noncomputable instance completeLattice [@DecidableRel α (· ≤ ·)] :
    CompleteLattice (Interval α) := by
  classical
  exact
      { Interval.lattice, Interval.boundedOrder with
        sSup := fun S =>
          if h : S ⊆ {⊥} then ⊥
          else
            some
              ⟨⟨⨅ (s : NonemptyInterval α) (_ : ↑s ∈ S), s.fst,
                  ⨆ (s : NonemptyInterval α) (_ : ↑s ∈ S), s.snd⟩, by
                obtain ⟨s, hs, ha⟩ := not_subset.1 h
                lift s to NonemptyInterval α using ha
                exact iInf₂_le_of_le s hs (le_iSup₂_of_le s hs s.fst_le_snd)⟩
        le_sSup := fun s s ha => by
          dsimp only -- Porting note: added
          split_ifs with h
          · exact (h ha).le
          cases s
          · exact bot_le
          · -- Porting note: This case was
            -- `exact WithBot.some_le_some.2 ⟨iInf₂_le _ ha, le_iSup₂_of_le _ ha le_rfl⟩`
            -- but there seems to be a defEq-problem at `iInf₂_le` that lean cannot resolve yet.
            apply WithBot.some_le_some.2
            constructor
            · apply iInf₂_le
              exact ha
            · exact le_iSup₂_of_le _ ha le_rfl
        sSup_le := fun s s ha => by
          dsimp only -- Porting note: added
          split_ifs with h
          · exact bot_le
          obtain ⟨b, hs, hb⟩ := not_subset.1 h
          lift s to NonemptyInterval α using ne_bot_of_le_ne_bot hb (ha _ hs)
          exact
            WithBot.coe_le_coe.2
              ⟨le_iInf₂ fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).1,
                iSup₂_le fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).2⟩
        sInf := fun S =>
          if h :
              ⊥ ∉ S ∧
                ∀ ⦃s : NonemptyInterval α⦄,
                  ↑s ∈ S → ∀ ⦃t : NonemptyInterval α⦄, ↑t ∈ S → s.fst ≤ t.snd then
            some
              ⟨⟨⨆ (s : NonemptyInterval α) (_ : ↑s ∈ S), s.fst,
                  ⨅ (s : NonemptyInterval α) (_ : ↑s ∈ S), s.snd⟩,
                iSup₂_le fun s hs => le_iInf₂ <| h.2 hs⟩
          else ⊥
        sInf_le := fun s₁ s ha => by
          dsimp only -- Porting note: added
          split_ifs with h
          · lift s to NonemptyInterval α using ne_of_mem_of_not_mem ha h.1
            -- Porting note: Lean failed to figure out the function `f` by itself,
            -- so I added it through manually
            let f := fun (s : NonemptyInterval α) (_ : ↑s ∈ s₁) => s.toProd.fst
            exact WithBot.coe_le_coe.2 ⟨le_iSup₂ (f := f) s ha, iInf₂_le s ha⟩
          · exact bot_le
        le_sInf := by
          intro S s ha
          cases s with
          | none => exact bot_le
          | some s =>
            dsimp -- Porting note: added
            split_ifs with h
            · exact WithBot.some_le_some.2
                ⟨iSup₂_le fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).1,
                  le_iInf₂ fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).2⟩
            · rw [not_and_or, not_not] at h
              rcases h with h | h
              · exact ha _ h
              · -- Porting note: ungolfed, due to identification problems
                -- between `toProd` and `toDualProd`. Original mathport output:
                -- cases h fun t hb c hc =>
                --   (WithBot.coe_le_coe.1 <| ha _ hb).1.trans <|
                --     s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2 }
                exfalso
                apply h
                intro b hb c hc
                have h₁ := (WithBot.coe_le_coe.1 <| ha _ hb).1
                repeat rw [NonemptyInterval.toDualProd_apply] at h₁
                

Proof state:
case neg.inr.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
κ : ι → Sort u_6
inst✝¹ : CompleteLattice α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
src✝¹ : Lattice (Interval α) := lattice
src✝ : BoundedOrder (Interval α) := boundedOrder
S : Set (Interval α)
s : NonemptyInterval α
ha : ∀ b ∈ S, some s ≤ b
h : ¬∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ S → ∀ ⦃t : NonemptyInterval α⦄, ↑t ∈ S → s.toProd.1 ≤ t.toProd.2
b : NonemptyInterval α
hb : ↑b ∈ S
c : NonemptyInterval α
hc : ↑c ∈ S
h₁ : (toDual s.toProd.1, s.toProd.2).1 ≤ (toDual b.toProd.1, b.toProd.2).1
⊢ b.toProd.1 ≤ c.toProd.2

Next tactic:
rw [OrderDual.toDual_le_toDual] at h₁