/-- Compactness criterion: a closed set of compact metric spaces is compact if the spaces have
a uniformly bounded diameter, and for all `ε` the number of balls of radius `ε` required
to cover the spaces is uniformly bounded. This is an equivalence, but we only prove the
interesting direction that these conditions imply compactness. -/
theorem totallyBounded {t : Set GHSpace} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
    (ulim : Tendsto u atTop (𝓝 0)) (hdiam : ∀ p ∈ t, diam (univ : Set (GHSpace.Rep p)) ≤ C)
    (hcov : ∀ p ∈ t, ∀ n : ℕ, ∃ s : Set (GHSpace.Rep p),
      (#s) ≤ K n ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)) :
    TotallyBounded t := by
  /- Let `δ>0`, and `ε = δ/5`. For each `p`, we construct a finite subset `s p` of `p`, which
    is `ε`-dense and has cardinality at most `K n`. Encoding the mutual distances of points
    in `s p`, up to `ε`, we will get a map `F` associating to `p` finitely many data, and making
    it possible to reconstruct `p` up to `ε`. This is enough to prove total boundedness. -/
  refine' Metric.totallyBounded_of_finite_discretization fun δ δpos => _
  let ε := 1 / 5 * δ
  have εpos : 0 < ε := mul_pos (by norm_num) δpos
  -- choose `n` for which `u n < ε`
  rcases Metric.tendsto_atTop.1 ulim ε εpos with ⟨n, hn⟩
  have u_le_ε : u n ≤ ε := by
    have := hn n le_rfl
    simp only [Real.dist_eq, add_zero, sub_eq_add_neg, neg_zero] at this
    exact le_of_lt (lt_of_le_of_lt (le_abs_self _) this)
  -- construct a finite subset `s p` of `p` which is `ε`-dense and has cardinal `≤ K n`
  have :
    ∀ p : GHSpace,
      ∃ s : Set p.Rep, ∃ N ≤ K n, ∃ _ : Equiv s (Fin N), p ∈ t → univ ⊆ ⋃ x ∈ s, ball x (u n) := by
    intro p
    by_cases hp : p ∉ t
    · have : Nonempty (Equiv (∅ : Set p.Rep) (Fin 0)) := by
        rw [← Fintype.card_eq];
        simp only [empty_card', Fintype.card_fin]
      use ∅, 0, bot_le, choice this
      -- porting note: unclear why this next line wasn't needed in Lean 3
      exact fun hp' => (hp hp').elim
    · rcases hcov _ (Set.not_not_mem.1 hp) n with ⟨s, ⟨scard, scover⟩⟩
      rcases Cardinal.lt_aleph0.1 (lt_of_le_of_lt scard (Cardinal.nat_lt_aleph0 _)) with ⟨N, hN⟩
      rw [hN, Cardinal.natCast_le] at scard
      have : #s = #(Fin N) := by rw [hN, Cardinal.mk_fin]
      cases' Quotient.exact this with E
      use s, N, scard, E
      simp only [scover, imp_true_iff]
  choose s N hN E hs using this
  -- Define a function `F` taking values in a finite type and associating to `p` enough data
  -- to reconstruct it up to `ε`, namely the (discretized) distances between elements of `s p`.
  let M := ⌊ε⁻¹ * max C 0⌋₊
  let F : GHSpace → Σ k : Fin (K n).succ, Fin k → Fin k → Fin M.succ := fun p =>
    ⟨⟨N p, lt_of_le_of_lt (hN p) (Nat.lt_succ_self _)⟩, fun a b =>
      ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊,
        (min_le_left _ _).trans_lt (Nat.lt_succ_self _)⟩⟩
  refine' ⟨_, _, fun p => F p, _⟩; infer_instance
  -- It remains to show that if `F p = F q`, then `p` and `q` are `ε`-close
  rintro ⟨p, pt⟩ ⟨q, qt⟩ hpq
  have Npq : N p = N q := Fin.ext_iff.1 (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
  have main : ghDist p.Rep q.Rep ≤ ε + ε / 2 + ε := by
    -- to prove the main inequality, argue that `s p` is `ε`-dense in `p`, and `s q` is `ε`-dense
    -- in `q`, and `s p` and `s q` are almost isometric. Then closeness follows
    -- from `ghDist_le_of_approx_subsets`
    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 (u n) := (hs p pt) (mem_univ _)
      rcases mem_iUnion₂.1 this with ⟨y, ys, hy⟩
      exact ⟨y, ys, le_trans (le_of_lt hy) u_le_ε⟩
    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 (u n) := (hs q qt) (mem_univ _)
      rcases mem_iUnion₂.1 this with ⟨y, ys, hy⟩
      let i : ℕ := E q ⟨y, ys⟩
      let hi := ((E q) ⟨y, ys⟩).2
      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_trans (le_of_lt hy) u_le_ε
    

Proof state:
case refine'_3
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set (GHSpace.Rep p)
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (Nat.succ (K n))) × (Fin ↑k → Fin ↑k → Fin (Nat.succ M)) :=
  fun p =>
    { fst := { val := N p, isLt := (_ : N p < Nat.succ (K n)) },
      snd := fun a b =>
        { val := min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊,
          isLt := (_ : min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊ < Nat.succ M) } }
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p => F ↑p) { val := p, property := pt } = (fun p => F ↑p) { val := q, property := qt }
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