-- Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
theorem closed_candidatesB : IsClosed (candidatesB X Y)

Proof state:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace 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 {f | f (x, i) ≤ ↑(GromovHausdorff.maxVar X Y)}

Tactic:
intro x

Result:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace 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 {f | f (x✝, x) ≤ ↑(GromovHausdorff.maxVar X Y)}

Proof state:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
⊢ IsClosed (GromovHausdorff.candidatesB X Y)

Tactic:
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

Result:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
⊢ IsClosed (GromovHausdorff.candidatesB X Y)

Proof state:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace 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
⊢ IsClosed {f | f (x, x) = 0}

Tactic:
apply I5 _

Result:
no goals

Proof state:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace 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 {f | f (x✝, x) ≤ ↑(GromovHausdorff.maxVar X Y)}

Tactic:
first
  |apply IsClosed.inter _ _
  |apply isClosed_iInter _
  |apply I1 _ _|apply I2 _ _|apply I3 _ _|apply I4 _ _ _|apply I5 _|apply I6 _ _|intro x

Result:
no goals

Proof state:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace 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 {f | f (x✝, x) ≤ ↑(GromovHausdorff.maxVar X Y)}

Tactic:
apply I6 _ _

Result:
no goals

Proof state:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace 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
⊢ IsClosed {f | f (inr x✝, inr x) = dist x✝ x}

Tactic:
apply I2 _ _

Result:
no goals

Proof state:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace 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 {f | f (x✝, x) = f (x, x✝)}

Tactic:
apply I3 _ _

Result:
no goals