Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: 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)
⊢ TotallyBounded t

Tactic:
refine Metric.totallyBounded_of_finite_discretization fun δ δpos => ?_
  let ε := 1 / 5 * δ
  have εpos : 0 < ε := mul_pos (by simp) 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, card_empty, Fintype.card_fin]
      use ∅, 0, bot_le, this.some
      exact fun hp' => (hp hp').elim
    · rcases hcov _ (Set.not_notMem.1 hp) n with ⟨s, ⟨scard, scover⟩⟩
      rcases Cardinal.lt_aleph0.1 (scard.trans_lt Cardinal.natCast_lt_aleph0) with ⟨N, hN⟩
      rw [hN, Nat.cast_le] at scard
      have : #s = #(Fin N) := by rw [hN, Cardinal.mk_fin]
      obtain ⟨E⟩ := Quotient.exact this
      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 [Ψ, Φ]; rw [C1, C2, C3]
      rw [this]
      exact le_trans (le_of_lt hy) u_le_ε
    · 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. -/
      intro x y
      have : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y) := rfl
      rw [this]
      -- 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 [i, Ψ, Equiv.apply_symm_apply, Fin.val_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) := by simp only [j, Ψ, Equiv.apply_symm_apply, Fin.val_cast]
      -- Express `dist x y` in terms of `F p`
      have Ap : ((F p).2 ⟨i, hip⟩ ⟨j, hjp⟩).1 = ⌊ε⁻¹ * dist x y⌋₊ :=
        calc
          ((F p).2 ⟨i, hip⟩ ⟨j, hjp⟩).1 = ((F p).2 ((E p) x) ((E p) y)).1 := by
            congr
          _ = min M ⌊ε⁻¹ * dist x y⌋₊ := by simp only [F, (E p).symm_apply_apply]
          _ = ⌊ε⁻¹ * dist x y⌋₊ := by
            refine min_eq_right (Nat.floor_mono ?_)
            refine mul_le_mul_of_nonneg_left (le_trans ?_ (le_max_left _ _)) (inv_pos.2 εpos).le
            change dist (x : p.Rep) y ≤ C
            refine (dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ _) (mem_univ _)).trans ?_
            exact hdiam p pt
      -- Express `dist (Φ x) (Φ y)` in terms of `F q`
      have Aq : ((F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩).1 = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊ :=
        calc
          ((F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩).1 = ((F q).2 ((E q) (Ψ x)) ((E q) (Ψ y))).1 := by
            congr!
          _ = min M ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊ := by simp only [F, (E q).symm_apply_apply]
          _ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊ := by
            refine min_eq_right (Nat.floor_mono ?_)
            refine mul_le_mul_of_nonneg_left (le_trans ?_ (le_max_left _ _)) (inv_pos.2 εpos).le
            change dist (Ψ x : q.Rep) (Ψ y) ≤ C
            refine (dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ _) (mem_univ _)).trans ?_
            exact hdiam q qt
      -- 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⟩).1 = ((F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩).1 := by
        have hpq' : (F p).snd ≍ (F q).snd := (Sigma.mk.inj_iff.1 hpq).2
        rw [Fin.heq_fun₂_iff Npq Npq] at hpq'
        rw [← hpq']
      have : ⌊ε⁻¹ * dist x y⌋ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋ := by
        rw [Ap, Aq] at this
        have D : 0 ≤ ⌊ε⁻¹ * dist x y⌋ :=
          floor_nonneg.2 (mul_nonneg (le_of_lt (inv_pos.2 εpos)) dist_nonneg)
        have D' : 0 ≤ ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋ :=
          floor_nonneg.2 (mul_nonneg (le_of_lt (inv_pos.2 εpos)) dist_nonneg)
        rw [← Int.toNat_of_nonneg D, ← Int.toNat_of_nonneg D', Int.floor_toNat, Int.floor_toNat,
          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_cancel₀ (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 _
  calc
    dist p q = ghDist p.Rep q.Rep := dist_ghDist p q
    _ ≤ ε + ε / 2 + ε := main
    _ = δ / 2 := by simp only [ε, one_div]; ring
    _ < δ := half_lt_self δpos

Goal state after: