∀ {α : Type u_1} [inst : Lattice α] (motive : Interval α → Interval α → Prop) (x x_1 : Interval α),
  (∀ (x : Interval α), motive none x) →
    (∀ (s : NonemptyInterval α), motive (some s) none) →
      (∀ (s t : NonemptyInterval α), motive (some s) (some t)) → motive x x_1