awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
f : X →* Y
hf : Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)
n : ℕ
ih : Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
⊢ Set.MapsTo (⇑f : (a : X) → Y) (U (n + 1)) (V (n + 1))

Tactic:
fun x hx ↦ hV (ih (hU3 n hx)) (map_mul f x x ▸ ih (hU2 n (Set.mul_mem_mul hx hx)))

Premises used: Eq.rec, map_mul, Set.MapsTo, Set.mul_mem_mul
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
f : X →* Y
hf : Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)
n : ℕ
ih : Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
⊢ Set.MapsTo (⇑f : (a : X) → Y) (U (n + 1)) (V (n + 1))

Tactic:
exact fun x hx ↦ hV (ih (hU3 n hx)) (map_mul f x x ▸ ih (hU2 n (Set.mul_mem_mul hx hx)))
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
f : X →* Y
hf : Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)
n : ℕ
zero : Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)
succ : ∀ (n : ℕ) (ih : Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)), Set.MapsTo (⇑f : (a : X) → Y) (U (n + 1)) (V (n + 1))
⊢ Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)

Tactic:
induction n with
    | zero => exact hf
    | succ n ih =>
      exact fun x hx ↦ hV (ih (hU3 n hx)) (map_mul f x x ▸ ih (hU2 n (Set.mul_mem_mul hx hx)))
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
⊢ ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)

Tactic:
have hU4 : ∀ f : X →* Y, Set.MapsTo f (U 0) (V 0) → ∀ n, Set.MapsTo f (U n) (V n) := by
    intro f hf n
    induction n with
    | zero => exact hf
    | succ n ih =>
      exact fun x hx ↦ hV (ih (hU3 n hx)) (map_mul f x x ▸ ih (hU2 n (Set.mul_mem_mul hx hx)))
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
apply :
  EquicontinuousAt (X := X) (α := Y)
    (fun (f : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))) ↦
      (⇑(↑f : X →* Y) : (a : X) → Y))
    1
⊢ LocallyCompactSpace (X →ₜ* Y)

Tactic:
apply locallyCompactSpace_of_equicontinuousAt (U 0) (V 0) hU0c (hVo.mem_of_mem trivial)

Premises used: trivial, ContinuousMonoidHom.locallyCompactSpace_of_equicontinuousAt, OfNat.ofNat, Filter.HasBasis.mem_of_mem
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
rw :
  ∀ (k : ℕ),
    True →
      ∀ᶠ (x : X) in 𝓝 1,
        ∀ (i : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))),
          Membership.mem.{u_8, u_8} (α := Y × Y) (γ := Set (Y × Y)) {x : Y × Y | x.2 / x.1 ∈ V k}
            ((((↑i : X →* Y) : (a : X) → Y) 1, ((↑i : X →* Y) : (a : X) → Y) x) : Y × Y)
⊢ EquicontinuousAt (X := X) (α := Y)
    (fun (f : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))) ↦
      (⇑(↑f : X →* Y) : (a : X) → Y))
    1

Tactic:
rw [hVo.uniformity_of_nhds_one.equicontinuousAt_iff_right]

Premises used: Filter.HasBasis.equicontinuousAt_iff_right, Filter.HasBasis.uniformity_of_nhds_one
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
⊢ ∀ (k : ℕ),
    True →
      ∀ᶠ (x : X) in 𝓝 1,
        ∀ (i : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))),
          Membership.mem.{u_8, u_8} (α := Y × Y) (γ := Set (Y × Y)) {x : Y × Y | x.2 / x.1 ∈ V k}
            ((((↑i : X →* Y) : (a : X) → Y) 1, ((↑i : X →* Y) : (a : X) → Y) x) : Y × Y)

Tactic:
fun n _ ↦ Filter.eventually_iff_exists_mem.mpr ⟨U n, hU1 n, fun x hx ⟨f, hf⟩ ↦ ?_⟩

Premises used: Iff.mpr, setOf, Membership.mem, HDiv.hDiv, Prod.mk, Prod.fst, _private.0.ContinuousMonoidHom.locallyCompactSpace_of_hasBasis.match_1, And.intro, Exists.intro, OfNat.ofNat, Subtype.val, Prod.snd, DFunLike.coe, Filter.eventually_iff_exists_mem
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
refine :
  ∀ (n : ℕ),
    True →
      ∀ (x : X) (hx : x ∈ U n) (x_1 : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8)))
        (f : X →* Y)
        (hf : Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} f),
        Membership.mem.{u_8, u_8} (α := Y × Y) (γ := Set (Y × Y)) {x : Y × Y | x.2 / x.1 ∈ V n}
          ((((↑(Subtype.mk (p := fun (x : X →* Y) ↦
                        Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} x) f
                        hf) :
                    X →* Y) :
                  (a : X) → Y)
                1,
              ((↑(Subtype.mk (p := fun (x : X →* Y) ↦
                        Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} x) f
                        hf) :
                    X →* Y) :
                  (a : X) → Y)
                x) :
            Y × Y)
⊢ ∀ (k : ℕ),
    True →
      ∀ᶠ (x : X) in 𝓝 1,
        ∀ (i : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))),
          Membership.mem.{u_8, u_8} (α := Y × Y) (γ := Set (Y × Y)) {x : Y × Y | x.2 / x.1 ∈ V k}
            ((((↑i : X →* Y) : (a : X) → Y) 1, ((↑i : X →* Y) : (a : X) → Y) x) : Y × Y)

Tactic:
refine fun n _ ↦ Filter.eventually_iff_exists_mem.mpr ⟨U n, hU1 n, fun x hx ⟨f, hf⟩ ↦ ?_⟩

Premises used: Iff.mpr, And.intro, Exists.intro, Filter.eventually_iff_exists_mem
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
n : ℕ
h : True
x : X
hx : x ∈ U n
x_1 : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))
f : X →* Y
hf : Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} f
rw :
  Membership.mem (α := Y) (V n)
    ((1,
          ((↑(Subtype.mk (p := fun (x : X →* Y) ↦
                    Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} x) f
                    hf) :
                X →* Y) :
              (a : X) → Y)
            x).2 :
      Y)
⊢ Membership.mem.{u_8, u_8} (α := Y × Y) (γ := Set (Y × Y)) {x : Y × Y | x.2 / x.1 ∈ V n}
    ((((↑(Subtype.mk (p := fun (x : X →* Y) ↦
                  Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} x) f hf) :
              X →* Y) :
            (a : X) → Y)
          1,
        ((↑(Subtype.mk (p := fun (x : X →* Y) ↦
                  Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} x) f hf) :
              X →* Y) :
            (a : X) → Y)
          x) :
      Y × Y)

Tactic:
rw [Set.mem_setOf_eq, map_one, div_one]

Premises used: Set.mem_setOf_eq, map_one, div_one
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
n : ℕ
h : True
x : X
hx : x ∈ U n
x_1 : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))
f : X →* Y
hf : Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} f
⊢ Membership.mem (α := Y) (V n)
    ((1,
          ((↑(Subtype.mk (p := fun (x : X →* Y) ↦
                    Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} x) f
                    hf) :
                X →* Y) :
              (a : X) → Y)
            x).2 :
      Y)

Tactic:
hU4 f hf n hx

Premises used: Set.Mem
-- Module: Mathlib.Topology.Algebra.Group.CompactOpen
Goal:
X : Type u_7
Y : Type u_8
inst : TopologicalSpace X
inst_1 : Group X
inst_2 : IsTopologicalGroup X
inst_3 : UniformSpace Y
inst_4 : CommGroup Y
inst_5 : IsUniformGroup Y
inst_6 : T0Space Y
inst_7 : CompactSpace Y
inst_8 : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : Filter.HasBasis (𝓝 1) (fun (x : ℕ) ↦ True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : Membership.mem (γ := Filter X) (𝓝 1) U0
U_aux : ℕ → (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7) :=
  fun (t : ℕ) ↦
    Nat.rec (motive := fun (x : ℕ) ↦ (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7))
      (Subtype.mk (p := fun (x : Set X) ↦
        Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x) U0 hU0o)
      (fun (x : ℕ) (S : (↑{S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} : Type u_7)) ↦
        let h := ⋯;
        Subtype.mk (p := fun (x : Set X) ↦
          Membership.mem (γ := Set (Set X)) {S : Set X | Membership.mem (γ := Filter X) (𝓝 1) S} x)
          (Classical.choose (p := fun (x : Set X) ↦
            Membership.mem (γ := Filter X) (𝓝 1) x ∧ IsClosed[inst] x ∧ x⁻¹ = x ∧ x * x ⊆ (↑S : Set X)) h)
          ⋯)
      t
U : ℕ → Set X := fun (n : ℕ) ↦ (↑(U_aux n) : Set X)
hU1 : ∀ (n : ℕ), Membership.mem (γ := Filter X) (𝓝 1) (U n)
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f : (a : X) → Y) (U n) (V n)
n : ℕ
h : True
x : X
hx : x ∈ U n
x_1 : (↑{f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} : Type (max u_7 u_8))
f : X →* Y
hf : Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} f
⊢ Membership.mem (α := Y) (V n)
    ((1,
          ((↑(Subtype.mk (p := fun (x : X →* Y) ↦
                    Membership.mem (γ := Set (X →* Y)) {f : X →* Y | Set.MapsTo (⇑f : (a : X) → Y) (U 0) (V 0)} x) f
                    hf) :
                X →* Y) :
              (a : X) → Y)
            x).2 :
      Y)

Tactic:
exact hU4 f hf n hx