awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
n : ℕ
g : G
hg : g ∈ s
hc : IsCompact (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))
⊢ IsClosed[inst_1] (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))

Tactic:
apply IsCompact.isClosed

Premises used: IsCompact.isClosed
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
n : ℕ
g : G
hg : g ∈ s
H : IsCompact.{u_2} (X := X) ((fun (g : G) ↦ g • x) '' (K n ∩ g • V))
⊢ IsCompact (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))

Tactic:
simpa only [F, smul_singleton] using H

Premises used: Set.smul_singleton
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
n : ℕ
g : G
hg : g ∈ s
hc : IsCompact.{u_2} (X := X) ((fun (g : G) ↦ g • x) '' (K n ∩ g • V))
⊢ IsCompact (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))

Tactic:
suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
        simpa only [F, smul_singleton] using H
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
n : ℕ
g : G
hg : g ∈ s
⊢ Continuous[inst, inst_1] fun (g : G) ↦ g • x

Tactic:
fun_prop
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
n : ℕ
g : G
hg : g ∈ s
apply : IsCompact (K n ∩ g • V)
⊢ IsCompact.{u_2} (X := X) ((fun (g : G) ↦ g • x) '' (K n ∩ g • V))

Tactic:
apply IsCompact.image ?_ (by fun_prop)

Premises used: IsCompact.image
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
n : ℕ
g : G
hg : g ∈ s
⊢ IsCompact (K n ∩ g • V)

Tactic:
(isCompact_compactCovering G n).inter_right (V_closed.smul g)

Premises used: isCompact_compactCovering, IsCompact.inter_right, IsClosed.smul
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
n : ℕ
g : G
hg : g ∈ s
⊢ IsCompact (K n ∩ g • V)

Tactic:
exact (isCompact_compactCovering G n).inter_right (V_closed.smul g)

Premises used: isCompact_compactCovering, IsCompact.inter_right, IsClosed.smul
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
apply : ∀ (y : X), Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) y
⊢ Eq.{u_2 + 1} (α := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) univ

Tactic:
apply eq_univ_iff_forall.2 (fun y ↦ ?_)

Premises used: Iff.mpr, Set.eq_univ_iff_forall
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
y : X
⊢ ∃ (h : G), h • x = y

Tactic:
exists_smul_eq G x y

Premises used: MulAction.exists_smul_eq
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
y : X
obtain : ∀ (h : G), Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) (h • x)
⊢ Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) y

Tactic:
obtain ⟨h, rfl⟩ : ∃ h, h • x = y := exists_smul_eq G x y

Premises used: MulAction.exists_smul_eq, rfl
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
h : G
⊢ ∃ (n : ℕ), h ∈ K n

Tactic:
exists_mem_compactCovering h

Premises used: Set.Mem, exists_mem_compactCovering
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
h : G
obtain : ∀ (n : ℕ) (hn : h ∈ K n), Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) (h • x)
⊢ Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) (h • x)

Tactic:
obtain ⟨n, hn⟩ : ∃ n, h ∈ K n := exists_mem_compactCovering h

Premises used: exists_mem_compactCovering
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
h : G
n : ℕ
hn : h ∈ K n
⊢ ∃ g ∈ s, h ∈ g • V

Tactic:
exists_set_mem_of_union_eq_top s _ hs _

Premises used: Set.exists_set_mem_of_union_eq_top
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
h : G
n : ℕ
hn : h ∈ K n
obtain : ∀ (g : G) (gs : g ∈ s) (hg : h ∈ g • V), Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) (h • x)
⊢ Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) (h • x)

Tactic:
obtain ⟨g, gs, hg⟩ : ∃ g ∈ s, h ∈ g • V := exists_set_mem_of_union_eq_top s _ hs _

Premises used: Set.exists_set_mem_of_union_eq_top
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
h : G
n : ℕ
hn : h ∈ K n
g : G
gs : g ∈ s
hg : h ∈ g • V
simp : ∃ (a : ℕ), ∃ a_1 ∈ s, ∃ (x_1 : G), (x_1 ∈ K a ∧ x_1 ∈ a_1 • V) ∧ x_1 • x = h • x
⊢ Membership.mem (γ := Set X) (⋃ (i : ℕ × (↑s : Type u_1)), F i) (h • x)

Tactic:
simp only [F, smul_singleton, mem_iUnion, mem_image, mem_inter_iff, Prod.exists,
        Subtype.exists, exists_prop]

Premises used: Set.mem_inter_iff, Set.mem_iUnion, Set.mem_image, Subtype.exists, exists_prop, Set.smul_singleton, Prod.exists
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
h : G
n : ℕ
hn : h ∈ K n
g : G
gs : g ∈ s
hg : h ∈ g • V
⊢ ∃ (a : ℕ), ∃ a_1 ∈ s, ∃ (x_1 : G), (x_1 ∈ K a ∧ x_1 ∈ a_1 • V) ∧ x_1 • x = h • x

Tactic:
⟨n, g, gs, h, ⟨hn, hg⟩, rfl⟩

Premises used: And.intro, Exists.intro, rfl
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
this_1 : Nonempty X
this : Encodable (↑s : Type u_1)
h : G
n : ℕ
hn : h ∈ K n
g : G
gs : g ∈ s
hg : h ∈ g • V
⊢ ∃ (a : ℕ), ∃ a_1 ∈ s, ∃ (x_1 : G), (x_1 ∈ K a ∧ x_1 ∈ a_1 • V) ∧ x_1 • x = h • x

Tactic:
exact ⟨n, g, gs, h, ⟨hn, hg⟩, rfl⟩
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
obtain :
  ∀ (n : ℕ) (g : G) (hg : g ∈ s) (hi : (interior (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))).Nonempty),
    HSMul.hSMul (β := Set X) U {x} ∈ 𝓝 x
⊢ HSMul.hSMul (β := Set X) U {x} ∈ 𝓝 x

Tactic:
obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · rintro ⟨n, ⟨g, hg⟩⟩
      apply IsCompact.isClosed
      suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
        simpa only [F, smul_singleton] using H
      apply IsCompact.image ?_ (by fun_prop)
      exact (isCompact_compactCovering G n).inter_right (V_closed.smul g)
    · apply eq_univ_iff_forall.2 (fun y ↦ ?_)
      obtain ⟨h, rfl⟩ : ∃ h, h • x = y := exists_smul_eq G x y
      obtain ⟨n, hn⟩ : ∃ n, h ∈ K n := exists_mem_compactCovering h
      obtain ⟨g, gs, hg⟩ : ∃ g ∈ s, h ∈ g • V := exists_set_mem_of_union_eq_top s _ hs _
      simp only [F, smul_singleton, mem_iUnion, mem_image, mem_inter_iff, Prod.exists,
        Subtype.exists, exists_prop]
      exact ⟨n, g, gs, h, ⟨hn, hg⟩, rfl⟩
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
n : ℕ
g : G
hg : g ∈ s
hi : (interior (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))).Nonempty
apply : interior (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg)) ⊆ interior (HSMul.hSMul (β := Set X) (g • V) {x})
⊢ (interior (HSMul.hSMul (β := Set X) (g • V) {x})).Nonempty

Tactic:
apply hi.mono

Premises used: Set.Nonempty.mono
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
n : ℕ
g : G
hg : g ∈ s
hi : (interior (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))).Nonempty
h : F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg) ⊆ HSMul.hSMul (β := Set X) (g • V) {x}
⊢ interior (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg)) ⊆ interior (HSMul.hSMul (β := Set X) (g • V) {x})

Tactic:
apply interior_mono

Premises used: interior_mono
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
n : ℕ
g : G
hg : g ∈ s
hi : (interior (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))).Nonempty
⊢ F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg) ⊆ HSMul.hSMul (β := Set X) (g • V) {x}

Tactic:
smul_subset_smul_right inter_subset_right

Premises used: Set.smul_subset_smul_right, Set.inter_subset_right
-- Module: Mathlib.Topology.Algebra.Group.OpenMapping
Goal:
G : Type u_1
X : Type u_2
inst : TopologicalSpace G
inst_1 : TopologicalSpace X
inst_2 : Group G
inst_3 : IsTopologicalGroup G
inst_4 : MulAction G X
inst_5 : SigmaCompactSpace G
inst_6 : BaireSpace X
inst_7 : T2Space X
inst_8 : ContinuousSMul G X
inst_9 : IsPretransitive G X
U : Set G
hU : Membership.mem (γ := Filter G) (𝓝 1) U
x : X
V : Set G
V_mem : Membership.mem (γ := Filter G) (𝓝 1) V
V_closed : IsClosed[inst] V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : s.Countable
hs : Eq.{u_1 + 1} (α := Set G) (⋃ g ∈ s, g • V) univ
K : ℕ → Set G := compactCovering G
F : ℕ × (↑s : Type u_1) → Set X := fun (p : ℕ × (↑s : Type u_1)) ↦ HSMul.hSMul (β := Set X) (K p.1 ∩ (↑p.2 : G) • V) {x}
n : ℕ
g : G
hg : g ∈ s
hi : (interior (F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg))).Nonempty
⊢ F (n, Subtype.mk (p := fun (x : G) ↦ x ∈ s) g hg) ⊆ HSMul.hSMul (β := Set X) (g • V) {x}

Tactic:
exact smul_subset_smul_right inter_subset_right

Premises used: Set.smul_subset_smul_right, Set.inter_subset_right