instance locallyFiniteOrder : LocallyFiniteOrder (WithTop α) where
  finsetIcc a b :=
    match a, b with
    | ⊤, ⊤ => {⊤}
    | ⊤, (b : α) => ∅
    | (a : α), ⊤ => insertNone (Ici a)
    | (a : α), (b : α) => (Icc a b).map Embedding.some
  finsetIco a b :=
    match a, b with
    | ⊤, _ => ∅
    | (a : α), ⊤ => (Ici a).map Embedding.some
    | (a : α), (b : α) => (Ico a b).map Embedding.some
  finsetIoc a b :=
    match a, b with
    | ⊤, _ => ∅
    | (a : α), ⊤ => insertNone (Ioi a)
    | (a : α), (b : α) => (Ioc a b).map Embedding.some
  finsetIoo a b :=
    match a, b with
    | ⊤, _ => ∅
    | (a : α), ⊤ => (Ioi a).map Embedding.some
    | (a : α), (b : α) => (Ioo a b).map Embedding.some
  -- Porting note: the proofs below got much worse
  finset_mem_Icc a b x :=
    match a, b, x with
    | ⊤, ⊤, x => mem_singleton.trans (le_antisymm_iff.trans and_comm)
    | ⊤, (b : α), x =>
      iff_of_false (not_mem_empty _) fun h => (h.1.trans h.2).not_lt <| coe_lt_top _
    | (a : α), ⊤, ⊤ => by simp [WithTop.some, WithTop.top, insertNone]
    | (a : α), ⊤, (x : α) => by
        simp only [some, le_eq_subset, some_le_some, le_top, and_true]
        rw [some_mem_insertNone]
        simp
    | (a : α), (b : α), ⊤ => by
        simp only [Embedding.some, mem_map, mem_Icc, and_false, exists_const, some, le_top,
          top_le_iff]
    | (a : α), (b : α), (x : α) => by
        simp only [some, le_eq_subset, Embedding.some, mem_map, mem_Icc, Embedding.coeFn_mk,
          some_le_some]
        -- This used to be in the above `simp` before leanprover/lean4#2644
        erw [aux]
  finset_mem_Ico a b x :=
    match a, b, x with
    | ⊤, b, x => iff_of_false (not_mem_empty _) fun h => not_top_lt <| h.1.trans_lt h.2
    | (a : α), ⊤, ⊤ => by simp [some, Embedding.some]
    | (a : α), ⊤, (x : α) => by
        simp only [some, Embedding.some, mem_map, mem_Ici, Embedding.coeFn_mk, some_le_some, aux,
          top, some_lt_none, and_true]
        -- This used to be in the above `simp` before leanprover/lean4#2644
        erw [aux]
    | (a : α), (b : α), ⊤ => by simp [some, Embedding.some]
    | (a : α), (b : α), (x : α) => by simp [some, Embedding.some, aux]
                                      -- This used to be in the above `simp` before
                                      -- leanprover/lean4#2644
                                      erw [aux]
  finset_mem_Ioc a b x :=
    match a, b, x with
    | ⊤, b, x => iff_of_false (not_mem_empty _) fun h => not_top_lt <| h.1.trans_le h.2
    | (a : α), ⊤, ⊤ => by simp [some, insertNone, top]
    | (a : α), ⊤, (x : α) => by simp [some, Embedding.some, insertNone, aux]
                                -- This used to be in the above `simp` before
                                -- leanprover/lean4#2644
                                erw [aux]
    | (a : α), (b : α), ⊤ => by simp [some, Embedding.some, insertNone]
    | (a : α), (b : α), (x : α) => by simp [some, Embedding.some, insertNone, aux]
                                      -- This used to be in the above `simp` before
                                      -- leanprover/lean4#2644
                                      erw [aux]
  finset_mem_Ioo a b x :=
    match a, b, x with
    | ⊤, b, x => iff_of_false (not_mem_empty _) fun h => not_top_lt <| h.1.trans h.2
    | (a : α), ⊤, ⊤ => by 

Proof state:
α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : OrderTop α
inst✝ : LocallyFiniteOrder α
a✝ b x : WithTop α
a : α
⊢ ⊤ ∈
      (fun a b =>
          match a, b with
          | none, x => ∅
          | Option.some a, none => Finset.map Embedding.some (Ioi a)
          | Option.some a, Option.some b => Finset.map Embedding.some (Ioo a b))
        ↑a ⊤ ↔
    ↑a < ⊤ ∧ ⊤ < ⊤

Next tactic:
simp [some, Embedding.some, insertNone]