set_option maxHeartbeats 300000 in
/-- The Gromov-Hausdorff distance defines a genuine distance on the Gromov-Hausdorff space. -/
instance : MetricSpace GHSpace where
  dist := dist
  -- porting note: why does Lean 4 want this?
  edist_dist _ _ := by exact ENNReal.coe_nnreal_eq _
  dist_self x := by
    rcases exists_rep x with ⟨y, hy⟩
    refine' le_antisymm _ _
    · apply csInf_le
      · exact ⟨0, by rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg⟩
      · simp only [mem_image, mem_prod, mem_setOf_eq, Prod.exists]
        exists y, y
        simpa only [and_self_iff, hausdorffDist_self_zero, eq_self_iff_true, and_true_iff]
    · apply le_csInf
      · exact Set.Nonempty.image _ <| Set.Nonempty.prod ⟨y, hy⟩ ⟨y, hy⟩
      · rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg
  dist_comm x y := by
    have A :
      (fun p : NonemptyCompacts ℓ_infty_ℝ × NonemptyCompacts ℓ_infty_ℝ =>
            hausdorffDist (p.1 : Set ℓ_infty_ℝ) p.2) ''
          { a | ⟦a⟧ = x } ×ˢ { b | ⟦b⟧ = y } =
        (fun p : NonemptyCompacts ℓ_infty_ℝ × NonemptyCompacts ℓ_infty_ℝ =>
              hausdorffDist (p.1 : Set ℓ_infty_ℝ) p.2) ∘
            Prod.swap ''
          { a | ⟦a⟧ = x } ×ˢ { b | ⟦b⟧ = y } := by
      congr
      funext
      simp only [comp_apply, Prod.fst_swap, Prod.snd_swap]
      congr
      simp (config := { singlePass := true }) only [hausdorffDist_comm]
    simp only [dist, A, image_comp, image_swap_prod]
  eq_of_dist_eq_zero {x} {y} hxy := by
    /- To show that two spaces at zero distance are isometric,
       we argue that the distance is realized by some coupling.
        In this coupling, the two spaces are at zero Hausdorff distance,
        i.e., they coincide. Therefore, the original spaces are isometric. -/
    rcases ghDist_eq_hausdorffDist x.Rep y.Rep with ⟨Φ, Ψ, Φisom, Ψisom, DΦΨ⟩
    rw [← dist_ghDist] at DΦΨ
    simp_rw [hxy] at DΦΨ -- porting note: I have no idea why this needed `simp_rw` versus `rw`
    have : range Φ = range Ψ := by
      have hΦ : IsCompact (range Φ) := isCompact_range Φisom.continuous
      have hΨ : IsCompact (range Ψ) := isCompact_range Ψisom.continuous
      apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm
      · exact hΦ.isClosed
      · exact hΨ.isClosed
      · exact hausdorffEdist_ne_top_of_nonempty_of_bounded (range_nonempty _) (range_nonempty _)
          hΦ.isBounded hΨ.isBounded
    have T : (range Ψ ≃ᵢ y.Rep) = (range Φ ≃ᵢ y.Rep) := by rw [this]
    have eΨ := cast T Ψisom.isometryEquivOnRange.symm
    have e := Φisom.isometryEquivOnRange.trans eΨ
    rw [← x.toGHSpace_rep, ← y.toGHSpace_rep, toGHSpace_eq_toGHSpace_iff_isometryEquiv]
    exact ⟨e⟩
  dist_triangle x y z := by
    /- To show the triangular inequality between `X`, `Y` and `Z`,
        realize an optimal coupling between `X` and `Y` in a space `γ1`,
        and an optimal coupling between `Y` and `Z` in a space `γ2`.
        Then, glue these metric spaces along `Y`. We get a new space `γ`
        in which `X` and `Y` are optimally coupled, as well as `Y` and `Z`.
        Apply the triangle inequality for the Hausdorff distance in `γ`
        to conclude. -/
    let X := x.Rep
    let Y := y.Rep
    let Z := z.Rep
    let γ1 := OptimalGHCoupling X Y
    let γ2 := OptimalGHCoupling Y Z
    let Φ : Y → γ1 := optimalGHInjr X Y
    have hΦ : Isometry Φ := isometry_optimalGHInjr X Y
    let Ψ : Y → γ2 := optimalGHInjl Y Z
    have hΨ : Isometry Ψ := isometry_optimalGHInjl Y Z
    have Comm : toGlueL hΦ hΨ ∘ optimalGHInjr X Y = toGlueR hΦ hΨ ∘ optimalGHInjl Y Z :=
      toGlue_commute hΦ hΨ
    

Proof state:
x y z : GHSpace
X : Type := GHSpace.Rep x
Y : Type := GHSpace.Rep y
Z : Type := GHSpace.Rep z
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : toGlueL hΦ hΨ ∘ optimalGHInjr X Y = toGlueR hΦ hΨ ∘ optimalGHInjl Y Z
⊢ dist x z ≤ dist x y + dist y z

Next tactic:
calc
      dist x z = dist (toGHSpace X) (toGHSpace Z) := by
        rw [x.toGHSpace_rep, z.toGHSpace_rep]
      _ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
            (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) :=
        (ghDist_le_hausdorffDist ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y))
          ((toGlueR_isometry hΦ hΨ).comp (isometry_optimalGHInjr Y Z)))
      _ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
              (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y)) +
            hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
              (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) := by
        refine' hausdorffDist_triangle <| hausdorffEdist_ne_top_of_nonempty_of_bounded
          (range_nonempty _) (range_nonempty _) _ _
        · exact (isCompact_range (Isometry.continuous
            ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y)))).isBounded
        · exact (isCompact_range (Isometry.continuous
            ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjr X Y)))).isBounded
      _ = hausdorffDist (toGlueL hΦ hΨ '' range (optimalGHInjl X Y))
              (toGlueL hΦ hΨ '' range (optimalGHInjr X Y)) +
            hausdorffDist (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
              (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) := by
        simp only [← range_comp, Comm, eq_self_iff_true, add_right_inj]
      _ = hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
            hausdorffDist (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) := by
        rw [hausdorffDist_image (toGlueL_isometry hΦ hΨ),
          hausdorffDist_image (toGlueR_isometry hΦ hΨ)]
      _ = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) := by
        rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
      _ = dist x y + dist y z := by rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]