Notation "a &&& b" := (if a then b else false)
 (at level 40, left associativity) : lazy_bool_scope.Notation "a ||| b" := (if a then true else b)
 (at level 50, left associativity) : lazy_bool_scope.

Lemma andb_lazy_alt : forall a b : bool, a && b = a &&& b.