l3lab/ntp-mathlib huggingface dataset:
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
      

Proof state:
x y : GHSpace
⊢ (fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y} =
    (fun p => hausdorffDist ↑p.1 ↑p.2) ∘ Prod.swap '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y}

Next tactic:
funext
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
      

Proof state:
case h
x y : GHSpace
x✝ : ℝ
⊢ ((fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y}) x✝ =
    ((fun p => hausdorffDist ↑p.1 ↑p.2) ∘ Prod.swap '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y}) x✝

Next tactic:
simp only [comp_apply, Prod.fst_swap, Prod.snd_swap]
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]
      

Proof state:
case h
x y : GHSpace
x✝ : ℝ
⊢ ((fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y}) x✝ =
    ((fun a => hausdorffDist ↑a.2 ↑a.1) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y}) x✝

Next tactic:
congr
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
      

Proof state:
case h.e_f
x y : GHSpace
x✝ : ℝ
⊢ (fun p => hausdorffDist ↑p.1 ↑p.2) = fun a => hausdorffDist ↑a.2 ↑a.1

Next tactic:
simp (config := { singlePass := true }) only [hausdorffDist_comm]
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]
    

Proof state:
x y : GHSpace
A :
  (fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y} =
    (fun p => hausdorffDist ↑p.1 ↑p.2) ∘ Prod.swap '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y}
⊢ dist x y = dist y x

Next tactic:
simp only [dist, A, image_comp, image_swap_prod]
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 

Proof state:
x✝¹ x✝ : GHSpace
⊢ (fun x y =>
        ↑{ val := sInf ((fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y}),
            property := (_ : 0 ≤ sInf ((fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = y})) })
      x✝¹ x✝ =
    ENNReal.ofReal (dist x✝¹ x✝)

Next tactic:
exact ENNReal.coe_nnreal_eq _
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. -/
    

Proof state:
x y : GHSpace
hxy : dist x y = 0
⊢ x = y

Next tactic:
rcases ghDist_eq_hausdorffDist x.Rep y.Rep with ⟨Φ, Ψ, Φisom, Ψisom, DΦΨ⟩
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⟩
    

Proof state:
case intro.intro.intro.intro
x y : GHSpace
hxy : dist x y = 0
Φ : GHSpace.Rep x → ↥(lp (fun n => ℝ) ⊤)
Ψ : GHSpace.Rep y → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : ghDist (GHSpace.Rep x) (GHSpace.Rep y) = hausdorffDist (range Φ) (range Ψ)
⊢ x = y

Next tactic:
rw [← dist_ghDist] at DΦΨ
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
    

Proof state:
case intro.intro.intro.intro
x y : GHSpace
hxy : dist x y = 0
Φ : GHSpace.Rep x → ↥(lp (fun n => ℝ) ⊤)
Ψ : GHSpace.Rep y → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : dist x y = hausdorffDist (range Φ) (range Ψ)
⊢ x = y

Next tactic:
simp_rw [hxy] at DΦΨ
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`
    

Proof state:
case intro.intro.intro.intro
x y : GHSpace
hxy : dist x y = 0
Φ : GHSpace.Rep x → ↥(lp (fun n => ℝ) ⊤)
Ψ : GHSpace.Rep y → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
⊢ x = y

Next tactic:
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
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
      

Proof state:
x y : GHSpace
hxy : dist x y = 0
Φ : GHSpace.Rep x → ↥(lp (fun n => ℝ) ⊤)
Ψ : GHSpace.Rep y → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
⊢ range Φ = range Ψ

Next tactic:
have hΦ : IsCompact (range Φ) := isCompact_range Φisom.continuous
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
      

Proof state:
x y : GHSpace
hxy : dist x y = 0
Φ : GHSpace.Rep x → ↥(lp (fun n => ℝ) ⊤)
Ψ : GHSpace.Rep y → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
hΦ : IsCompact (range Φ)
⊢ range Φ = range Ψ

Next tactic:
have hΨ : IsCompact (range Ψ) := isCompact_range Ψisom.continuous
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
      

Proof state:
x y : GHSpace
hxy : dist x y = 0
Φ : GHSpace.Rep x → ↥(lp (fun n => ℝ) ⊤)
Ψ : GHSpace.Rep y → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
⊢ range Φ = range Ψ

Next tactic:
apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm