l3lab/ntp-mathlib huggingface dataset:
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  

Proof state:
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ r

Next tactic:
refine' hausdorffDist_le_of_mem_dist _ A _
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · 

Proof state:
case refine'_1
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
⊢ 0 ≤ r

Next tactic:
inhabit X
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    

Proof state:
case refine'_1
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
inhabited_h : Inhabited X
⊢ 0 ≤ r

Next tactic:
rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    

Proof state:
case refine'_1.intro.intro
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
inhabited_h : Inhabited X
y : OptimalGHCoupling X Y
hy : dist (optimalGHInjl X Y default) y ≤ r
⊢ 0 ≤ r

Next tactic:
exact le_trans dist_nonneg hy
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    exact le_trans dist_nonneg hy
  · 

Proof state:
case refine'_2
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
⊢ ∀ x ∈ range (optimalGHInjr X Y), ∃ y ∈ range (optimalGHInjl X Y), dist x y ≤ r

Next tactic:
rintro _ ⟨z, rfl⟩
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    exact le_trans dist_nonneg hy
  · rintro _ ⟨z, rfl⟩
    

Proof state:
case refine'_2.intro
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
⊢ ∃ y ∈ range (optimalGHInjl X Y), dist (optimalGHInjr X Y z) y ≤ r

Next tactic:
have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    exact le_trans dist_nonneg hy
  · rintro _ ⟨z, rfl⟩
    have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr
    

Proof state:
case refine'_2.intro
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
I1 : ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r
⊢ ∃ y ∈ range (optimalGHInjl X Y), dist (optimalGHInjr X Y z) y ≤ r

Next tactic:
have I2 :
        ⨅ x, optimalGHDist X Y (inl x, inr z) ≤ ⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux2 _ 0) (mem_range_self _)
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    exact le_trans dist_nonneg hy
  · rintro _ ⟨z, rfl⟩
    have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr
    have I2 :
        ⨅ x, optimalGHDist X Y (inl x, inr z) ≤ ⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by 

Proof state:
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
I1 : ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r
⊢ BddAbove (range fun y => ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y))

Next tactic:
simpa using HD_bound_aux2 _ 0
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    exact le_trans dist_nonneg hy
  · rintro _ ⟨z, rfl⟩
    have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr
    have I2 :
        ⨅ x, optimalGHDist X Y (inl x, inr z) ≤ ⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux2 _ 0) (mem_range_self _)
    

Proof state:
case refine'_2.intro
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
I1 : ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r
I2 :
  ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr z) ≤ ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y)
⊢ ∃ y ∈ range (optimalGHInjl X Y), dist (optimalGHInjr X Y z) y ≤ r

Next tactic:
have I : ⨅ x, optimalGHDist X Y (inl x, inr z) < r := lt_of_le_of_lt I2 I1
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    exact le_trans dist_nonneg hy
  · rintro _ ⟨z, rfl⟩
    have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr
    have I2 :
        ⨅ x, optimalGHDist X Y (inl x, inr z) ≤ ⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux2 _ 0) (mem_range_self _)
    have I : ⨅ x, optimalGHDist X Y (inl x, inr z) < r := lt_of_le_of_lt I2 I1
    

Proof state:
case refine'_2.intro
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
I1 : ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r
I2 :
  ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr z) ≤ ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y)
I : ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr z) < r
⊢ ∃ y ∈ range (optimalGHInjl X Y), dist (optimalGHInjr X Y z) y ≤ r

Next tactic:
rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
/-- For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
  refine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)
  have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
  refine' hausdorffDist_le_of_mem_dist _ A _
  · inhabit X
    rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
    exact le_trans dist_nonneg hy
  · rintro _ ⟨z, rfl⟩
    have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr
    have I2 :
        ⨅ x, optimalGHDist X Y (inl x, inr z) ≤ ⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux2 _ 0) (mem_range_self _)
    have I : ⨅ x, optimalGHDist X Y (inl x, inr z) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    

Proof state:
case refine'_2.intro.intro.intro.intro
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
I1 : ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r
I2 :
  ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr z) ≤ ⨆ y, ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr y)
I : ⨅ x, (GromovHausdorff.optimalGHDist X Y) (inl x, inr z) < r
z' : X
hr' : (fun x => (GromovHausdorff.optimalGHDist X Y) (inl x, inr z)) z' < r
⊢ ∃ y ∈ range (optimalGHInjl X Y), dist (optimalGHInjr X Y z) y ≤ r

Next tactic:
refine' ⟨optimalGHInjl X Y z', mem_range_self _, le_of_lt _⟩