/-- 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
    show ∀ x y : s p, |dist x y - dist (Φ x) (Φ y)| ≤ ε
    · /- the distance between `x` and `y` is encoded in `F p`, and the distance between
            `Φ x` and `Φ y` (two points of `s q`) is encoded in `F q`, all this up to `ε`.
            As `F p = F q`, the distances are almost equal. -/
      -- porting note : we have to circumvent the absence of `change … with … `
      intro x y
      -- have : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y) := rfl
      rw [show dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y) from rfl]
      -- introduce `i`, that codes both `x` and `Φ x` in `fin (N p) = fin (N q)`
      let i : ℕ := E p x
      have hip : i < N p := ((E p) x).2
      have hiq : i < N q := by rwa [Npq] at hip
      have i' : i = (E q) (Ψ x) := by simp only [Equiv.apply_symm_apply, Fin.coe_cast]
      -- introduce `j`, that codes both `y` and `Φ y` in `fin (N p) = fin (N q)`
      let j : ℕ := E p y
      have hjp : j < N p := ((E p) y).2
      have hjq : j < N q := by rwa [Npq] at hjp
      have j' : j = ((E q) (Ψ y)).1 := by
        simp only [Equiv.apply_symm_apply, Fin.coe_cast]
      -- Express `dist x y` in terms of `F p`
      have : (F p).2 ((E p) x) ((E p) y) = ⌊ε⁻¹ * dist x y⌋ := by
        simp only [(E p).symm_apply_apply]
      have Ap : (F p).2 ⟨i, hip⟩ ⟨j, hjp⟩ = ⌊ε⁻¹ * dist x y⌋ := by rw [← this]
      -- Express `dist (Φ x) (Φ y)` in terms of `F q`
      have : (F q).2 ((E q) (Ψ x)) ((E q) (Ψ y)) = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋ := by
        simp only [(E q).symm_apply_apply]
      have Aq : (F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋ := by
        rw [← this]
        -- porting note: `congr` fails to make progress
        refine congr_arg₂ (F q).2 ?_ ?_ <;> ext1
        exacts [i', j']
      -- use the equality between `F p` and `F q` to deduce that the distances have equal
      -- integer parts
      have : (F p).2 ⟨i, hip⟩ ⟨j, hjp⟩ = (F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩ := by
        have hpq' : HEq (F p).snd (F q).snd := (Sigma.mk.inj_iff.1 hpq).2
        rw [Fin.heq_fun₂_iff Npq Npq] at hpq'
        rw [← hpq']
        -- porting note : new version above, because `change … with…` is not implemented
        -- we want to `subst hpq` where `hpq : F p = F q`, except that `subst` only works
        -- with a constant, so replace `F q` (and everything that depends on it) by a constant `f`
        -- then `subst`
        -- revert hiq hjq
        -- change N q with (F q).1
        -- generalize F q = f at hpq ⊢
        -- subst hpq
        -- rfl
      rw [Ap, Aq] at this
      -- deduce that the distances coincide up to `ε`, by a straightforward computation
      -- that should be automated
      have I :=
        calc
          |ε⁻¹| * |dist x y - dist (Ψ x) (Ψ y)| = |ε⁻¹ * (dist x y - dist (Ψ x) (Ψ y))| :=
            (abs_mul _ _).symm
          _ = |ε⁻¹ * dist x y - ε⁻¹ * dist (Ψ x) (Ψ y)| := by congr; ring
          _ ≤ 1 := le_of_lt (abs_sub_lt_one_of_floor_eq_floor this)
      calc
        |dist x y - dist (Ψ x) (Ψ y)| = ε * ε⁻¹ * |dist x y - dist (Ψ x) (Ψ y)| := by
          rw [mul_inv_cancel (ne_of_gt εpos), one_mul]
        _ = ε * (|ε⁻¹| * |dist x y - dist (Ψ x) (Ψ y)|) := by
          rw [abs_of_nonneg (le_of_lt (inv_pos.2 εpos)), mul_assoc]
        _ ≤ ε * 1 := (mul_le_mul_of_nonneg_left I (le_of_lt εpos))
        _ = ε := mul_one _
  

Proof state:
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)
main : ghDist (GHSpace.Rep p) (GHSpace.Rep q) ≤ ε + ε / 2 + ε
⊢ dist p q ≤ δ

Next tactic:
calc
    dist p q = ghDist p.Rep q.Rep := dist_ghDist p q
    _ ≤ ε + ε / 2 + ε := main
    _ = δ := by ring