∀ {α : Type u_1} [inst : Lattice α] [inst_1 : DecidableLE α] (s : NonemptyInterval α) (t c : Interval α),
  ↑s ≤ t →
    ↑s ≤ c →
      ↑s ≤
        match t, c with
        | none, x => ⊥
        | x, none => ⊥
        | some s, some t =>
          if h : s.toProd.1 ≤ t.toProd.2 ∧ t.toProd.1 ≤ s.toProd.2 then
            ↑{ fst := s.toProd.1 ⊔ t.toProd.1, snd := s.toProd.2 ⊓ t.toProd.2, fst_le_snd := ⋯ }
          else ⊥