mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hs : s.Nontrivial
hb : BddBelow (uncurry dist '' s.offDiag)
x✝ : ℝ
⊢ x✝ ≤ s.infsep ↔ x✝ ≤ ⨅ d, uncurry dist ↑d

Tactic:
hs.le_infsep_iff,

Goal state after:
case pos
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hs : s.Nontrivial
hb : BddBelow (uncurry dist '' s.offDiag)
x✝ : ℝ
⊢ (∀ x ∈ s, ∀ y ∈ s, x ≠ y → x✝ ≤ dist x y) ↔ x✝ ≤ ⨅ d, uncurry dist ↑d
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case right
X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
h : IsometryRel.setoid p (NonemptyCompacts.kuratowskiEmbedding X)
e : ↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥p
f : X ≃ᵢ ↥p
⊢ Subtype.val '' range ⇑f = ↑p

Tactic:
f.range_eq_univ,

Goal state after:
case right
X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
h : IsometryRel.setoid p (NonemptyCompacts.kuratowskiEmbedding X)
e : ↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥p
f : X ≃ᵢ ↥p
⊢ Subtype.val '' univ = ↑p
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case pos
α : Type u_1
inst✝² : PseudoMetricSpace α
s : Set α
inst✝¹ : Decidable s.Nontrivial
inst✝ : Fintype ↑s
hs : s.Nontrivial
x✝ : ℝ
⊢ x✝ ≤ s.infsep ↔ x✝ ≤ s.offDiag.toFinset.inf' ⋯ (uncurry dist)

Tactic:
hs.le_infsep_iff,

Goal state after:
case pos
α : Type u_1
inst✝² : PseudoMetricSpace α
s : Set α
inst✝¹ : Decidable s.Nontrivial
inst✝ : Fintype ↑s
hs : s.Nontrivial
x✝ : ℝ
⊢ (∀ x ∈ s, ∀ y ∈ s, x ≠ y → x✝ ≤ dist x y) ↔ x✝ ≤ s.offDiag.toFinset.inf' ⋯ (uncurry dist)
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hsf : s.Finite
hs : s.Nontrivial
x✝ : ℝ
⊢ x✝ ≤ s.infsep ↔ x✝ ≤ ⋯.toFinset.inf' ⋯ (uncurry dist)

Tactic:
hs.le_infsep_iff,

Goal state after:
case pos
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hsf : s.Finite
hs : s.Nontrivial
x✝ : ℝ
⊢ (∀ x ∈ s, ∀ y ∈ s, x ≠ y → x✝ ≤ dist x y) ↔ x✝ ≤ ⋯.toFinset.inf' ⋯ (uncurry dist)
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
e : ↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding Y)
⊢ Nonempty (X ≃ᵢ Y)

Tactic:
have I :
      (NonemptyCompacts.kuratowskiEmbedding X ≃ᵢ NonemptyCompacts.kuratowskiEmbedding Y) =
        (range (kuratowskiEmbedding X) ≃ᵢ range (kuratowskiEmbedding Y)) := by
      dsimp only [NonemptyCompacts.kuratowskiEmbedding]; rfl

Goal state after:
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
e : ↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding Y)
I :
  (↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding Y)) =
    (↑(range (kuratowskiEmbedding X)) ≃ᵢ ↑(range (kuratowskiEmbedding Y)))
⊢ Nonempty (X ≃ᵢ Y)
Module: Mathlib.Topology.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace (DedekindCut α)
inst✝ : OrderTopology (DedekindCut α)
⊢ Continuous fun a ↦ principal a

Tactic:
rw [OrderTopology.continuous_iff]
  refine fun c ↦ ⟨?_, ?_⟩
  · have : IsOpen (⋃ a ∈ c.right, Ioi a) := isOpen_biUnion fun _ _ ↦ isOpen_Ioi
    convert this
    ext
    simp [lt_principal_iff]
  · have : IsOpen (⋃ a ∈ c.left, Iio a) := isOpen_biUnion fun _ _ ↦ isOpen_Iio
    convert this
    ext
    simp [principal_lt_iff]

Goal state after:
Module: Mathlib.Topology.Order.NhdsSet

Goal state before:
pp: case inr
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
a b : α
h : a ≤ b
hlt : a < b
⊢ 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b)

Tactic:
rw [← Ioc_insert_left h, nhdsSet_insert, nhdsSet_Ioc hlt, sup_assoc]

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

Goal state before:
pp: case refine_1.h₁
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
A :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y →
        hausdorffDist ↑p ↑q < diam univ + 1 + diam univ →
          hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
B :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y → hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
⊢ ((fun p ↦ hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = toGHSpace X} ×ˢ {b | ⟦b⟧ = toGHSpace Y}).Nonempty

pp: case h₂
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
A :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y →
        hausdorffDist ↑p ↑q < diam univ + 1 + diam univ →
          hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
B :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y → hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
⊢ ∀ b ∈ (fun p ↦ hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = toGHSpace X} ×ˢ {b | ⟦b⟧ = toGHSpace Y},
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ b

Tactic:
· refine (Set.Nonempty.prod ?_ ?_).image _ <;> exact ⟨_, rfl⟩

Goal state after:
case h₂
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
A :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y →
        hausdorffDist ↑p ↑q < diam univ + 1 + diam univ →
          hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
B :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y → hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
⊢ ∀ b ∈ (fun p ↦ hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = toGHSpace X} ×ˢ {b | ⟦b⟧ = toGHSpace Y},
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ b
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case refine_3
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
F : OptimalGHCoupling X Y → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding (OptimalGHCoupling X Y)
Φ : X → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjl X Y
Ψ : Y → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjr X Y
⊢ ghDist X Y = hausdorffDist (F '' optimalGHInjl X Y '' univ) (Ψ '' univ)

Tactic:
image_univ,

Goal state after:
case refine_3
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
F : OptimalGHCoupling X Y → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding (OptimalGHCoupling X Y)
Φ : X → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjl X Y
Ψ : Y → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjr X Y
⊢ ghDist X Y = hausdorffDist (F '' range (optimalGHInjl X Y)) (Ψ '' univ)
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case refine_3
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
F : OptimalGHCoupling X Y → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding (OptimalGHCoupling X Y)
Φ : X → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjl X Y
Ψ : Y → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjr X Y
⊢ ghDist X Y = hausdorffDist (F '' range (optimalGHInjl X Y)) (F '' optimalGHInjr X Y '' univ)

Tactic:
image_univ,

Goal state after:
case refine_3
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
F : OptimalGHCoupling X Y → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding (OptimalGHCoupling X Y)
Φ : X → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjl X Y
Ψ : Y → ↥(lp (fun x ↦ ℝ) ∞) := F ∘ optimalGHInjr X Y
⊢ ghDist X Y = hausdorffDist (F '' range (optimalGHInjl X Y)) (F '' range (optimalGHInjr X Y))
Module: Mathlib.Topology.Separation.LinearUpperLowerSetTopology

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : Topology.IsUpperSet α
s t : Set α
a : α
ha : a ∈ s
b : α
hsct : Disjoint {a} (closure {b})
hcst : a < b
hb : b ∈ t
| Disjoint {a} (closure {b})

Tactic:
equals b < a => simp

Goal state after:
Module: Mathlib.Topology.Separation.PerfectlyNormal

Goal state before:
pp: case neg.refine_2.e_s.refine_2.a
X : Type u_1
inst✝ : TopologicalSpace X
h✝ : ∀ (s : Set X), IsClosed[inst✝] s → ∃ f, s = ⇑f ⁻¹' {0} ∧ ∀ (x : X), f x ∈ Icc 0 1
s : Set X
hs : IsClosed[inst✝] s
hse : s.Nonempty
f : C(X, ℝ)
hf : s = ⇑f ⁻¹' {0}
hfr : ∀ (x : X), f x ∈ Icc 0 1
x : ℝ
h : x ∈ ⋂ i, range ⇑f ∩ Iio (1 / (↑i + 1))
⊢ x ≤ 0

Tactic:
simp only [mem_iInter, mem_inter_iff, mem_Iio] at h
              exact ge_of_tendsto' tendsto_one_div_add_atTop_nhds_zero_nat (fun n => (h n).2.le)

Goal state after:
Module: Mathlib.Topology.Sheaves.EtaleSpace

Goal state before:
pp: X : TopCat
C : Type u
inst✝⁴ : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
inst✝¹ : Limits.HasColimits C
F : Presheaf C X
inst✝ : Limits.PreservesFilteredColimits (forget C)
U : Opens ↑X
hF_bij : ∀ (x : ↑X) (hx : x ∈ U), Bijective ⇑(ConcreteCategory.hom (F.germ U x hx))
x : ↑X
hx : x ∈ U
base : ↑X
s : ToType (F.stalk base)
hs : { base := base, germ := s } ∈ EtaleSpace.base ⁻¹' ↑U
⊢ ⟨{ base := base,
        germ :=
          (ConcreteCategory.hom (F.germ U base ⋯)) (surjInv ⋯ ((ConcreteCategory.hom (F.germ U x hx)) (surjInv ⋯ s))) },
      ⋯⟩ =
    ⟨{ base := base, germ := s }, hs⟩

Tactic:
congr 2

Goal state after:
X : TopCat
C : Type u
inst✝⁴ : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
inst✝¹ : Limits.HasColimits C
F : Presheaf C X
inst✝ : Limits.PreservesFilteredColimits (forget C)
U : Opens ↑X
hF_bij : ∀ (x : ↑X) (hx : x ∈ U), Bijective ⇑(ConcreteCategory.hom (F.germ U x hx))
x : ↑X
hx : x ∈ U
base : ↑X
s : ToType (F.stalk base)
hs : { base := base, germ := s } ∈ EtaleSpace.base ⁻¹' ↑U
⊢ (ConcreteCategory.hom (F.germ U base ⋯)) (surjInv ⋯ ((ConcreteCategory.hom (F.germ U x hx)) (surjInv ⋯ s))) = s
Module: Mathlib.Topology.Sheaves.Alexandrov

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasLimits C
X : TopCat
inst✝¹ : Preorder ↑X
inst✝ : Topology.IsUpperSet ↑X
F : ↑X ⥤ C
⊢ TopCat.Presheaf.IsSheaf (principalsKanExtension F)

Tactic:
rw [isSheaf_iff_isSheafOpensLeCover]
  intro ι Us
  constructor
  apply isLimit

Goal state after:
Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: X : TopCat
C : Type v
inst✝¹ : Category.{w, v} C
F : Presheaf C X
Y : TopCat
inst✝ : F.IsFlasque
f : X ⟶ Y
U V : (Opens ↑Y)ᵒᵖ
i : U ⟶ V
⊢ Epi (((pushforward C f).obj F).map i)

Tactic:
simp only [pushforward_obj_obj, pushforward_obj_map]

Goal state after:
X : TopCat
C : Type v
inst✝¹ : Category.{w, v} C
F : Presheaf C X
Y : TopCat
inst✝ : F.IsFlasque
f : X ⟶ Y
U V : (Opens ↑Y)ᵒᵖ
i : U ⟶ V
⊢ Epi (F.map ((Opens.map f).map i.unop).op)
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g ≤ HD g + dist f g

Tactic:
HD,

Goal state after:
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g ≤ max (⨆ x, ⨅ y, g (inl x, inr y)) (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g ≤ HD g + dist f g

Tactic:
HD,

Goal state after:
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g ≤ max (⨆ x, ⨅ y, g (inl x, inr y)) (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : QuasiSober X
inst✝¹ : PrespectralSpace X
inst✝ : QuasiSeparatedSpace X
𝔅 : Set (Set X) := constructibleTopologySubbasis X
𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}
s : Set (Set X)
hs : s ∈ 𝒮
B : Set (Set X)
hsB : s ⊆ B
hB : Maximal (fun x ↦ x ∈ 𝒮) B
B' : Set (Set X) := {s | s ∈ B ∧ IsClosed[inst✝⁴] s}
Z : Set X := ⋂₀ B'
hZ_closed : IsClosed[inst✝⁴] Z
hZ_nonempty : Z.Nonempty
hZ_irred : ¬IsPreirreducible Z
Y₁ Y₂ : Set X
hY₁_cl : IsClosed[inst✝⁴] Y₁
hY₂_cl : IsClosed[inst✝⁴] Y₂
hc₁ : IsCompact Y₁ᶜ
hc₂ : IsCompact Y₂ᶜ
hsub : Z ⊆ Y₁ ∪ Y₂
x₁ : X
hx₁ : x₁ ∈ Z ∩ Y₁ᶜ
x₂ : X
hx₂ : x₂ ∈ Z ∩ Y₂ᶜ
hY₁ : Y₁ ∈ 𝔅
hY₂ : Y₂ ∈ 𝔅
hY₁_inter_B : Y₁ ∩ ⋂₀ B = ∅
hY₂_inter_B : Y₂ ∩ ⋂₀ B = ∅
hY₁B : insert Y₁ B ∉ 𝒮
hY₂B : insert Y₂ B ∈ 𝒮
⊢ False

Tactic:
grind [show insert Y₂ B ⊆ B from hB.le_of_ge hY₂B (Set.subset_insert Y₂ B)]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Ultra.Completion

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : UniformSpace X
inst✝¹ : UniformSpace Y
inst✝ : IsUltraUniformity X
⊢ IsUltraUniformity (SeparationQuotient X)

Tactic:
have := IsUltraUniformity.hasBasis.map
    (Prod.map SeparationQuotient.mk (SeparationQuotient.mk (X := X)))
  rw [← SeparationQuotient.uniformity_eq] at this
  apply mk_of_hasBasis this
  · exact fun _ ⟨_, hU, _⟩ ↦ by rw [id_eq]; infer_instance
  · rintro U ⟨hU', _, hU⟩
    constructor
    rintro x y z
    simp only [id_eq, Set.mem_image, Prod.exists, Prod.map_apply, Prod.mk.injEq,
      forall_exists_index, and_imp]
    rintro a b hab rfl rfl c d hcd hc rfl
    have hbc : (b, c) ∈ U := by
      rw [eq_comm, SeparationQuotient.mk_eq_mk, inseparable_iff_ker_uniformity,
          Filter.mem_ker] at hc
      exact hc _ hU'
    exact ⟨a, d, U.trans (U.trans hab hbc) hcd, by simp, by simp⟩

Goal state after:
Module: Mathlib.Topology.UniformSpace.OfCompactT2

Goal state before:
pp: γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)
F : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s
V : Set (γ × γ)
V_in : V ∈ 𝓝Δ
H : ¬F ⊓ 𝓟 Vᶜ = ⊥
this : (F ⊓ 𝓟 Vᶜ).NeBot
x y : γ
hxy : ClusterPt (x, y) (F ⊓ 𝓟 Vᶜ)
clV : ClusterPt (x, y) (𝓟 Vᶜ)
h : Inseparable x y
⊢ False

Tactic:
← mem_closure_iff_clusterPt,

Goal state after:
γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)
F : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s
V : Set (γ × γ)
V_in : V ∈ 𝓝Δ
H : ¬F ⊓ 𝓟 Vᶜ = ⊥
this : (F ⊓ 𝓟 Vᶜ).NeBot
x y : γ
hxy : ClusterPt (x, y) (F ⊓ 𝓟 Vᶜ)
clV : (x, y) ∈ closure[instTopologicalSpaceProd] Vᶜ
h : Inseparable x y
⊢ False