/-- A `PseudoEMetricSpace` is always a paracompact space.
Formalization is based on [MR0236876]. -/
instance (priority := 100) instParacompactSpace [PseudoEMetricSpace α] : ParacompactSpace α := by
  /- We start with trivial observations about `1 / 2 ^ k`. Here and below we use `1 / 2 ^ k` in
    the comments and `2⁻¹ ^ k` in the code. -/
  have pow_pos : ∀ k : ℕ, (0 : ℝ≥0∞) < 2⁻¹ ^ k := fun k =>
    ENNReal.pow_pos (ENNReal.inv_pos.2 ENNReal.two_ne_top) _
  have hpow_le : ∀ {m n : ℕ}, m ≤ n → (2⁻¹ : ℝ≥0∞) ^ n ≤ 2⁻¹ ^ m := @fun m n h =>
    pow_le_pow_right_of_le_one' (ENNReal.inv_le_one.2 ENNReal.one_lt_two.le) h
  have h2pow : ∀ n : ℕ, 2 * (2⁻¹ : ℝ≥0∞) ^ (n + 1) = 2⁻¹ ^ n := fun n => by
    simp [pow_succ, ← mul_assoc, ENNReal.mul_inv_cancel two_ne_zero two_ne_top]
  -- Consider an open covering `S : Set (Set α)`
  refine' ⟨fun ι s ho hcov => _⟩
  simp only [iUnion_eq_univ_iff] at hcov
  -- choose a well founded order on `S`
  -- porting note: todo: add lemma that claims `∃ i : LinearOrder ι, WellFoundedLT ι`
  let _ : LinearOrder ι := by classical exact linearOrderOfSTO WellOrderingRel
  have wf : WellFounded ((· < ·) : ι → ι → Prop) := @IsWellFounded.wf ι WellOrderingRel _
  -- Let `ind x` be the minimal index `s : S` such that `x ∈ s`.
  set ind : α → ι := fun x => wf.min { i : ι | x ∈ s i } (hcov x)
  have mem_ind : ∀ x, x ∈ s (ind x) := fun x => wf.min_mem _ (hcov x)
  have nmem_of_lt_ind : ∀ {x i}, i < ind x → x ∉ s i := @fun x i hlt hxi =>
    wf.not_lt_min _ (hcov x) hxi hlt
  /- The refinement `D : ℕ → ι → Set α` is defined recursively. For each `n` and `i`, `D n i`
    is the union of balls `ball x (1 / 2 ^ n)` over all points `x` such that

    * `ind x = i`;
    * `x` does not belong to any `D m j`, `m < n`;
    * `ball x (3 / 2 ^ n) ⊆ s i`;

    We define this sequence using `Nat.strongRecOn'`, then restate it as `Dn` and `memD`.
  -/
  set D : ℕ → ι → Set α := fun n =>
    Nat.strongRecOn' n fun n D' i =>
      ⋃ (x : α) (hxs : ind x = i) (hb : ball x (3 * 2⁻¹ ^ n) ⊆ s i) (hlt :
        ∀ (m : ℕ) (H : m < n), ∀ (j : ι), x ∉ D' m H j), ball x (2⁻¹ ^ n) with hD
  have Dn : ∀ n i, D n i = ⋃ (x : α) (hxs : ind x = i) (hb : ball x (3 * 2⁻¹ ^ n) ⊆ s i)
      (hlt : ∀ m < n, ∀ (j : ι), x ∉ D m j), ball x (2⁻¹ ^ n) := fun n s => by
    simp only [hD]
    rw [Nat.strongRecOn'_beta]
  have memD : ∀ {n i y},
      y ∈ D n i ↔ ∃ x : α, ind x = i ∧ ball x (3 * 2⁻¹ ^ n) ⊆ s i ∧
        (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < 2⁻¹ ^ n := by
    intro n i y
    rw [Dn n i]
    simp only [mem_iUnion, mem_ball, exists_prop]
  -- The sets `D n i` cover the whole space. Indeed, for each `x` we can choose `n` such that
  -- `ball x (3 / 2 ^ n) ⊆ s (ind x)`, then either `x ∈ D n i`, or `x ∈ D m i` for some `m < n`.
  have Dcov : ∀ x, ∃ n i, x ∈ D n i := fun x => by
    obtain ⟨n, hn⟩ : ∃ n : ℕ, ball x (3 * 2⁻¹ ^ n) ⊆ s (ind x) := by
      -- This proof takes 5 lines because we can't import `specific_limits` here
      rcases isOpen_iff.1 (ho <| ind x) x (mem_ind x) with ⟨ε, ε0, hε⟩
      have : 0 < ε / 3 := ENNReal.div_pos_iff.2 ⟨ε0.lt.ne', ENNReal.coe_ne_top⟩
      rcases ENNReal.exists_inv_two_pow_lt this.ne' with ⟨n, hn⟩
      refine' ⟨n, Subset.trans (ball_subset_ball _) hε⟩
      simpa only [div_eq_mul_inv, mul_comm] using (ENNReal.mul_lt_of_lt_div hn).le
    by_contra! h
    apply h n (ind x)
    exact memD.2 ⟨x, rfl, hn, fun _ _ _ => h _ _, mem_ball_self (pow_pos _)⟩
  -- Each `D n i` is a union of open balls, hence it is an open set
  have Dopen : ∀ n i, IsOpen (D n i) := fun n i => by
    rw [Dn]
    iterate 4 refine' isOpen_iUnion fun _ => _
    exact isOpen_ball
  -- the covering `D n i` is a refinement of the original covering: `D n i ⊆ s i`
  have HDS : ∀ n i, D n i ⊆ s i := fun n i x => by
    rw [memD]
    rintro ⟨y, rfl, hsub, -, hyx⟩
    refine hsub (hyx.trans_le <| le_mul_of_one_le_left' ?_)
    norm_num1
  -- Let us show the rest of the properties. Since the definition expects a family indexed
  -- by a single parameter, we use `ℕ × ι` as the domain.
  refine' ⟨ℕ × ι, fun ni => D ni.1 ni.2, fun _ => Dopen _ _, _, _, fun ni => ⟨ni.2, HDS _ _⟩⟩
  -- The sets `D n i` cover the whole space as we proved earlier
  · refine' iUnion_eq_univ_iff.2 fun x => _
    rcases Dcov x with ⟨n, i, h⟩
    exact ⟨⟨n, i⟩, h⟩
  /- Let us prove that the covering `D n i` is locally finite. Take a point `x` and choose
    `n`, `i` so that `x ∈ D n i`. Since `D n i` is an open set, we can choose `k` so that
    `B = ball x (1 / 2 ^ (n + k + 1)) ⊆ D n i`. -/
  · intro x
    rcases Dcov x with ⟨n, i, hn⟩
    have : D n i ∈ 𝓝 x := IsOpen.mem_nhds (Dopen _ _) hn
    rcases (nhds_basis_uniformity uniformity_basis_edist_inv_two_pow).mem_iff.1 this with
      ⟨k, -, hsub : ball x (2⁻¹ ^ k) ⊆ D n i⟩
    set B := ball x (2⁻¹ ^ (n + k + 1))
    refine' ⟨B, ball_mem_nhds _ (pow_pos _), _⟩
    -- The sets `D m i`, `m > n + k`, are disjoint with `B`
    have Hgt : ∀ m ≥ n + k + 1, ∀ (i : ι), Disjoint (D m i) B := fun m hm i => by
      rw [disjoint_iff_inf_le]
      rintro y ⟨hym, hyx⟩
      rcases memD.1 hym with ⟨z, rfl, _hzi, H, hz⟩
      have : z ∉ ball x (2⁻¹ ^ k) := fun hz' => H n (by linarith) i (hsub hz')
      apply this
      calc
        edist z x ≤ edist y z + edist y x := edist_triangle_left _ _ _
        _ < 2⁻¹ ^ m + 2⁻¹ ^ (n + k + 1) := (ENNReal.add_lt_add hz hyx)
        _ ≤ 2⁻¹ ^ (k + 1) + 2⁻¹ ^ (k + 1) :=
          (add_le_add (hpow_le <| by linarith) (hpow_le <| by linarith))
        _ = 2⁻¹ ^ k := by rw [← two_mul, h2pow]
    -- For each `m ≤ n + k` there is at most one `j` such that `D m j ∩ B` is nonempty.
    have Hle : ∀ m ≤ n + k, Set.Subsingleton { j | (D m j ∩ B).Nonempty } := by
      rintro m hm j₁ ⟨y, hyD, hyB⟩ j₂ ⟨z, hzD, hzB⟩
      by_contra! h' : j₁ ≠ j₂
      wlog h : j₁ < j₂ generalizing j₁ j₂ y z
      · exact this z hzD hzB y hyD hyB h'.symm (h'.lt_or_lt.resolve_left h)
      rcases memD.1 hyD with ⟨y', rfl, hsuby, -, hdisty⟩
      rcases memD.1 hzD with ⟨z', rfl, -, -, hdistz⟩
      suffices : edist z' y' < 3 * 2⁻¹ ^ m
      exact nmem_of_lt_ind h (hsuby this)
      calc
        edist z' y' ≤ edist z' x + edist x y' := edist_triangle _ _ _
        _ ≤ edist z z' + edist z x + (edist y x + edist y y') :=
          (add_le_add (edist_triangle_left _ _ _) (edist_triangle_left _ _ _))
        _ < 2⁻¹ ^ m + 2⁻¹ ^ (n + k + 1) + (2⁻¹ ^ (n + k + 1) + 2⁻¹ ^ m) := by
          apply_rules [ENNReal.add_lt_add]
        _ = 2 * (2⁻¹ ^ m + 2⁻¹ ^ (n + k + 1)) := by simp only [two_mul, add_comm]
        _ ≤ 2 * (2⁻¹ ^ m + 2⁻¹ ^ (m + 1)) := by
          gcongr 2 * (_ + ?_); exact hpow_le (add_le_add hm le_rfl)
        _ = 3 * 2⁻¹ ^ m := by
          rw [mul_add, h2pow, ← two_add_one_eq_three, add_mul, one_mul]
    -- Finally, we glue `Hgt` and `Hle`
    have : (⋃ (m ≤ n + k) (i ∈ { i : ι | (D m i ∩ B).Nonempty }), {(m, i)}).Finite :=
      (finite_le_nat _).biUnion' fun i hi =>
        (Hle i hi).finite.biUnion' fun _ _ => finite_singleton _
    refine' this.subset fun I hI => _
    simp only [mem_iUnion]
    refine' ⟨I.1, _, I.2, hI, rfl⟩
    

Proof state:
case refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝ : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m
h2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen (s a)
hcov : ∀ (x : α), ∃ i, x ∈ s i
x✝ : LinearOrder ι := linearOrderOfSTO WellOrderingRel
wf : WellFounded fun x x_1 => x < x_1
ind : α → ι := fun x => WellFounded.min wf {i | x ∈ s i} (_ : ∃ i, x ∈ s i)
mem_ind : ∀ (x : α), x ∈ s (ind x)
nmem_of_lt_ind : ∀ {x : α} {i : ι}, i < ind x → x ∉ s i
D : ℕ → ι → Set α :=
  fun n =>
    Nat.strongRecOn' n fun n D' i =>
      ⋃ x,
        ⋃ (_ : ind x = i),
          ⋃ (_ : ball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), ball x (2⁻¹ ^ n)
hD :
  D = fun n =>
    Nat.strongRecOn' n fun n D' i =>
      ⋃ x,
        ⋃ (_ : ind x = i),
          ⋃ (_ : ball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), ball x (2⁻¹ ^ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ x,
        ⋃ (_ : ind x = i), ⋃ (_ : ball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), ball x (2⁻¹ ^ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔ ∃ x, ind x = i ∧ ball x (3 * 2⁻¹ ^ n) ⊆ s i ∧ (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < 2⁻¹ ^ n
Dcov : ∀ (x : α), ∃ n i, x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
hn : x ∈ D n i
this✝ : D n i ∈ 𝓝 x
k : ℕ
hsub : ball x (2⁻¹ ^ k) ⊆ D n i
B : Set α := ball x (2⁻¹ ^ (n + k + 1))
Hgt : ∀ m ≥ n + k + 1, ∀ (i : ι), Disjoint (D m i) B
Hle : ∀ m ≤ n + k, Set.Subsingleton {j | Set.Nonempty (D m j ∩ B)}
this : Set.Finite (⋃ m, ⋃ (_ : m ≤ n + k), ⋃ i ∈ {i | Set.Nonempty (D m i ∩ B)}, {(m, i)})
I : ℕ × ι
hI : I ∈ {i | Set.Nonempty ((fun ni => D ni.1 ni.2) i ∩ B)}
⊢ I.1 ≤ n + k

Next tactic:
exact not_lt.1 fun hlt => (Hgt I.1 hlt I.2).le_bot hI.choose_spec