l3lab/ntp-mathlib huggingface dataset:
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    first
      |apply IsClosed.inter _ _
      |apply isClosed_iInter _
      |apply I1 _ _|

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.ProdSpaceFun X Y
x✝ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ z, {f | f (x, z) ≤ f (x, i) + f (i, z)})

Next tactic:
apply I2 _ _
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    first
      |apply IsClosed.inter _ _
      |apply isClosed_iInter _
      |apply I1 _ _|apply I2 _ _|

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.ProdSpaceFun X Y
x✝ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ z, {f | f (x, z) ≤ f (x, i) + f (i, z)})

Next tactic:
apply I3 _ _
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    first
      |apply IsClosed.inter _ _
      |apply isClosed_iInter _
      |apply I1 _ _|apply I2 _ _|apply I3 _ _|

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.ProdSpaceFun X Y
x✝ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ z, {f | f (x, z) ≤ f (x, i) + f (i, z)})

Next tactic:
apply I4 _ _ _
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    first
      |apply IsClosed.inter _ _
      |apply isClosed_iInter _
      |apply I1 _ _|apply I2 _ _|apply I3 _ _|apply I4 _ _ _|

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.ProdSpaceFun X Y
x✝ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ z, {f | f (x, z) ≤ f (x, i) + f (i, z)})

Next tactic:
apply I5 _
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    first
      |apply IsClosed.inter _ _
      |apply isClosed_iInter _
      |apply I1 _ _|apply I2 _ _|apply I3 _ _|apply I4 _ _ _|apply I5 _|

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.ProdSpaceFun X Y
x✝ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ z, {f | f (x, z) ≤ f (x, i) + f (i, z)})

Next tactic:
apply I6 _ _
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    first
      |apply IsClosed.inter _ _
      |apply isClosed_iInter _
      |apply I1 _ _|apply I2 _ _|apply I3 _ _|apply I4 _ _ _|apply I5 _|apply I6 _ _|

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.ProdSpaceFun X Y
x✝ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ z, {f | f (x, z) ≤ f (x, i) + f (i, z)})

Next tactic:
intro x
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    

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.ProdSpaceFun X Y
x✝¹ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x✝ x : X ⊕ Y
⊢ IsClosed (⋂ z, {f | f (x✝, z) ≤ f (x✝, x) + f (x, z)})

Next tactic:
first
      |apply IsClosed.inter _ _
      |apply isClosed_iInter _
      |apply I1 _ _|apply I2 _ _|apply I3 _ _|apply I4 _ _ _|apply I5 _|apply I6 _ _|intro x
/-- To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. -/
private theorem closed_candidatesB : IsClosed (candidatesB X Y) := by
  have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
    isClosed_eq continuous_eval_const continuous_const
  have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
    isClosed_eq continuous_eval_const continuous_eval_const
  have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
    isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
  have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
    isClosed_eq continuous_eval_const continuous_const
  have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
    isClosed_le continuous_eval_const continuous_const
  have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
      ⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
      ⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
      ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
    ext
    simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
  rw [this]
  repeat'
    first
      |

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.ProdSpaceFun X Y
x✝¹ y z t : X ⊕ Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
this :
  GromovHausdorff.candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)}
x✝ x : X ⊕ Y
⊢ IsClosed (⋂ z, {f | f (x✝, z) ≤ f (x✝, x) + f (x, z)})

Next tactic:
apply IsClosed.inter _ _