/-- 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
  

Proof state:
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)
⊢ ∃ β t, ∃ (_ : ∀ (b : β), IsOpen (t b)) (_ : ⋃ b, t b = univ), LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a

Next tactic:
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]