P l2 -> l1 = l2.