mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: X : Type u_2
inst✝² : TopologicalSpace X
x : X
N : Type u_3
inst✝¹ : DecidableEq N
inst✝ : Unique N
p q : ↑(Ω^ N X x)
t : ↑I
⊢ ({ toFun := fun p ↦ { toFun := fun t ↦ p fun x ↦ t, continuous_toFun := ⋯, source' := ⋯, target' := ⋯ },
          invFun := fun p ↦ ⟨{ toFun := fun c ↦ p (c default), continuous_toFun := ⋯ }, ⋯⟩, left_inv := ⋯,
          right_inv := ⋯ }
        ⟨{
            toFun := fun t ↦
              if ↑(t default) ≤ 2⁻¹ then
                q (Function.update t default (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(t default))))
              else p (Function.update t default (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(t default) - 1))),
            continuous_toFun := ⋯ },
          ⋯⟩)
      t =
    if ↑t ≤ 2⁻¹ then
      (Path.extend
          ({ toFun := fun p ↦ { toFun := fun t ↦ p fun x ↦ t, continuous_toFun := ⋯, source' := ⋯, target' := ⋯ },
              invFun := fun p ↦ ⟨{ toFun := fun c ↦ p (c default), continuous_toFun := ⋯ }, ⋯⟩, left_inv := ⋯,
              right_inv := ⋯ }
            q))
        (2 * ↑t)
    else
      (Path.extend
          ({ toFun := fun p ↦ { toFun := fun t ↦ p fun x ↦ t, continuous_toFun := ⋯, source' := ⋯, target' := ⋯ },
              invFun := fun p ↦ ⟨{ toFun := fun c ↦ p (c default), continuous_toFun := ⋯ }, ⋯⟩, left_inv := ⋯,
              right_inv := ⋯ }
            p))
        (2 * ↑t - 1)

Tactic:
refine ite_congr rfl (fun _ ↦ congrArg q ?_)
    fun _ ↦ congrArg p ?_

Goal state after:
case refine_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
N : Type u_3
inst✝¹ : DecidableEq N
inst✝ : Unique N
p q : ↑(Ω^ N X x)
t : ↑I
x✝ : ↑t ≤ 2⁻¹
⊢ Function.update (fun x ↦ t) default (Set.projIcc 0 1 transAt._proof_3 (2 * ↑((fun x ↦ t) default))) = fun x ↦
    Set.projIcc 0 1 Path.extend._proof_2 (2 * ↑t)

case refine_2
X : Type u_2
inst✝² : TopologicalSpace X
x : X
N : Type u_3
inst✝¹ : DecidableEq N
inst✝ : Unique N
p q : ↑(Ω^ N X x)
t : ↑I
x✝ : ¬↑t ≤ 2⁻¹
⊢ Function.update (fun x ↦ t) default (Set.projIcc 0 1 transAt._proof_3 (2 * ↑((fun x ↦ t) default) - 1)) = fun x ↦
    Set.projIcc 0 1 Path.extend._proof_2 (2 * ↑t - 1)
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: case refine_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
N : Type u_3
inst✝¹ : DecidableEq N
inst✝ : Unique N
p q : ↑(Ω^ N X x)
t : ↑I
x✝ : ↑t ≤ 2⁻¹
i : N
⊢ ↑(Function.update (fun x ↦ t) default (Set.projIcc 0 1 transAt._proof_3 (2 * ↑((fun x ↦ t) default))) default) =
    ↑(Set.projIcc 0 1 Path.extend._proof_2 (2 * ↑t))

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : EMetricSpace α
⊢ 2 ≠ ∞

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: case refine_2
X : Type u_2
inst✝² : TopologicalSpace X
x : X
N : Type u_3
inst✝¹ : DecidableEq N
inst✝ : Unique N
p q : ↑(Ω^ N X x)
t : ↑I
x✝ : ¬↑t ≤ 2⁻¹
i : N
⊢ ↑(Function.update (fun x ↦ t) default (Set.projIcc 0 1 transAt._proof_3 (2 * ↑((fun x ↦ t) default) - 1)) default) =
    ↑(Set.projIcc 0 1 Path.extend._proof_2 (2 * ↑t - 1))

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋂ n, closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ m, ⋃ (_ : m ≥ n), ↑(s m))
⊢ ∃ x, Tendsto s atTop (𝓝 x)

Tactic:
let t : Closeds α := ⟨t0, isClosed_iInter fun _ => isClosed_closure⟩

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋂ n, closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ m, ⋃ (_ : m ≥ n), ↑(s m))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
⊢ ∃ x, Tendsto s atTop (𝓝 x)
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
⊢ 1 = 2 * 0 + 1 + 2 * 0

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
A : Set X
ε : ℝ≥0
h : ∀ i ⊆ A, IsSeparated (↑ε) i → i = ∅
x : X
hx : x ∈ A
⊢ IsSeparated ↑ε {x}

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
A : Set X
⊢ IsCover ε A A

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
A : Set X
⊢ A ⊆ A

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : MetricSpace Y
x y : X ⊕ Y
inst✝³ : CompactSpace X
inst✝² : Nonempty X
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
⊢ diam univ + (dist default default + 1 + dist default default) + diam univ = 1 * diam univ + 1 + 1 * diam univ

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: E : Type u_3
inst✝ : EMetricSpace E
A : Set E
⊢ A.encard ≤ ⨆ (_ : A ⊆ A), ⨆ (_ : IsSeparated (↑0) A), A.encard

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: case a
α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := ⋯
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋯
t : Closeds α := ⋯
n : ℕ
x : α
hx : x ∈ s n
l : ℕ
z : ↥(s (n + l))
⊢ n + l ≤ n + l

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: case a
α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := ⋯
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋯
t : Closeds α := ⋯
n : ℕ
x : α
hx : x ∈ s n
l : ℕ
z : ↥(s (n + l))
⊢ n + l ≤ n + l + 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
A : Set X
ε : ℝ≥0
h_nonempty : A.Nonempty
hA : ediam A ≤ ↑ε
a : X
ha : a ∈ A
⊢ {a}.encard ≤ 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : ProdSpaceFun X Y
fA : f ∈ candidates X Y
x : X
y : Y
⊢ ↑(maxVar X Y) = ↑(maxVar X Y) * 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : ProdSpaceFun X Y
fA : f ∈ candidates X Y
x : Y
y : X
⊢ ↑(maxVar X Y) = ↑(maxVar X Y) * 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
x : X
⊢ {x}.Nonempty

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
x : X
⊢ ediam {x} ≤ ↑ε

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
x : X
⊢ {x}.encard = 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
x : X
⊢ {x} ⊆ {x}

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
x : X
⊢ IsSeparated ↑ε {x}

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
x : X
⊢ 1 ≤ {x}.encard

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Kuratowski

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
ι : Type u_1
f : α → ↥(lp (fun x ↦ ℝ) ∞)
K : ℝ≥0
g : ι → α → ℝ
hgl : ∀ (i : ι), LipschitzWith K (g i)
hfl : ∀ (i : ι), LipschitzOnWith K (fun a ↦ ↑(f a) i) ∅
hgeq : ∀ (i : ι), EqOn (fun x ↦ ↑(f x) i) (g i) ∅
⊢ EqOn f 0 ∅

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
A : Set X
ε : ℝ≥0
h : packingNumber ε A ≠ ⊤
a : X
ha : a ∈ A
⊢ IsSeparated ↑ε {a}

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: case inr
X : Type u_1
inst✝ : PseudoEMetricSpace X
A : Set X
ε : ℝ≥0
h : packingNumber ε A ≠ ⊤
hA : A.Nonempty
⊢ ∃ C ⊆ A, C.Finite ∧ IsSeparated (↑ε) C ∧ C.encard = packingNumber ε A

Tactic:
have : Nonempty { s : Set X // s ⊆ A ∧ IsSeparated ε s } := by
    obtain ⟨a, ha⟩ := hA
    exact ⟨⟨{a}, by simp [ha], by simp⟩⟩

Goal state after:
case inr
X : Type u_1
inst✝ : PseudoEMetricSpace X
A : Set X
ε : ℝ≥0
h : packingNumber ε A ≠ ⊤
hA : A.Nonempty
this : Nonempty { s // s ⊆ A ∧ IsSeparated (↑ε) s }
⊢ ∃ C ⊆ A, C.Finite ∧ IsSeparated (↑ε) C ∧ C.encard = packingNumber ε A
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: 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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed
    ((((((⋂ 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) ≤ ↑(maxVar X Y)})

Tactic:
apply IsClosed.inter _ _

Goal state after:
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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed
    (((((⋂ 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 : 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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed (⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)})
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
A C : Set X
ε : ℝ≥0
h_subset : C ⊆ A
h_sep : IsSeparated (↑ε) C
h : packingNumber ε A ≠ ⊤
⊢ C.encard ≤ C.encard

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋂ n, closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ m, ⋃ (_ : m ≥ n), ↑(s m))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
I2 : ∀ (n : ℕ), ∀ x ∈ t0, ∃ y ∈ s n, edist x y ≤ 2 * B n
main : ∀ (n : ℕ), edist (s n) t ≤ 2 * B n
ε : ℝ≥0∞
εpos : ε > 0
⊢ 2⁻¹ < 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋂ n, closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ m, ⋃ (_ : m ≥ n), ↑(s m))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
I2 : ∀ (n : ℕ), ∀ x ∈ t0, ∃ y ∈ s n, edist x y ≤ 2 * B n
main : ∀ (n : ℕ), edist (s n) t ≤ 2 * B n
ε : ℝ≥0∞
εpos : ε > 0
⊢ 2 ≠ ∞

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: case inl
X : Type u_1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0
⊢ coveringNumber (2 * ε) ∅ ≤ externalCoveringNumber ε ∅

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: 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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed
    (((((⋂ 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})

Tactic:
apply IsClosed.inter _ _

Goal state after:
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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed
    ((((⋂ 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 : 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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed (⋂ x, {f | f (x, x) = 0})
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
g : ↑C → X
hg_mem : ∀ (x : ↑C), g x ∈ A
hg : ∀ (x : ↑C), f (g x) = ↑x
x✝ : Y
⊢ x✝ ∈ C ↔ x✝ ∈ f '' range g

Tactic:
simp

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
g : ↑C → X
hg_mem : ∀ (x : ↑C), g x ∈ A
hg : ∀ (x : ↑C), f (g x) = ↑x
x✝ : Y
⊢ x✝ ∈ C ↔ ∃ a, ∃ (h : a ∈ C), f (g ⟨a, ⋯⟩) = x✝
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: 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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed
    ((((⋂ 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)})

Tactic:
apply IsClosed.inter _ _

Goal state after:
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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed
    (((⋂ 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 : 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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed (⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)})
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case none
α : Type u_1
inst✝ : EDist α
ι : Type u_3
s : ι → Set α
⊢ (⋃ i ∈ none, s i).einfsep = ⨅ i ∈ none, (s i).einfsep

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case some
α : Type u_1
inst✝ : EDist α
ι : Type u_3
s : ι → Set α
val✝ : ι
⊢ (⋃ i ∈ some val✝, s i).einfsep = ⨅ i ∈ some val✝, (s i).einfsep

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: 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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed
    (((⋂ 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)})

Tactic:
apply IsClosed.inter _ _

Goal state after:
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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed ((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y})

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) ≤ ↑(maxVar X Y)}
this :
  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) ≤ ↑(maxVar X Y)}
⊢ IsClosed (⋂ x, ⋂ y, {f | f (x, y) = f (y, x)})