set_option maxHeartbeats 400000 in
/-- The optimal coupling constructed above realizes exactly the Gromov-Hausdorff distance,
essentially by design. -/
theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
    {Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y := by
  inhabit X; inhabit Y
  /- we only need to check the inequality `≤`, as the other one follows from the previous lemma.
       As the Gromov-Hausdorff distance is an infimum, we need to check that the Hausdorff distance
       in the optimal coupling is smaller than the Hausdorff distance of any coupling.
       First, we check this for couplings which already have small Hausdorff distance: in this
       case, the induced "distance" on `X ⊕ Y` belongs to the candidates family introduced in the
       definition of the optimal coupling, and the conclusion follows from the optimality
       of the optimal coupling within this family.
    -/
  have A :
    ∀ p q : NonemptyCompacts ℓ_infty_ℝ,
      ⟦p⟧ = toGHSpace X →
        ⟦q⟧ = toGHSpace Y →
          hausdorffDist (p : Set ℓ_infty_ℝ) q < diam (univ : Set X) + 1 + diam (univ : Set Y) →
            hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤
              hausdorffDist (p : Set ℓ_infty_ℝ) q := by
    intro p q hp hq bound
    rcases eq_toGHSpace_iff.1 hp with ⟨Φ, ⟨Φisom, Φrange⟩⟩
    rcases eq_toGHSpace_iff.1 hq with ⟨Ψ, ⟨Ψisom, Ψrange⟩⟩
    have I : diam (range Φ ∪ range Ψ) ≤ 2 * diam (univ : Set X) + 1 + 2 * diam (univ : Set Y) := by
      rcases exists_mem_of_nonempty X with ⟨xX, _⟩
      have : ∃ y ∈ range Ψ, dist (Φ xX) y < diam (univ : Set X) + 1 + diam (univ : Set Y) := by
        rw [Ψrange]
        have : Φ xX ∈ ↑p := Φrange.subst (mem_range_self _)
        exact
          exists_dist_lt_of_hausdorffDist_lt this bound
            (hausdorffEdist_ne_top_of_nonempty_of_bounded p.nonempty q.nonempty
              p.isCompact.isBounded q.isCompact.isBounded)
      rcases this with ⟨y, hy, dy⟩
      rcases mem_range.1 hy with ⟨z, hzy⟩
      rw [← hzy] at dy
      have DΦ : diam (range Φ) = diam (univ : Set X) := Φisom.diam_range
      have DΨ : diam (range Ψ) = diam (univ : Set Y) := Ψisom.diam_range
      calc
        diam (range Φ ∪ range Ψ) ≤ diam (range Φ) + dist (Φ xX) (Ψ z) + diam (range Ψ) :=
          diam_union (mem_range_self _) (mem_range_self _)
        _ ≤
            diam (univ : Set X) + (diam (univ : Set X) + 1 + diam (univ : Set Y)) +
              diam (univ : Set Y) := by
          rw [DΦ, DΨ]
          gcongr
          -- apply add_le_add (add_le_add le_rfl (le_of_lt dy)) le_rfl
        _ = 2 * diam (univ : Set X) + 1 + 2 * diam (univ : Set Y) := by ring
    let f : Sum X Y → ℓ_infty_ℝ := fun x =>
      match x with
      | inl y => Φ y
      | inr z => Ψ z
    let F : Sum X Y × Sum X Y → ℝ := fun p => dist (f p.1) (f p.2)
    -- check that the induced "distance" is a candidate
    have Fgood : F ∈ candidates X Y := by
      simp only [candidates, forall_const, and_true_iff, add_comm, eq_self_iff_true, dist_eq_zero,
        and_self_iff, Set.mem_setOf_eq]
      repeat' 

Proof state:
case left.left.left.right
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n => ℝ) ⊤)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n => ℝ) ⊤) :=
  fun x =>
    match x with
    | inl y => Φ y
    | inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p => dist (f p.1) (f p.2)
⊢ ∀ (x y : Y), dist (Ψ x) (Ψ y) = dist x y

Next tactic:
constructor