Section AllB.
Universe u.
Variable T : Type@{u}.
Variable p : T -> bool.
Fixpoint allb@{} (ls : list T) : bool :=
match ls with
| nil => true
| l :: ls =>
if p l then allb ls else false
end.
Fixpoint anyb@{} (ls : list T) : bool :=
match ls with
| nil => false
| l :: ls =>
if p l then true else anyb ls
end.
End AllB.

Lemma Forall_cons_iff@{u} : forall (T : Type@{u}) (P : T -> Prop) a b,
Forall P (a :: b) <-> (P a /\ Forall P b).