namespace nat inductive less (a : nat) : nat → Prop := | base : less a (succ a) | step : Π {b}, less a b → less a (succ b) end nat open nat check less.rec_on namespace foo1 protected definition foo2.bar := 10 end foo1 example : foo1.foo2.bar = 10 := rfl open foo1 example : foo2.bar = 10 := rfl