mathlib-initiative/mathlib-tactics huggingface dataset:
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} ∪ F)).Nonempty

Tactic:
]

Goal state after:
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} ∪ F)).Nonempty
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : 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✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : 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
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
t : β
hinfi_le : ∀ y0 ∈ Y, ∃ x ∈ X, f x y0 ≤ t
a : E
ha : a ∈ X
t' : β
htt' : t < t'
y1 y2 : { x // x ∈ Y }
ht : ∀ x ∈ X, t < max (f x ↑y1) (f x ↑y2)
ha' : IsMinOn (fun x ↦ max (f x ↑y1) (f x ↑y2)) X a
ht' : t' < max (f a ↑y1) (f a ↑y2)
J1 : Set ↑(segment ℝ ↑y1 ↑y2) := setOfPred_sublevelLeft_subset X Y f t t' y1 y2
mem_J1_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J1 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y1
φ : ↑(segment ℝ ↑y1 ↑y2) ≃ₜ ↑(segment ℝ ↑y2 ↑y1) := Homeomorph.setCongr ⋯
J2 : Set ↑(segment ℝ ↑y1 ↑y2) := ⇑φ ⁻¹' setOfPred_sublevelLeft_subset X Y f t t' y2 y1
mem_J2_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J2 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
h_mem_Y : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), ↑z ∈ Y
hJ1J2 : J1 ∩ J2 = ∅
z : { x // x ∈ segment ℝ ↑y1 ↑y2 }
h2 : sublevelLeft X f t' ↑z ⊆ sublevelLeft X f t' ↑y2
⊢ z ∈ J2

Tactic:
mem_J2_iff,

Goal state after:
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : 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✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : 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
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
t : β
hinfi_le : ∀ y0 ∈ Y, ∃ x ∈ X, f x y0 ≤ t
a : E
ha : a ∈ X
t' : β
htt' : t < t'
y1 y2 : { x // x ∈ Y }
ht : ∀ x ∈ X, t < max (f x ↑y1) (f x ↑y2)
ha' : IsMinOn (fun x ↦ max (f x ↑y1) (f x ↑y2)) X a
ht' : t' < max (f a ↑y1) (f a ↑y2)
J1 : Set ↑(segment ℝ ↑y1 ↑y2) := setOfPred_sublevelLeft_subset X Y f t t' y1 y2
mem_J1_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J1 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y1
φ : ↑(segment ℝ ↑y1 ↑y2) ≃ₜ ↑(segment ℝ ↑y2 ↑y1) := Homeomorph.setCongr ⋯
J2 : Set ↑(segment ℝ ↑y1 ↑y2) := ⇑φ ⁻¹' setOfPred_sublevelLeft_subset X Y f t t' y2 y1
mem_J2_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J2 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
h_mem_Y : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), ↑z ∈ Y
hJ1J2 : J1 ∩ J2 = ∅
z : { x // x ∈ segment ℝ ↑y1 ↑y2 }
h2 : sublevelLeft X f t' ↑z ⊆ sublevelLeft X f t' ↑y2
⊢ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
Module: Mathlib.Topology.UniformSpace.CompareReals

Goal state before:
pp: s : Set (ℚ × ℚ)
⊢ (∃ i, 0 < i ∧ {p | AbsoluteValue.abs (p.2 - p.1) < i} ⊆ s) ↔ ∃ i, 0 < i ∧ {p | dist p.1 p.2 < ↑i} ⊆ s

Tactic:
]

Goal state after:
s : Set (ℚ × ℚ)
⊢ (∃ i, 0 < i ∧ {p | AbsoluteValue.abs (p.2 - p.1) < i} ⊆ s) ↔ ∃ i, 0 < i ∧ {p | dist p.1 p.2 < ↑i} ⊆ s
Module: Mathlib.Topology.UniformSpace.CompareReals

Goal state before:
pp: ⊢ IsUniformInducing Rat.cast

Tactic:
]

Goal state after:
⊢ IsUniformInducing Rat.cast
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : 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✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : 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
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
t : β
hinfi_le : ∀ y0 ∈ Y, ∃ x ∈ X, f x y0 ≤ t
a : E
ha : a ∈ X
t' : β
htt' : t < t'
y1 y2 : { x // x ∈ Y }
ht : ∀ x ∈ X, t < max (f x ↑y1) (f x ↑y2)
ha' : IsMinOn (fun x ↦ max (f x ↑y1) (f x ↑y2)) X a
ht' : t' < max (f a ↑y1) (f a ↑y2)
J1 : Set ↑(segment ℝ ↑y1 ↑y2) := setOfPred_sublevelLeft_subset X Y f t t' y1 y2
mem_J1_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J1 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y1
φ : ↑(segment ℝ ↑y1 ↑y2) ≃ₜ ↑(segment ℝ ↑y2 ↑y1) := Homeomorph.setCongr ⋯
J2 : Set ↑(segment ℝ ↑y1 ↑y2) := ⇑φ ⁻¹' setOfPred_sublevelLeft_subset X Y f t t' y2 y1
mem_J2_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J2 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
h_mem_Y : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), ↑z ∈ Y
hJ1J2 : J1 ∩ J2 = ∅
z : { x // x ∈ segment ℝ ↑y1 ↑y2 }
h2 : sublevelLeft X f t' ↑z ⊆ sublevelLeft X f t' ↑y2
⊢ sublevelLeft X f t' ↑y2 ⊆ sublevelLeft X f t' ↑y2

Tactic:
]

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
s : Set α
hF_cont : ∀ (i : ι), ContinuousOn (F i) s
hF_mono : ∀ x ∈ s, Monotone fun x_1 ↦ F x_1 x
hf : ContinuousOn f s
h_tendsto : ∀ x ∈ s, Tendsto (fun x_1 ↦ F x_1 x) atTop (𝓝 (f x))
⊢ TendstoLocallyUniformly (fun i x ↦ F i ↑x) (f ∘ Subtype.val) atTop

Tactic:
]

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
s : Set α
hF_cont : ∀ (i : ι), ContinuousOn (F i) s
hF_mono : ∀ x ∈ s, Monotone fun x_1 ↦ F x_1 x
hf : ContinuousOn f s
h_tendsto : ∀ x ∈ s, Tendsto (fun x_1 ↦ F x_1 x) atTop (𝓝 (f x))
⊢ TendstoLocallyUniformly (fun i x ↦ F i ↑x) (f ∘ Subtype.val) atTop
Module: Mathlib.Topology.Sion

Goal state before:
pp: case inl
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : 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✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : 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
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
t : β
hinfi_le : ∀ y0 ∈ Y, ∃ x ∈ X, f x y0 ≤ t
a : E
ha : a ∈ X
t' : β
htt' : t < t'
y1 y2 : { x // x ∈ Y }
ht : ∀ x ∈ X, t < max (f x ↑y1) (f x ↑y2)
ha' : IsMinOn (fun x ↦ max (f x ↑y1) (f x ↑y2)) X a
ht' : t' < max (f a ↑y1) (f a ↑y2)
J1 : Set ↑(segment ℝ ↑y1 ↑y2) := setOfPred_sublevelLeft_subset X Y f t t' y1 y2
mem_J1_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J1 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y1
φ : ↑(segment ℝ ↑y1 ↑y2) ≃ₜ ↑(segment ℝ ↑y2 ↑y1) := Homeomorph.setCongr ⋯
J2 : Set ↑(segment ℝ ↑y1 ↑y2) := ⇑φ ⁻¹' setOfPred_sublevelLeft_subset X Y f t t' y2 y1
mem_J2_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J2 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
h_mem_Y : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), ↑z ∈ Y
hJ1J2 : ∀ (x : ↑(segment ℝ ↑y1 ↑y2)), x ∉ J1 ∩ J2
hJ1_union_J2 : Subtype.val '' (J1 ∪ J2) = segment ℝ ↑y1 ↑y2
this :
  ∀ (u v : Set ↑(segment ℝ ↑y1 ↑y2)), IsClosed u → IsClosed v → univ ⊆ u ∪ v → univ ∩ (u ∩ v) = ∅ → univ ⊆ u ∨ univ ⊆ v
hJ1 : IsClosed J1
hJ2 : IsClosed J2
h_univ : univ ⊆ J1 ∪ J2
h_inter : univ ∩ (J1 ∩ J2) = ∅
h1 : univ ⊆ J1
⊢ sublevelLeft X f t' ↑y2 ⊆ sublevelLeft X f t' ↑y2

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: case inr
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : 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✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : 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
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
t : β
hinfi_le : ∀ y0 ∈ Y, ∃ x ∈ X, f x y0 ≤ t
a : E
ha : a ∈ X
t' : β
htt' : t < t'
y1 y2 : { x // x ∈ Y }
ht : ∀ x ∈ X, t < max (f x ↑y1) (f x ↑y2)
ha' : IsMinOn (fun x ↦ max (f x ↑y1) (f x ↑y2)) X a
ht' : t' < max (f a ↑y1) (f a ↑y2)
J1 : Set ↑(segment ℝ ↑y1 ↑y2) := setOfPred_sublevelLeft_subset X Y f t t' y1 y2
mem_J1_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J1 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y1
φ : ↑(segment ℝ ↑y1 ↑y2) ≃ₜ ↑(segment ℝ ↑y2 ↑y1) := Homeomorph.setCongr ⋯
J2 : Set ↑(segment ℝ ↑y1 ↑y2) := ⇑φ ⁻¹' setOfPred_sublevelLeft_subset X Y f t t' y2 y1
mem_J2_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J2 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
h_mem_Y : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), ↑z ∈ Y
hJ1J2 : ∀ (x : ↑(segment ℝ ↑y1 ↑y2)), x ∉ J1 ∩ J2
hJ1_union_J2 : Subtype.val '' (J1 ∪ J2) = segment ℝ ↑y1 ↑y2
this :
  ∀ (u v : Set ↑(segment ℝ ↑y1 ↑y2)), IsClosed u → IsClosed v → univ ⊆ u ∪ v → univ ∩ (u ∩ v) = ∅ → univ ⊆ u ∨ univ ⊆ v
hJ1 : IsClosed J1
hJ2 : IsClosed J2
h_univ : univ ⊆ J1 ∪ J2
h_inter : univ ∩ (J1 ∩ J2) = ∅
h2 : univ ⊆ J2
⊢ sublevelLeft X f t' ↑y1 ⊆ sublevelLeft X f t' ↑y1

Tactic:
]

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

Goal state before:
pp: X : Type u_1
V : SetRel X X
inst✝¹ : V.IsSymm
inst✝ : V.IsTrans
x y : X
hy : y ∈ ball x V
⊢ y ∈ ball x V

Tactic:
]

Goal state after:
X : Type u_1
V : SetRel X X
inst✝¹ : V.IsSymm
inst✝ : V.IsTrans
x y : X
hy : y ∈ ball x V
⊢ y ∈ ball x V
Module: Mathlib.Topology.UniformSpace.Ultra.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
s : Set X
⊢ (∃ V ∈ 𝓤 X, ball x V ⊆ s) ↔ ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ ball x V ⊆ s

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
s : Set X
⊢ (∃ V ∈ 𝓤 X, ball x V ⊆ s) ↔ ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ ball x V ⊆ s
Module: Mathlib.Topology.UniformSpace.Ultra.Basic

Goal state before:
pp: case mp
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
s : Set X
V : Set (X × X)
V_in : ∃ i, (i ∈ 𝓤 X ∧ i.IsSymm ∧ i.IsTrans) ∧ id i ⊆ V
V_sub : ball x V ⊆ s
⊢ ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ ball x V ⊆ s

Tactic:
]

Goal state after:
case mp
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
s : Set X
V : Set (X × X)
V_in : ∃ i, (i ∈ 𝓤 X ∧ i.IsSymm ∧ i.IsTrans) ∧ id i ⊆ V
V_sub : ball x V ⊆ s
⊢ ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ ball x V ⊆ s
Module: Mathlib.Topology.UniformSpace.Ultra.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : UniformSpace X
x : X
V : SetRel X X
inst✝ : V.IsTrans
h' : V ∈ 𝓤 X
⊢ ∀ x_1 ∈ ball x V, ∃ V_1 ∈ 𝓤 X, ball x_1 V_1 ⊆ ball x V

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : UniformSpace X
x : X
V : SetRel X X
inst✝ : V.IsTrans
h' : V ∈ 𝓤 X
⊢ ∀ x_1 ∈ ball x V, ∃ V_1 ∈ 𝓤 X, ball x_1 V_1 ⊆ ball x V
Module: Mathlib.Topology.UniformSpace.Ultra.Basic

Goal state before:
pp: X : Type u_1
inst✝² : UniformSpace X
x : X
V : SetRel X X
inst✝¹ : V.IsSymm
inst✝ : V.IsTrans
h' : V ∈ 𝓤 X
⊢ ∀ x_1 ∈ (ball x V)ᶜ, ∃ V_1 ∈ 𝓤 X, ball x_1 V_1 ⊆ (ball x V)ᶜ

Tactic:
]

Goal state after:
X : Type u_1
inst✝² : UniformSpace X
x : X
V : SetRel X X
inst✝¹ : V.IsSymm
inst✝ : V.IsTrans
h' : V ∈ 𝓤 X
⊢ ∀ x_1 ∈ (ball x V)ᶜ, ∃ V_1 ∈ 𝓤 X, ball x_1 V_1 ⊆ (ball x V)ᶜ
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun n ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
n : ℕ
⊢ dist (X3 n) (X3 (n + 1)) ≤ (1 / 2) ^ n

Tactic:
]

Goal state after:
X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun n ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
n : ℕ
⊢ dist (X3 n) (X3 (n + 1)) ≤ (1 / 2) ^ n
Module: Mathlib.Topology.UniformSpace.Ultra.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
inst✝ : DiscreteUniformity X
this : (uniformity X).HasBasis (fun x ↦ True) fun x ↦ SetRel.id
⊢ IsUltraUniformity X

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
inst✝ : DiscreteUniformity X
this : (uniformity X).HasBasis (fun x ↦ True) fun x ↦ SetRel.id
⊢ IsUltraUniformity X
Module: Mathlib.Topology.UniformSpace.Ultra.Constructions

Goal state before:
pp: X : Type u_1
this✝ : UniformSpace X := ⊤
this : (uniformity X).HasBasis (fun x ↦ True) fun x ↦ Set.univ
⊢ IsUltraUniformity X

Tactic:
]

Goal state after:
X : Type u_1
this✝ : UniformSpace X := ⊤
this : (uniformity X).HasBasis (fun x ↦ True) fun x ↦ Set.univ
⊢ IsUltraUniformity X
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
this :
  (𝓤 (SeparationQuotient X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans) fun i ↦
    Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
⊢ IsUltraUniformity (SeparationQuotient X)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : UniformSpace X
inst✝¹ : UniformSpace Y
inst✝ : IsUltraUniformity X
this :
  (𝓤 (SeparationQuotient X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans) fun i ↦
    Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
⊢ IsUltraUniformity (SeparationQuotient X)
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
this :
  (𝓤 (SeparationQuotient X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans) fun i ↦
    Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
x✝¹ : SetRel X X
x✝ : x✝¹ ∈ 𝓤 X ∧ x✝¹.IsSymm ∧ x✝¹.IsTrans
left✝ : x✝¹ ∈ 𝓤 X
hU : x✝¹.IsSymm
right✝ : x✝¹.IsTrans
⊢ SetRel.IsSymm (Prod.map SeparationQuotient.mk SeparationQuotient.mk '' x✝¹)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : UniformSpace X
inst✝¹ : UniformSpace Y
inst✝ : IsUltraUniformity X
this :
  (𝓤 (SeparationQuotient X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans) fun i ↦
    Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
x✝¹ : SetRel X X
x✝ : x✝¹ ∈ 𝓤 X ∧ x✝¹.IsSymm ∧ x✝¹.IsTrans
left✝ : x✝¹ ∈ 𝓤 X
hU : x✝¹.IsSymm
right✝ : x✝¹.IsTrans
⊢ SetRel.IsSymm (Prod.map SeparationQuotient.mk SeparationQuotient.mk '' x✝¹)
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
this :
  (𝓤 (SeparationQuotient X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans) fun i ↦
    Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
U : SetRel X X
hU' : U ∈ 𝓤 X
left✝ : U.IsSymm
hU : U.IsTrans
a b : X
hab : (a, b) ∈ U
c d : X
hcd : (c, d) ∈ U
hc : ∀ s ∈ 𝓤 X, (b, c) ∈ s
⊢ (b, c) ∈ U

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : UniformSpace X
inst✝¹ : UniformSpace Y
inst✝ : IsUltraUniformity X
this :
  (𝓤 (SeparationQuotient X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans) fun i ↦
    Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
U : SetRel X X
hU' : U ∈ 𝓤 X
left✝ : U.IsSymm
hU : U.IsTrans
a b : X
hab : (a, b) ∈ U
c d : X
hcd : (c, d) ∈ U
hc : ∀ s ∈ 𝓤 X, (b, c) ∈ s
⊢ (b, c) ∈ U
Module: Mathlib.Topology.UniformSpace.Ultra.Completion

Goal state before:
pp: X : Type u_1
inst✝ : UniformSpace X
⊢ IsUltraUniformity (SeparationQuotient (CauchyFilter X)) ↔ IsUltraUniformity (UniformSpace.Completion X)

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : UniformSpace X
⊢ IsUltraUniformity (SeparationQuotient (CauchyFilter X)) ↔ IsUltraUniformity (UniformSpace.Completion X)
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this✝ : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun n ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
n : ℕ
⊢ Nonempty (↥(X3 n) ≃ᵢ (u n).Rep)

Tactic:
]

Goal state after:
X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this✝ : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun n ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
n : ℕ
⊢ Nonempty (↥(X3 n) ≃ᵢ (u n).Rep)