mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
⊢ Continuous[inst✝⁴, inst✝²] f → ScottContinuousOn D f

Tactic:
refine fun hf t h₀ d₁ d₂ a d₃ ↦
      ⟨(monotone_of_continuous (D := D) hf).mem_upperBounds_image d₃.1,
      fun b hb ↦ ?_⟩
    by_contra h
    let u := (Iic b)ᶜ
    have hu : IsOpen (f ⁻¹' u) := isClosed_Iic.isOpen_compl.preimage hf
    rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)] at hu
    obtain ⟨c, hcd, hfcb⟩ := hu.2 h₀ d₁ d₂ d₃ h
    simp only [upperBounds, mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
      mem_ofPred] at hb
    exact hfcb <| hb _ hcd

Goal state after:
Module: Mathlib.Topology.OmegaCompletePartialOrder

Goal state before:
pp: α : Type u_1
inst✝² : OmegaCompletePartialOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α (range fun c ↦ range ⇑c)
f : α → Prop
⊢ ScottContinuousOn (range fun c ↦ range ⇑c) f ↔ Continuous[inst✝¹, _] f

Tactic:
scottContinuousOn_iff_continuous (fun a b hab => by
    use Chain.pair a b hab; exact OmegaCompletePartialOrder.Chain.range_pair a b hab)

Goal state after:
α : Type u_1
inst✝² : OmegaCompletePartialOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α (range fun c ↦ range ⇑c)
f : α → Prop
⊢ Continuous[inst✝¹, _] f ↔ Continuous[inst✝¹, _] f
Module: Mathlib.Topology.Order.Hom.Basic

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝⁷ : TopologicalSpace α
inst✝⁶ : Preorder α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace γ
inst✝² : Preorder γ
inst✝¹ : TopologicalSpace δ
inst✝ : Preorder δ
f g : α →Co β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by
    obtain ⟨⟨_, _⟩, _⟩ := f
    obtain ⟨⟨_, _⟩, _⟩ := g
    congr

Goal state after:
Module: Mathlib.Topology.Order.Hom.Esakia

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : Preorder γ
inst✝ : Preorder δ
f g : PseudoEpimorphism α β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by
    obtain ⟨⟨_, _⟩, _⟩ := f
    obtain ⟨⟨_, _⟩, _⟩ := g
    congr

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

Goal state before:
pp: α : Type ?u.3
inst✝ : CompleteLattice α
T : Set α
hG : OrderGenerates T
a : αᵒᵈ
⊢ a ≤ OrderDual.toDual (kernel (hull T (OrderDual.ofDual a)))

Tactic:
obtain ⟨S, rfl⟩ := hG a
    rw [OrderDual.le_toDual, kernel, kernel]
    exact sInf_le_sInf <| image_val_mono fun c hcS => by
      rw [hull, mem_preimage, mem_Ici]
      exact sInf_le (mem_image_of_mem Subtype.val hcS)

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

Goal state before:
pp: x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
⊢ dist x z ≤ dist x y + dist y z

Tactic:
let Ψ : Y → γ2 := optimalGHInjl Y Z

Goal state after:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
⊢ dist x z ≤ dist x y + dist y z
Module: Mathlib.Topology.Sheaves.Abelian

Goal state before:
pp: X✝ : TopCat
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
p₀ : ↑X
S : ShortComplex (Sheaf C X)
h : S.ShortExact
⊢ (S.map (forget C X ⋙ stalkFunctor C p₀)).ShortExact

Tactic:
have := ((forget C X ⋙ stalkFunctor C p₀).preservesFiniteColimits_tfae.out 3 0).mp
      (inferInstance : (PreservesFiniteColimits _))

Goal state after:
X✝ : TopCat
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
p₀ : ↑X
S : ShortComplex (Sheaf C X)
h : S.ShortExact
this :
  ∀ (S : ShortComplex (Sheaf C X)),
    S.ShortExact → (S.map (forget C X ⋙ stalkFunctor C p₀)).Exact ∧ Epi ((forget C X ⋙ stalkFunctor C p₀).map S.g)
⊢ (S.map (forget C X ⋙ stalkFunctor C p₀)).ShortExact
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
y : ↥U
g : ToType (F.stalk x)
V : Opens ↑X
f : ToType (F.obj (op V))
hyV : ↑y ∈ V
hgf : (ConcreteCategory.hom (F.germ U ↑y ⋯)) (surjInv ⋯ g) = (ConcreteCategory.hom (F.germ V ↑y ⋯)) f
W : Opens ↑X
hyW : ↑y ∈ W
ιWU : W ⟶ U
ιWV : W ⟶ V
hW : (ConcreteCategory.hom (F.map ιWU.op)) (surjInv ⋯ g) = (ConcreteCategory.hom (F.map ιWV.op)) f
z : ↥U
⊢ z ∈ Subtype.val ⁻¹' ↑W →
    ∃ (h : ↑z ∈ V), (ConcreteCategory.hom (F.germ U ↑z ⋯)) (surjInv ⋯ g) = (ConcreteCategory.hom (F.germ V (↑z) h)) f

Tactic:
hz

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
y : ↥U
g : ToType (F.stalk x)
V : Opens ↑X
f : ToType (F.obj (op V))
hyV : ↑y ∈ V
hgf : (ConcreteCategory.hom (F.germ U ↑y ⋯)) (surjInv ⋯ g) = (ConcreteCategory.hom (F.germ V ↑y ⋯)) f
W : Opens ↑X
hyW : ↑y ∈ W
ιWU : W ⟶ U
ιWV : W ⟶ V
hW : (ConcreteCategory.hom (F.map ιWU.op)) (surjInv ⋯ g) = (ConcreteCategory.hom (F.map ιWV.op)) f
z : ↥U
hz : z ∈ Subtype.val ⁻¹' ↑W
⊢ ∃ (h : ↑z ∈ V), (ConcreteCategory.hom (F.germ U ↑z ⋯)) (surjInv ⋯ g) = (ConcreteCategory.hom (F.germ V (↑z) h)) f
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ (X_1 : (CategoryTheory.Pairwise ι)ᵒᵖ) →
    ((Functor.const (CategoryTheory.Pairwise ι)ᵒᵖ).obj c.pt).obj X_1 ⟶ ((diagram U).op ⋙ F).obj X_1

Tactic:
by
        intro x
        induction x with | op x => ?_
        rcases x with (⟨i⟩ | ⟨i, j⟩)
        · exact c.π.app WalkingParallelPair.zero ≫ Pi.π _ i
        · exact c.π.app WalkingParallelPair.one ≫ Pi.π _ (i, j)

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt ≫
        ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app j =
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app j

Tactic:
rintro ⟨_ | _⟩

Goal state after:
case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt ≫
      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.zero =
    ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.zero

case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt ≫
      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.one =
    ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.one
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
        ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app j =
      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app j

Tactic:
rintro ⟨_ | _⟩

Goal state after:
case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.zero =
    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.zero

case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.one =
    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.one
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝⁸ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝² : TopologicalSpace F
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
cY : Convex ℝ Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
a : E
b b' : β
y y' : ↑Y
ha : ∀ x ∈ X, max (f a ↑y) (f a ↑y') ≤ max (f x ↑y) (f x ↑y')
hb : ∀ y ∈ Y, ∃ x ∈ X, f x y ≤ b
hb' : b' < max (f a ↑y) (f a ↑y')
hbb' : b < b'
J : Set ↑(segment ℝ ↑y ↑y') := setOfPred_sublevelLeft_subset X Y f b b' y y'
z : ↑(segment ℝ ↑y ↑y')
⊢ ClusterPt z (𝓟 J) → z ∈ J

Tactic:
hz

Goal state after:
E : Type u_1
F : Type u_2
β : Type u_3
inst✝⁸ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝² : TopologicalSpace F
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
cY : Convex ℝ Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
a : E
b b' : β
y y' : ↑Y
ha : ∀ x ∈ X, max (f a ↑y) (f a ↑y') ≤ max (f x ↑y) (f x ↑y')
hb : ∀ y ∈ Y, ∃ x ∈ X, f x y ≤ b
hb' : b' < max (f a ↑y) (f a ↑y')
hbb' : b < b'
J : Set ↑(segment ℝ ↑y ↑y') := setOfPred_sublevelLeft_subset X Y f b b' y y'
z : ↑(segment ℝ ↑y ↑y')
hz : ClusterPt z (𝓟 J)
⊢ z ∈ J
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case neg.refine_2
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 = ∅
A₁ : Set (Set X)
hA₁ : A₁ ⊆ insert Y₁ B
hA₁' : A₁.Finite
hA₁'' : ⋂₀ A₁ = ∅
A₂ : Set (Set X)
hA₂ : A₂ ⊆ insert Y₂ B
hA₂' : A₂.Finite
hA₂'' : ⋂₀ A₂ = ∅
this : ¬(Z ∩ ⋂₀ (A₁ \ {Y₁} ∪ A₂ \ {Y₂})).Nonempty
F : Set (Set X)
hsub : F ⊆ B'
hF : F.Finite
⊢ (⋂₀ (A₁ \ {Y₁} ∪ A₂ \ {Y₂}) ∩ ⋂₀ F).Nonempty

Tactic:
rw [← Set.sInter_union]
      refine hB.prop.2.1 (_ ∪ F) ?_ <| (hA₁'.sdiff.union hA₂'.sdiff).union hF
      grind [Set.sdiff_singleton_subset_iff, Set.union_subset_iff]

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
G : Type u_3
inst✝⁵ : Preorder ι
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup G
inst✝² : Lattice G
inst✝¹ : HasSolidNorm G
inst✝ : IsOrderedAddMonoid G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F i)
hF_mono : Monotone F
hf : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
x✝ : atTop.NeBot
F_le_f : ∀ (x : α) (n : ι), F n x ≤ f x
ε : ℝ
ε_pos : ε > 0
x : α
h_tendsto : ∀ (x : α), Tendsto (fun e ↦ ‖F e x - f x‖) atTop (𝓝 0)
n : ι
hn : ‖F n x - f x‖ < ε
m : ι
hnm : n ≤ m
z : α
⊢ ‖F n z - f z‖ < ε → ‖F m z - f z‖ < ε

Tactic:
hz

Goal state after:
ι : Type u_1
α : Type u_2
G : Type u_3
inst✝⁵ : Preorder ι
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup G
inst✝² : Lattice G
inst✝¹ : HasSolidNorm G
inst✝ : IsOrderedAddMonoid G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F i)
hF_mono : Monotone F
hf : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
x✝ : atTop.NeBot
F_le_f : ∀ (x : α) (n : ι), F n x ≤ f x
ε : ℝ
ε_pos : ε > 0
x : α
h_tendsto : ∀ (x : α), Tendsto (fun e ↦ ‖F e x - f x‖) atTop (𝓝 0)
n : ι
hn : ‖F n x - f x‖ < ε
m : ι
hnm : n ≤ m
z : α
hz : ‖F n z - f z‖ < ε
⊢ ‖F m z - f z‖ < ε
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:
rw [← mem_closure_iff_clusterPt, (h.prod .rfl).mem_closed_iff isClosed_closure,
        closure_compl] at clV

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 : (y, y) ∈ (interior V)ᶜ
h : Inseparable x y
⊢ False