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

Goal state before:
pp: X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝² : IsCompact s
this✝¹ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this✝ : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
F : ↑s → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding ↑s
this : hausdorffDist (F '' range Φ') (F '' range Ψ') = hausdorffDist (range Φ') (range Ψ')
A : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Φ', isCompact' := ⋯, nonempty' := ⋯ }
B : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Ψ', isCompact' := ⋯, nonempty' := ⋯ }
⊢ ⟦A⟧ = toGHSpace X

Tactic:
rw [eq_toGHSpace_iff]

Goal state after:
X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝² : IsCompact s
this✝¹ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this✝ : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
F : ↑s → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding ↑s
this : hausdorffDist (F '' range Φ') (F '' range Ψ') = hausdorffDist (range Φ') (range Ψ')
A : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Φ', isCompact' := ⋯, nonempty' := ⋯ }
B : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Ψ', isCompact' := ⋯, nonempty' := ⋯ }
⊢ ∃ Ψ, Isometry Ψ ∧ range Ψ = ↑A
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝² : IsCompact s
this✝¹ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this✝ : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
F : ↑s → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding ↑s
this : hausdorffDist (F '' range Φ') (F '' range Ψ') = hausdorffDist (range Φ') (range Ψ')
A : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Φ', isCompact' := ⋯, nonempty' := ⋯ }
B : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Ψ', isCompact' := ⋯, nonempty' := ⋯ }
AX : ⟦A⟧ = toGHSpace X
⊢ ⟦B⟧ = toGHSpace Y

Tactic:
rw [eq_toGHSpace_iff]

Goal state after:
X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝² : IsCompact s
this✝¹ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this✝ : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
F : ↑s → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding ↑s
this : hausdorffDist (F '' range Φ') (F '' range Ψ') = hausdorffDist (range Φ') (range Ψ')
A : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Φ', isCompact' := ⋯, nonempty' := ⋯ }
B : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞) := { carrier := F '' range Ψ', isCompact' := ⋯, nonempty' := ⋯ }
AX : ⟦A⟧ = toGHSpace X
⊢ ∃ Ψ, Isometry Ψ ∧ range Ψ = ↑B
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Preorder α
⊢ Topology.IsLowerSet α

Tactic:
by
  let := lowerSet α
  exact ⟨rfl⟩

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

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
a : α
⊢ closure[inst✝¹] {a} = Iic a

Tactic:
rw [closure_eq_lowerClosure, lowerClosure_singleton]
  rfl

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

Goal state before:
pp: 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
⊢ ScottContinuousOn D f ↔ Continuous[inst✝⁴, inst✝²] f

Tactic:
refine ⟨fun h ↦ continuous_def.2 fun u hu ↦ ?_, ?_⟩

Goal state after:
case refine_1
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
h : ScottContinuousOn D f
u : Set β
hu : IsOpen[inst✝²] u
⊢ IsOpen[inst✝⁴] (f ⁻¹' u)

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
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun n ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun n ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
⊢ IsClosed (range Φ)

Tactic:
exact hΦ.isClosed

Goal state after:
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)), S.ShortExact → (S.map (forget C X ⋙ stalkFunctor C p₀)).ShortExact

Tactic:
intro S

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)
⊢ S.ShortExact → (S.map (forget C X ⋙ stalkFunctor C p₀)).ShortExact
Module: Mathlib.Topology.Sheaves.Alexandrov

Goal state before:
pp: X✝ : Type v
inst✝⁶ : TopologicalSpace X✝
inst✝⁵ : Preorder X✝
inst✝⁴ : Topology.IsUpperSet X✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasLimits C
F✝ : X✝ ⥤ C
X : TopCat
inst✝¹ : Preorder ↑X
inst✝ : Topology.IsUpperSet ↑X
F : ↑X ⥤ C
α : Type v
Us : α → Opens ↑X
⊢ ∀ (s : Cone ((ObjectProperty.ι fun V ↦ ∃ i, V ≤ Us i).op ⋙ principalsKanExtension F))
    (m : s.pt ⟶ ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).pt),
    (∀ (j : (ObjectProperty.FullSubcategory fun V ↦ ∃ i, V ≤ Us i)ᵒᵖ),
        m ≫ ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).π.app j = s.π.app j) →
      m = limit.lift (generator (iSup Us) ⋙ F) (lowerCone Us s)

Tactic:
intro S

Goal state after:
X✝ : Type v
inst✝⁶ : TopologicalSpace X✝
inst✝⁵ : Preorder X✝
inst✝⁴ : Topology.IsUpperSet X✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasLimits C
F✝ : X✝ ⥤ C
X : TopCat
inst✝¹ : Preorder ↑X
inst✝ : Topology.IsUpperSet ↑X
F : ↑X ⥤ C
α : Type v
Us : α → Opens ↑X
S : Cone ((ObjectProperty.ι fun V ↦ ∃ i, V ≤ Us i).op ⋙ principalsKanExtension F)
⊢ ∀ (m : S.pt ⟶ ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).pt),
    (∀ (j : (ObjectProperty.FullSubcategory fun V ↦ ∃ i, V ≤ Us i)ᵒᵖ),
        m ≫ ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).π.app j = S.π.app j) →
      m = limit.lift (generator (iSup Us) ⋙ F) (lowerCone Us S)
Module: Mathlib.Topology.Sheaves.MayerVietoris

Goal state before:
pp: case refine_2
T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
⊢ ∀ (s : PullbackCone sq.f₂₄ sq.f₃₄), homOfLE ⋯ ≫ sq.f₁₂ = s.fst

Tactic:
intros

Goal state after:
case refine_2
T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
s✝ : PullbackCone sq.f₂₄ sq.f₃₄
⊢ homOfLE ⋯ ≫ sq.f₁₂ = s✝.fst
Module: Mathlib.Topology.Sheaves.MayerVietoris

Goal state before:
pp: case refine_3
T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
⊢ ∀ (s : PullbackCone sq.f₂₄ sq.f₃₄), homOfLE ⋯ ≫ sq.f₁₃ = s.snd

Tactic:
intros

Goal state after:
case refine_3
T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
s✝ : PullbackCone sq.f₂₄ sq.f₃₄
⊢ homOfLE ⋯ ≫ sq.f₁₃ = s✝.snd
Module: Mathlib.Topology.Sheaves.MayerVietoris

Goal state before:
pp: case refine_4
T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
⊢ ∀ (s : PullbackCone sq.f₂₄ sq.f₃₄) (m : s.pt ⟶ sq.X₁), m ≫ sq.f₁₂ = s.fst → m ≫ sq.f₁₃ = s.snd → m = homOfLE ⋯

Tactic:
intros

Goal state after:
case refine_4
T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
s✝ : PullbackCone sq.f₂₄ sq.f₃₄
m✝ : s✝.pt ⟶ sq.X₁
x✝¹ : m✝ ≫ sq.f₁₂ = s✝.fst
x✝ : m✝ ≫ sq.f₁₃ = s✝.snd
⊢ m✝ = homOfLE ⋯
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
⊢ dist ↑x ↑y ≤ C

Tactic:
refine (dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ _) (mem_univ _)).trans ?_

Goal state after:
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
⊢ diam univ ≤ C
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
⊢ dist ↑(Ψ x) ↑(Ψ y) ≤ C

Tactic:
refine (dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ _) (mem_univ _)).trans ?_

Goal state after:
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
⊢ diam univ ≤ C
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝² : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
b : β
y y' : ↑Y
z : ↑(segment ℝ ↑y ↑y')
x : ↑X
hfx' :
  ∀ ⦃x_1 : F⦄,
    x_1 ∈ {x_2 | x_2 ∈ Y ∧ min (f ↑x ↑y) (f ↑x ↑y') ≤ (fun y ↦ f (↑x) y) x_2} →
      ∀ ⦃y_1 : F⦄,
        y_1 ∈ {x_2 | x_2 ∈ Y ∧ min (f ↑x ↑y) (f ↑x ↑y') ≤ (fun y ↦ f (↑x) y) x_2} →
          segment ℝ x_1 y_1 ⊆ {x_2 | x_2 ∈ Y ∧ min (f ↑x ↑y) (f ↑x ↑y') ≤ (fun y ↦ f (↑x) y) x_2}
hx : x ∈ sublevelLeft X f b ↑z
⊢ min (f ↑x ↑y) (f ↑x ↑y') ≤ b

Tactic:
specialize hfx' ⟨y.prop, inf_le_left⟩ ⟨y'.prop, inf_le_right⟩ z.prop

Goal state after:
E : Type u_1
F : Type u_2
β : Type u_3
inst✝² : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
b : β
y y' : ↑Y
z : ↑(segment ℝ ↑y ↑y')
x : ↑X
hfx' : ↑z ∈ {x_1 | x_1 ∈ Y ∧ min (f ↑x ↑y) (f ↑x ↑y') ≤ (fun y ↦ f (↑x) y) x_1}
hx : x ∈ sublevelLeft X f b ↑z
⊢ min (f ↑x ↑y) (f ↑x ↑y') ≤ b
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
n : ℕ
h : smallInductiveDimension X ≤ ↑n
⊢ ↑n < ↑(n + 1)

Tactic:
exact_mod_cast n.lt_add_one

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
n : ℕ
h : HasSmallInductiveDimensionLT X (n + 1)
⊢ ↑n < ↑(n + 1)

Tactic:
exact_mod_cast n.lt_add_one

Goal state after:
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) ↔ s ∈ uniformity ℚ

Tactic:
Metric.uniformity_basis_dist_rat.mem_iff

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.Path

Goal state before:
pp: X : Type u_1
inst✝ : UniformSpace X
x y z : X
U : Set (X × X)
hU : U ∈ 𝓤 X
x✝² x✝¹ : Path x y × Path y z
t : ↑I
fst✝¹ : Path x y
snd✝¹ : Path y z
fst✝ : Path x y
snd✝ : Path y z
x✝ :
  ((fst✝¹, snd✝¹), fst✝, snd✝) ∈
    entourageProd {γ | ∀ (t : ↑I), (γ.1 t, γ.2 t) ∈ (U, U).1} {γ | ∀ (t : ↑I), (γ.1 t, γ.2 t) ∈ (U, U).2}
h₁ : (fst✝¹, fst✝) ∈ {γ | ∀ (t : ↑I), (γ.1 t, γ.2 t) ∈ (U, U).1}
h₂ : (snd✝¹, snd✝) ∈ {γ | ∀ (t : ↑I), (γ.1 t, γ.2 t) ∈ (U, U).2}
⊢ ((Function.uncurry trans (fst✝¹, snd✝¹), Function.uncurry trans (fst✝, snd✝)).1 t,
      (Function.uncurry trans (fst✝¹, snd✝¹), Function.uncurry trans (fst✝, snd✝)).2 t) ∈
    U

Tactic:
by_cases ht : (t : ℝ) ≤ 2⁻¹ <;> simp [Path.trans_apply, ht, h₁ _, h₂ _]

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
⊢ ball x V = ball y V

Tactic:
by
  refine le_antisymm (ball_subset_of_mem ?_) (ball_subset_of_mem hy)
  rwa [← mem_ball_symmetry]

Goal state after:
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 ∈ 𝓤 X, ball x V ⊆ s) → ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ ball x V ⊆ s

Tactic:
rintro ⟨V, V_in, V_sub⟩
    rw [IsUltraUniformity.hasBasis.mem_iff'] at V_in
    obtain ⟨U, ⟨U_in, U_sym, U_trans⟩, U_sub⟩ := V_in
    refine ⟨U, U_in, U_sym, U_trans, (UniformSpace.ball_mono U_sub _).trans V_sub⟩

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
x✝¹ : SetRel X X
x✝ : x✝¹ ∈ 𝓤 X ∧ x✝¹.IsSymm ∧ x✝¹.IsTrans
left✝¹ : x✝¹ ∈ 𝓤 X
left✝ : x✝¹.IsSymm
hU : x✝¹.IsTrans
⊢ SetRel.IsTrans ((CauchyFilter.gen ∘ id) x✝¹)

Tactic:
by simpa using CauchyFilter.isTrans_gen

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 : (y, y) ∈ closure[instTopologicalSpaceProd] Vᶜ
h : Inseparable x y
⊢ False

Tactic:
closure_compl

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
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
cX : Convex ℝ X
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
x : E
hx : x ∈ X
y : F
hy : y ∈ Y
⊢ f x y ≤ sup_y x

Tactic:
· exact (hsup_y x hx).1 ⟨y, hy, rfl⟩

Goal state after: