/-- The Gromov-Hausdorff space is second countable. -/
instance : SecondCountableTopology GHSpace := by
  refine' secondCountable_of_countable_discretization fun δ δpos => _
  let ε := 2 / 5 * δ
  have εpos : 0 < ε := mul_pos (by norm_num) δpos
  have : ∀ p : GHSpace, ∃ s : Set p.Rep, s.Finite ∧ univ ⊆ ⋃ x ∈ s, ball x ε := fun p => by
    simpa only [subset_univ, true_and] using
      finite_cover_balls_of_compact (α := p.Rep) isCompact_univ εpos
  -- for each `p`, `s p` is a finite `ε`-dense subset of `p` (or rather the metric space
  -- `p.rep` representing `p`)
  choose s hs using this
  have : ∀ p : GHSpace, ∀ t : Set p.Rep, t.Finite → ∃ n : ℕ, ∃ _ : Equiv t (Fin n), True := by
    intro p t ht
    letI : Fintype t := Finite.fintype ht
    exact ⟨Fintype.card t, Fintype.equivFin t, trivial⟩
  choose N e _ using this
  -- cardinality of the nice finite subset `s p` of `p.rep`, called `N p`
  let N := fun p : GHSpace => N p (s p) (hs p).1
  -- equiv from `s p`, a nice finite subset of `p.rep`, to `fin (N p)`, called `E p`
  let E := fun p : GHSpace => e p (s p) (hs p).1
  -- A function `F` associating to `p : GHSpace` the data of all distances between points
  -- in the `ε`-dense set `s p`.
  let F : GHSpace → Σ n : ℕ, Fin n → Fin n → ℤ := fun p =>
    ⟨N p, fun a b => ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
  refine' ⟨Σ n, Fin n → Fin n → ℤ, by infer_instance, F, fun p q hpq => _⟩
  /- As the target space of F is countable, it suffices to show that two points
    `p` and `q` with `F p = F q` are at distance `≤ δ`.
    For this, we construct a map `Φ` from `s p ⊆ p.rep` (representing `p`)
    to `q.rep` (representing `q`) which is almost an isometry on `s p`, and
    with image `s q`. For this, we compose the identification of `s p` with `fin (N p)`
    and the inverse of the identification of `s q` with `fin (N q)`. Together with
    the fact that `N p = N q`, this constructs `Ψ` between `s p` and `s q`, and then
    composing with the canonical inclusion we get `Φ`. -/
  have Npq : N p = N q := (Sigma.mk.inj_iff.1 hpq).1
  let Ψ : s p → s q := fun x => (E q).symm (Fin.cast Npq ((E p) x))
  let Φ : s p → q.Rep := fun x => Ψ x
  -- Use the almost isometry `Φ` to show that `p.rep` and `q.rep`
  -- are within controlled Gromov-Hausdorff distance.
  have main : ghDist p.Rep q.Rep ≤ ε + ε / 2 + ε := by
    refine' ghDist_le_of_approx_subsets Φ _ _ _
    show ∀ x : p.Rep, ∃ y ∈ s p, dist x y ≤ ε
    · -- by construction, `s p` is `ε`-dense
      intro x
      have : x ∈ ⋃ y ∈ s p, ball y ε := (hs p).2 (mem_univ _)
      rcases mem_iUnion₂.1 this with ⟨y, ys, hy⟩
      exact ⟨y, ys, le_of_lt hy⟩
    show ∀ x : q.Rep, ∃ z : s p, dist x (Φ z) ≤ ε
    · -- by construction, `s q` is `ε`-dense, and it is the range of `Φ`
      intro x
      have : x ∈ ⋃ y ∈ s q, ball y ε := (hs q).2 (mem_univ _)
      rcases mem_iUnion₂.1 this with ⟨y, ys, hy⟩
      let i : ℕ := E q ⟨y, ys⟩
      let hi := ((E q) ⟨y, ys⟩).is_lt
      have ihi_eq : (⟨i, hi⟩ : Fin (N q)) = (E q) ⟨y, ys⟩ := by rw [Fin.ext_iff, Fin.val_mk]
      have hiq : i < N q := hi
      have hip : i < N p := by rwa [Npq.symm] at hiq
      let z := (E p).symm ⟨i, hip⟩
      use z
      have C1 : (E p) z = ⟨i, hip⟩ := (E p).apply_symm_apply ⟨i, hip⟩
      have C2 : Fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩ := rfl
      have C3 : (E q).symm ⟨i, hi⟩ = ⟨y, ys⟩ := by
        rw [ihi_eq]; exact (E q).symm_apply_apply ⟨y, ys⟩
      have : Φ z = y := by simp only; rw [C1, C2, C3]
      rw [this]
      exact le_of_lt hy
    

Proof state:
case refine'_3
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set (GHSpace.Rep p)
hs : ∀ (p : GHSpace), Set.Finite (s p) ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N✝ : (p : GHSpace) → (t : Set (GHSpace.Rep p)) → Set.Finite t → ℕ
e : (p : GHSpace) → (t : Set (GHSpace.Rep p)) → (a : Set.Finite t) → ↑t ≃ Fin (N✝ p t a)
x✝ : ∀ (p : GHSpace) (t : Set (GHSpace.Rep p)), Set.Finite t → True
N : GHSpace → ℕ := fun p => N✝ p (s p) (_ : Set.Finite (s p))
E : (p : GHSpace) → ↑(s p) ≃ Fin (N✝ p (s p) (_ : Set.Finite (s p))) := fun p => e p (s p) (_ : Set.Finite (s p))
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) :=
  fun p => { fst := N p, snd := fun a b => ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋ }
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x => (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → GHSpace.Rep q := fun x => ↑(Ψ x)
⊢ ∀ (x y : ↑(s p)), |dist x y - dist (Φ x) (Φ y)| ≤ ε

Next tactic:
show ∀ x y : s p, |dist x y - dist (Φ x) (Φ y)| ≤ ε