mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∈ xs
hy : y ∈ xs
⊢ x = y

pp: case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∈ xs
hy : y ∉ xs
⊢ x = y

pp: case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∈ xs
⊢ x = y

pp: case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∉ xs
⊢ x = y

Tactic:
·

Goal state after:
case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∈ xs
hy : y ∈ xs
⊢ x = y
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: case inl
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
f : X → α
⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ⊥ (𝓝 (UniformFun.ofFun f)) ↔ Tendsto F ⊥ (𝓝 f)

pp: case inr
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
ℱ : Filter ι
f : X → α
ℱ_ne : ℱ.NeBot
⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f)) ↔ Tendsto F ℱ (𝓝 f)

Tactic:
·

Goal state after:
case inl
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
f : X → α
⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ⊥ (𝓝 (UniformFun.ofFun f)) ↔ Tendsto F ⊥ (𝓝 f)
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: case inr.mp
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
ℱ : Filter ι
f : X → α
ℱ_ne : ℱ.NeBot
H : Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f))
⊢ Tendsto F ℱ (𝓝 f)

pp: case inr.mpr
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
ℱ : Filter ι
f : X → α
ℱ_ne : ℱ.NeBot
H : Tendsto F ℱ (𝓝 f)
⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f))

Tactic:
·

Goal state after:
case inr.mp
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
ℱ : Filter ι
f : X → α
ℱ_ne : ℱ.NeBot
H : Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f))
⊢ Tendsto F ℱ (𝓝 f)
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: case inr.mpr
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
ℱ : Filter ι
f : X → α
ℱ_ne : ℱ.NeBot
H : Tendsto F ℱ (𝓝 f)
⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f))

Tactic:
·

Goal state after:
case inr.mpr
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
ℱ : Filter ι
f : X → α
ℱ_ne : ℱ.NeBot
H : Tendsto F ℱ (𝓝 f)
⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f))
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case pos.e_a
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
i : ℕ
h : ys[i]? = some (List.applyId (xs.zip ys) y)
hx : xs[i]? = some x
j : ℕ
hy : xs[j]? = some y
h₂ : xs.length = ys.length
⊢ ys[i]? = ys[j]?

pp: α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
i : ℕ
h : ys[i]? = some (List.applyId (xs.zip ys) y)
hx : xs[i]? = some x
j : ℕ
hy : xs[j]? = some y
h₂ : xs.length = ys.length
⊢ i < ys.length

Tactic:
·

Goal state after:
case pos.e_a
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
i : ℕ
h : ys[i]? = some (List.applyId (xs.zip ys) y)
hx : xs[i]? = some x
j : ℕ
hy : xs[j]? = some y
h₂ : xs.length = ys.length
⊢ ys[i]? = ys[j]?
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: ι : Type u_1
X : Type u_2
α : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
F : ι → X → α
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
ℱ : Filter ι
f : X → α
ℱ_ne : ℱ.NeBot
H : Tendsto F ℱ (𝓝 f)
S : Set (X → α) := closure[Pi.topologicalSpace] (range F)
𝒢 : Filter ↑S := comap Subtype.val (map F ℱ)
⊢ Equicontinuous (id ∘ Subtype.val)

Tactic:
by rwa [equicontinuous_iff_range] at F_eqcont

Goal state after:
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
i : ℕ
h : ys[i]? = some (List.applyId (xs.zip ys) y)
hx : xs[i]? = some x
j : ℕ
hy : xs[j]? = some y
h₂ : xs.length = ys.length
⊢ i < ys.length

Tactic:
·

Goal state after:
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
i : ℕ
h : ys[i]? = some (List.applyId (xs.zip ys) y)
hx : xs[i]? = some x
j : ℕ
hy : xs[j]? = some y
h₂ : xs.length = ys.length
⊢ i < ys.length
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∈ xs
hy : y ∉ xs
⊢ x = y

pp: case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∈ xs
⊢ x = y

pp: case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∉ xs
⊢ x = y

Tactic:
·

Goal state after:
case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∈ xs
hy : y ∉ xs
⊢ x = y
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∈ xs
⊢ x = y

pp: case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∉ xs
⊢ x = y

Tactic:
·

Goal state after:
case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∈ xs
⊢ x = y
Module: Mathlib.Topology.Algebra.Group.CompactOpen

Goal state before:
pp: case mp
X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : IsUniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := ⋯
S2 : Set (X →ₜ* Y) := ⋯
S3 : Set C(X, Y) := ⋯
S4 : Set (X → Y) := ⋯
h : Equicontinuous fun x ↦ ⇑↑x
x✝ : X → Y
⊢ x✝ ∈ S4 → x✝ ∈ DFunLike.coe '' S1

pp: case mpr
X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : IsUniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := ⋯
S2 : Set (X →ₜ* Y) := ⋯
S3 : Set C(X, Y) := ⋯
S4 : Set (X → Y) := ⋯
h : Equicontinuous fun x ↦ ⇑↑x
x✝ : X → Y
⊢ x✝ ∈ DFunLike.coe '' S1 → x✝ ∈ S4

Tactic:
·

Goal state after:
case mp
X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : IsUniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := ⋯
S2 : Set (X →ₜ* Y) := ⋯
S3 : Set C(X, Y) := ⋯
S4 : Set (X → Y) := ⋯
h : Equicontinuous fun x ↦ ⇑↑x
x✝ : X → Y
⊢ x✝ ∈ S4 → x✝ ∈ DFunLike.coe '' S1
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∉ xs
⊢ x = y

Tactic:
·

Goal state after:
case neg
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : List.applyId (xs.zip ys) x = List.applyId (xs.zip ys) y
hx : x ∉ xs
hy : y ∉ xs
⊢ x = y
Module: Mathlib.Topology.Algebra.Group.CompactOpen

Goal state before:
pp: case mpr
X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : IsUniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := ⋯
S2 : Set (X →ₜ* Y) := ⋯
S3 : Set C(X, Y) := ⋯
S4 : Set (X → Y) := ⋯
h : Equicontinuous fun x ↦ ⇑↑x
x✝ : X → Y
⊢ x✝ ∈ DFunLike.coe '' S1 → x✝ ∈ S4

Tactic:
·

Goal state after:
case mpr
X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : IsUniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := ⋯
S2 : Set (X →ₜ* Y) := ⋯
S3 : Set C(X, Y) := ⋯
S4 : Set (X → Y) := ⋯
h : Equicontinuous fun x ↦ ⇑↑x
x✝ : X → Y
⊢ x✝ ∈ DFunLike.coe '' S1 → x✝ ∈ S4
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case h₀
α : Type u
inst✝ : DecidableEq α
xs : List ((_ : α) × α)
xs₀ : List α
h₀ : map Sigma.fst xs = xs₀
xs₁ : List α
h₁ : map Sigma.snd xs = xs₁
hxs : xs = List.toFinmap' (xs₀.zip xs₁)
hperm : map Sigma.fst (List.toFinmap' (xs₀.zip xs₁)) ~ map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))
hnodup : (map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))).Nodup
⊢ xs₀.Nodup

pp: case h₁
α : Type u
inst✝ : DecidableEq α
xs : List ((_ : α) × α)
xs₀ : List α
h₀ : map Sigma.fst xs = xs₀
xs₁ : List α
h₁ : map Sigma.snd xs = xs₁
hxs : xs = List.toFinmap' (xs₀.zip xs₁)
hperm : map Sigma.fst (List.toFinmap' (xs₀.zip xs₁)) ~ map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))
hnodup : (map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))).Nodup
⊢ xs₀ ~ xs₁

Tactic:
·

Goal state after:
case h₀
α : Type u
inst✝ : DecidableEq α
xs : List ((_ : α) × α)
xs₀ : List α
h₀ : map Sigma.fst xs = xs₀
xs₁ : List α
h₁ : map Sigma.snd xs = xs₁
hxs : xs = List.toFinmap' (xs₀.zip xs₁)
hperm : map Sigma.fst (List.toFinmap' (xs₀.zip xs₁)) ~ map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))
hnodup : (map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))).Nodup
⊢ xs₀.Nodup
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case h₁
α : Type u
inst✝ : DecidableEq α
xs : List ((_ : α) × α)
xs₀ : List α
h₀ : map Sigma.fst xs = xs₀
xs₁ : List α
h₁ : map Sigma.snd xs = xs₁
hxs : xs = List.toFinmap' (xs₀.zip xs₁)
hperm : map Sigma.fst (List.toFinmap' (xs₀.zip xs₁)) ~ map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))
hnodup : (map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))).Nodup
⊢ xs₀ ~ xs₁

Tactic:
·

Goal state after:
case h₁
α : Type u
inst✝ : DecidableEq α
xs : List ((_ : α) × α)
xs₀ : List α
h₀ : map Sigma.fst xs = xs₀
xs₁ : List α
h₁ : map Sigma.snd xs = xs₁
hxs : xs = List.toFinmap' (xs₀.zip xs₁)
hperm : map Sigma.fst (List.toFinmap' (xs₀.zip xs₁)) ~ map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))
hnodup : (map Sigma.snd (List.toFinmap' (xs₀.zip xs₁))).Nodup
⊢ xs₀ ~ xs₁
Module: Mathlib.Topology.Algebra.Group.OpenMapping

Goal state before:
pp: case hc
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed[inst✝⁹] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := ⋯
F : ℕ × ↑s → Set X := ⋯
this✝ : Nonempty X
this : Encodable ↑s
⊢ ∀ (i : ℕ × ↑s), IsClosed[inst✝⁸] (F i)

pp: case hU
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed[inst✝⁹] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := ⋯
F : ℕ × ↑s → Set X := ⋯
this✝ : Nonempty X
this : Encodable ↑s
⊢ ⋃ i, F i = univ

Tactic:
·

Goal state after:
case hc
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed[inst✝⁹] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := ⋯
F : ℕ × ↑s → Set X := ⋯
this✝ : Nonempty X
this : Encodable ↑s
⊢ ∀ (i : ℕ × ↑s), IsClosed[inst✝⁸] (F i)
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
⊢ Tendsto (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((UniformOnFun.ofFun 𝔖) f)) ↔ Tendsto F ℱ (𝓝 f)

Tactic:
_root_.nhds_iInf,

Goal state after:
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
⊢ Tendsto (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (⨅ i ∈ 𝔖, 𝓝 ((UniformOnFun.ofFun 𝔖) f)) ↔ Tendsto F ℱ (⨅ i ∈ 𝔖, 𝓝 f)
Module: Mathlib.Topology.Algebra.Group.CompactOpen

Goal state before:
pp: X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : IsUniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := {f | Set.MapsTo (⇑f) U W}
S2 : Set (X →ₜ* Y) := {f | Set.MapsTo (⇑f) U W}
S3 : Set C(X, Y) := _root_.toContinuousMap '' S2
S4 : Set (X → Y) := DFunLike.coe '' S3
h : Equicontinuous fun x ↦ ⇑↑x
hS4 : IsClosed[Pi.topologicalSpace] S4
hS2 : (interior S2).Nonempty
⊢ LocallyCompactSpace (X →ₜ* Y)

Tactic:
exact TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
    ⟨⟨S2, (isInducing_toContinuousMap X Y).isCompact_iff.mpr
      (ArzelaAscoli.isCompact_of_equicontinuous S3 hS4.isCompact h)⟩, hS2⟩

Goal state after:
Module: Mathlib.Topology.Algebra.Group.OpenMapping

Goal state before:
pp: case hU
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed[inst✝⁹] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := ⋯
F : ℕ × ↑s → Set X := ⋯
this✝ : Nonempty X
this : Encodable ↑s
⊢ ⋃ i, F i = univ

Tactic:
·

Goal state after:
case hU
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed[inst✝⁹] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := ⋯
F : ℕ × ↑s → Set X := ⋯
this✝ : Nonempty X
this : Encodable ↑s
⊢ ⋃ i, F i = univ
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: case inl
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
H : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(UniformOnFun.ofFun 𝔖) ∘ F))
h✝ : IsEmpty α
⊢ IsClosed[Pi.topologicalSpace] (range ((⋃₀ 𝔖).domRestrict ∘ F))

pp: case inr
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
H : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(UniformOnFun.ofFun 𝔖) ∘ F))
h✝ : Nonempty α
⊢ IsClosed[Pi.topologicalSpace] (range ((⋃₀ 𝔖).domRestrict ∘ F))

Tactic:
·

Goal state after:
case inl
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
H : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(UniformOnFun.ofFun 𝔖) ∘ F))
h✝ : IsEmpty α
⊢ IsClosed[Pi.topologicalSpace] (range ((⋃₀ 𝔖).domRestrict ∘ F))
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion

Goal state before:
pp: G : GrpCat
P : ProfiniteGrp.{u}
f : G ⟶ GrpCat.of ↑P.toProfinite.toTop
X Y : OpenNormalSubgroup ↑P.toProfinite.toTop
g : X ⟶ Y
x :
  (j : FiniteIndexNormalSubgroup ↑G) →
    ↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙ compHausToTop).obj j)
hx :
  x ∈
    {u |
      ∀ {i j : FiniteIndexNormalSubgroup ↑G} (f : i ⟶ j),
        (ConcreteCategory.hom
              ((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙ compHausToTop).map f))
            (u i) =
          u j}
⊢ (Hom.hom
        (((Functor.const (OpenNormalSubgroup ↑P.toProfinite.toTop)).obj (limitCone (diagram G)).pt).map g ≫
          (limitCone (diagram G)).π.app (preimage f Y) ≫ ofFiniteGrpHom (quotientMap f Y)))
      ⟨x, hx⟩ =
    (Hom.hom (((limitCone (diagram G)).π.app (preimage f X) ≫ ofFiniteGrpHom (quotientMap f X)) ≫ P.diagram.map g))
      ⟨x, hx⟩

Tactic:
change quotientMap f Y (x <| preimage f Y) =
        P.diagram.map g (quotientMap _ _ <| x <| preimage f X)

Goal state after:
G : GrpCat
P : ProfiniteGrp.{u}
f : G ⟶ GrpCat.of ↑P.toProfinite.toTop
X Y : OpenNormalSubgroup ↑P.toProfinite.toTop
g : X ⟶ Y
x :
  (j : FiniteIndexNormalSubgroup ↑G) →
    ↑((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙ compHausToTop).obj j)
hx :
  x ∈
    {u |
      ∀ {i j : FiniteIndexNormalSubgroup ↑G} (f : i ⟶ j),
        (ConcreteCategory.hom
              ((((diagram G ⋙ forget₂ ProfiniteGrp.{u} Profinite) ⋙ profiniteToCompHaus) ⋙ compHausToTop).map f))
            (u i) =
          u j}
⊢ (ConcreteCategory.hom (quotientMap f Y)) (x (preimage f Y)) =
    (Hom.hom (P.diagram.map g)) ((ConcreteCategory.hom (quotientMap f X)) (x (preimage f X)))
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion

Goal state before:
pp: G : GrpCat
P : ProfiniteGrp.{u}
f : G ⟶ GrpCat.of ↑P.toProfinite.toTop
e : P ≅ limit P.diagram := P.isoLimittoFiniteQuotientFunctor
⊢ eta G ≫ (forget₂ ProfiniteGrp.{u} GrpCat).map (lift f) = f

Tactic:
rw [← (forget₂ ProfiniteGrp GrpCat).mapIso e |>.cancel_iso_hom_right]

Goal state after:
G : GrpCat
P : ProfiniteGrp.{u}
f : G ⟶ GrpCat.of ↑P.toProfinite.toTop
e : P ≅ limit P.diagram := P.isoLimittoFiniteQuotientFunctor
⊢ (eta G ≫ (forget₂ ProfiniteGrp.{u} GrpCat).map (lift f)) ≫ ((forget₂ ProfiniteGrp.{u} GrpCat).mapIso e).hom =
    f ≫ ((forget₂ ProfiniteGrp.{u} GrpCat).mapIso e).hom
Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun x ↦ y ^ x) ↔ MapClusterPt x atTop fun x ↦ y ^ x
tfae_3_to_4 : x ∈ closure[inst✝²] (range fun x ↦ y ^ x) → x ∈ closure[inst✝²] (range fun x ↦ y ^ x)
tfae_4_to_1 : x ∈ closure[inst✝²] (range fun x ↦ y ^ x) → MapClusterPt x atTop fun x ↦ y ^ x
tfae_1_to_3 : (MapClusterPt x atTop fun x ↦ y ^ x) → x ∈ closure[inst✝²] (range fun x ↦ y ^ x)
⊢ [MapClusterPt x atTop fun x ↦ y ^ x, MapClusterPt x atTop fun x ↦ y ^ x, x ∈ closure[inst✝²] (range fun x ↦ y ^ x),
      x ∈ closure[inst✝²] (range fun x ↦ y ^ x)].TFAE

Tactic:
tfae_finish

Goal state after:
Module: Mathlib.Topology.Algebra.Group.OpenMapping

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g✝ : G
U : Set G
hU : U ∈ 𝓝 g✝
g : G
⊢ g • x = ((fun x_1 ↦ x_1 • g✝ • x) ∘ fun x ↦ x * g✝⁻¹) g

Tactic:
simp [smul_smul]

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

Goal state before:
pp: X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ IsInducing (⇑(UniformOnFun.ofFun ?m.83) ∘ Subtype.val ∘ ⇑(Equiv.Set.image DFunLike.coe S ⋯))

pp: case 𝔖_covers
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ ⋃₀ ?m.83 = univ

pp: case 𝔖_compact
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ ∀ K ∈ ?m.83, IsCompact K

pp: case F_eqcont
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ ∀ K ∈ ?m.83, EquicontinuousOn (Subtype.val ∘ ⇑(Equiv.Set.image DFunLike.coe S ⋯)) K

pp: X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ Set (Set X)

Tactic:
·

Goal state after:
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ IsInducing (⇑(UniformOnFun.ofFun ?m.83) ∘ Subtype.val ∘ ⇑(Equiv.Set.image DFunLike.coe S ⋯))
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: case 𝔖_covers
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ ⋃₀ {K | IsCompact K} = univ

pp: case 𝔖_compact
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ ∀ K ∈ {K | IsCompact K}, IsCompact K

pp: case F_eqcont
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ ∀ K ∈ {K | IsCompact K}, EquicontinuousOn (Subtype.val ∘ ⇑(Equiv.Set.image DFunLike.coe S ⋯)) K

Tactic:
·

Goal state after:
case 𝔖_covers
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ ⋃₀ {K | IsCompact K} = univ