∀ {α : Type u_1} [inst : PartialOrder α] (motive : Interval α → Interval α → Prop) (s t : Interval α),
  (∀ (x : Interval α), motive none x) →
    (∀ (s : NonemptyInterval α), motive (some s) none) →
      (∀ (val val_1 : NonemptyInterval α), motive (some val) (some val_1)) → motive s t