awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
A B : OpenNormalSubgroup G
h : (↑A.toOpenSubgroup : Subgroup G) = (↑B.toOpenSubgroup : Subgroup G)
x : G
⊢ x ∈ (↑A.toOpenSubgroup : Subgroup G).carrier ↔ x ∈ (↑B.toOpenSubgroup : Subgroup G).carrier
have carrier : x ∈ (↑B.toOpenSubgroup : Subgroup G).carrier ↔ x ∈ (↑B.toOpenSubgroup : Subgroup G).carrier
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
U V : Set G
h : (fun (S : Set G) ↦ ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ S * T ⊆ W) U
h_1 : (fun (S : Set G) ↦ ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ S * T ⊆ W) V
T₁ : Set G
hT₁ : Membership.mem (γ := Filter G) (𝓝 1) T₁
mem1 : U * T₁ ⊆ W
T₂ : Set G
hT₂ : Membership.mem (γ := Filter G) (𝓝 1) T₂
mem2 : V * T₂ ⊆ W
⊢ (U ∪ V) * (T₁ ∩ T₂) ⊆ W
have rw : U * (T₁ ∩ T₂) ∪ V * (T₁ ∩ T₂) ⊆ W
Tactic: rw [union_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
⊢ ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ W * T ⊆ W
have apply : ∀ x ∈ W, ∃ t ∈ 𝓝[W] x, ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ t * T ⊆ W
Tactic: apply WClopen.isClosed.isCompact.induction_on (p := fun S ↦ ∃ T ∈ 𝓝 (1 : G), S * T ⊆ W)
    ⟨Set.univ, by simp only [univ_mem, empty_mul, empty_subset, and_self]⟩
    (fun _ _ huv ⟨T, hT, mem⟩ ↦ ⟨T, hT, (mul_subset_mul_right huv).trans mem⟩)
    fun U V ⟨T₁, hT₁, mem1⟩ ⟨T₂, hT₂, mem2⟩ ↦ ⟨T₁ ∩ T₂, inter_mem hT₁ hT₂, by
      rw [union_mul]
      exact union_subset (mul_subset_mul_left inter_subset_left |>.trans mem1)
        (mul_subset_mul_left inter_subset_right |>.trans mem2) ⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (hypothesis)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
x : G
memW : x ∈ W
⊢ ∃ t ∈ 𝓝[W] x, ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ t * T ⊆ W
have this : Membership.mem (γ := Set (G × G)) ((fun (p : G × G) ↦ p.1 * p.2) ⁻¹' W) (x, 1)
Tactic: have : (x, 1) ∈ (fun p ↦ p.1 * p.2) ⁻¹' W := by simp [memW]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
x : G
memW : x ∈ W
this : Membership.mem (γ := Set (G × G)) ((fun (p : G × G) ↦ p.1 * p.2) ⁻¹' W) (x, 1)
⊢ ∃ t ∈ 𝓝[W] x, ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ t * T ⊆ W
have rcases : ∀ (U V : Set G) (Uopen : IsOpen[inst] U) (Vopen : IsOpen[inst] V) (xmemU : x ∈ U) (onememV : 1 ∈ V)
  (prodsub : U ×ˢ V ⊆ (fun (p : G × G) ↦ p.1 * p.2) ⁻¹' W),
  ∃ t ∈ 𝓝[W] x, ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ t * T ⊆ W
Tactic: rcases isOpen_prod_iff.mp (continuous_mul.isOpen_preimage W <| WClopen.2) x 1 this with
    ⟨U, V, Uopen, Vopen, xmemU, onememV, prodsub⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (hypothesis)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
x : G
memW : x ∈ W
this : Membership.mem (γ := Set (G × G)) ((fun (p : G × G) ↦ p.1 * p.2) ⁻¹' W) (x, 1)
U V : Set G
Uopen : IsOpen[inst] U
Vopen : IsOpen[inst] V
xmemU : x ∈ U
onememV : 1 ∈ V
prodsub : U ×ˢ V ⊆ (fun (p : G × G) ↦ p.1 * p.2) ⁻¹' W
⊢ ∃ t ∈ 𝓝[W] x, ∃ (T : Set G), Membership.mem (γ := Filter G) (𝓝 1) T ∧ t * T ⊆ W
have h6 : U * V ⊆ W
Tactic: have h6 : U * V ⊆ W := mul_subset_iff.mpr (fun _ hx _ hy ↦ prodsub (mk_mem_prod hx hy))
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
⊢ ∃ (T : Set G), mulInvClosureNhd T W
have rcases : ∀ (S : Set G) (Smemnhds : Membership.mem (γ := Filter G) (𝓝 1) S) (mulclose : W * S ⊆ W),
  ∃ (T : Set G), mulInvClosureNhd T W
Tactic: rcases exist_mul_closure_nhds WClopen with ⟨S, Smemnhds, mulclose⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : Membership.mem (γ := Filter G) (𝓝 1) S
mulclose : W * S ⊆ W
⊢ ∃ (T : Set G), mulInvClosureNhd T W
have rcases : ∀ (U : Set G) (UsubS : U ⊆ S) (Uopen : IsOpen[inst] U) (onememU : 1 ∈ U), ∃ (T : Set G), mulInvClosureNhd T W
Tactic: rcases mem_nhds_iff.mp Smemnhds with ⟨U, UsubS, Uopen, onememU⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : Membership.mem (γ := Filter G) (𝓝 1) S
mulclose : W * S ⊆ W
U : Set G
UsubS : U ⊆ S
Uopen : IsOpen[inst] U
onememU : 1 ∈ U
⊢ ∃ (T : Set G), mulInvClosureNhd T W
have h : mulInvClosureNhd (U ∩ U⁻¹) W
Tactic: use U ∩ U⁻¹
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : Membership.mem (γ := Filter G) (𝓝 1) S
mulclose : W * S ⊆ W
U : Set G
UsubS : U ⊆ S
Uopen : IsOpen[inst] U
onememU : 1 ∈ U
⊢ mulInvClosureNhd (U ∩ U⁻¹) W
have h : Membership.mem (γ := Filter G) (𝓝 1) (U ∩ U⁻¹)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : Membership.mem (γ := Filter G) (𝓝 1) S
mulclose : W * S ⊆ W
U : Set G
UsubS : U ⊆ S
Uopen : IsOpen[inst] U
onememU : 1 ∈ U
h : Membership.mem (γ := Filter G) (𝓝 1) (U ∩ U⁻¹)
⊢ mulInvClosureNhd (U ∩ U⁻¹) W
have h₁ : (U ∩ U⁻¹)⁻¹ = U ∩ U⁻¹
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : Membership.mem (γ := Filter G) (𝓝 1) S
mulclose : W * S ⊆ W
U : Set G
UsubS : U ⊆ S
Uopen : IsOpen[inst] U
onememU : 1 ∈ U
h : Membership.mem (γ := Filter G) (𝓝 1) (U ∩ U⁻¹)
h₁ : (U ∩ U⁻¹)⁻¹ = U ∩ U⁻¹
⊢ mulInvClosureNhd (U ∩ U⁻¹) W
have h₂ : IsOpen[inst] (U ∩ U⁻¹)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_1
inst : TopologicalSpace G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : Membership.mem (γ := Filter G) (𝓝 1) S
mulclose : W * S ⊆ W
U : Set G
UsubS : U ⊆ S
Uopen : IsOpen[inst] U
onememU : 1 ∈ U
h : Membership.mem (γ := Filter G) (𝓝 1) (U ∩ U⁻¹)
h₁ : (U ∩ U⁻¹)⁻¹ = U ∩ U⁻¹
h₂ : IsOpen[inst] (U ∩ U⁻¹)
⊢ mulInvClosureNhd (U ∩ U⁻¹) W
have h₃ : W * (U ∩ U⁻¹) ⊆ W
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
⊢ ∃ (H : OpenSubgroup G), (↑H : Set G) ⊆ W
have rcases : ∀ (V : Set G) (hV : mulInvClosureNhd V W), ∃ (H : OpenSubgroup G), (↑H : Set G) ⊆ W
Tactic: rcases exists_mulInvClosureNhd WClopen with ⟨V, hV⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
a b : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) a
hb : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) b
⊢ Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) (a * b)
have rcases : ∀ (k : ℕ) (hk : a ∈ V ^ (k + 1)), Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) (a * b)
Tactic: rcases mem_iUnion.mp ha with ⟨k, hk⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
a b : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) a
hb : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) b
k : ℕ
hk : a ∈ V ^ (k + 1)
⊢ Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) (a * b)
have rcases : ∀ (l : ℕ) (hl : b ∈ V ^ (l + 1)), Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) (a * b)
Tactic: rcases mem_iUnion.mp hb with ⟨l, hl⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
a b : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) a
hb : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) b
k : ℕ
hk : a ∈ V ^ (k + 1)
l : ℕ
hl : b ∈ V ^ (l + 1)
⊢ Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) (a * b)
have apply : ∃ (i : ℕ), a * b ∈ V ^ (i + 1)
Tactic: apply mem_iUnion.mpr
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
a b : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) a
hb : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) b
k : ℕ
hk : a ∈ V ^ (k + 1)
l : ℕ
hl : b ∈ V ^ (l + 1)
⊢ ∃ (i : ℕ), a * b ∈ V ^ (i + 1)
have h : a * b ∈ V ^ (k + 1 + l + 1)
Tactic: use k + 1 + l
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
a b : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) a
hb : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) b
k : ℕ
hk : a ∈ V ^ (k + 1)
l : ℕ
hl : b ∈ V ^ (l + 1)
⊢ a * b ∈ V ^ (k + 1 + l + 1)
have h : a * b ∈ V ^ (k + 1) * V ^ (l + 1)
Tactic: rw [add_assoc, pow_add]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
⊢ Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) 1
have apply : ∃ (i : ℕ), 1 ∈ V ^ (i + 1)
Tactic: apply mem_iUnion.mpr
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
⊢ ∃ (i : ℕ), 1 ∈ V ^ (i + 1)
have h : 1 ∈ V ^ HAdd.hAdd (α := ℕ) 0 1
Tactic: use 0
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
x : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) x
⊢ Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) x⁻¹
have rcases : ∀ (k : ℕ) (hk : x ∈ V ^ (k + 1)), Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) x⁻¹
Tactic: rcases mem_iUnion.mp ha with ⟨k, hk⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
x : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) x
k : ℕ
hk : x ∈ V ^ (k + 1)
⊢ Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) x⁻¹
have apply : ∃ (i : ℕ), x⁻¹ ∈ V ^ (i + 1)
Tactic: apply mem_iUnion.mpr
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
x : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) x
k : ℕ
hk : x ∈ V ^ (k + 1)
⊢ ∃ (i : ℕ), x⁻¹ ∈ V ^ (i + 1)
have h : x⁻¹ ∈ V ^ (k + 1)
Tactic: use k
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
x : G
ha : Membership.mem (γ := Set G) (⋃ (n : ℕ), V ^ (n + 1)) x
k : ℕ
hk : x ∈ V ^ (k + 1)
⊢ x⁻¹ ∈ V ^ (k + 1)
have h : x⁻¹ ∈ V⁻¹ ^ (k + 1)
Tactic: rw [← hV.inv]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
⊢ IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
have refine : ∀ (n : ℕ), IsOpen[inst_1] (V ^ (n + 1))
Tactic: refine isOpen_iUnion (fun n ↦ ?_)
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
n : ℕ
⊢ IsOpen[inst_1] (V ^ (n + 1))
have rw : IsOpen[inst_1] (V ^ n * V)
Tactic: rw [pow_succ]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (hypothesis)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
⊢ ∃ (H : OpenSubgroup G), (↑H : Set G) ⊆ W
have this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
Tactic: have : IsOpen (⋃ n, V ^ (n + 1)) := by
    refine isOpen_iUnion (fun n ↦ ?_)
    rw [pow_succ]
    exact hV.isOpen.mul_left
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
⊢ ∃ (H : OpenSubgroup G), (↑H : Set G) ⊆ W
have h : (↑{ toSubgroup := S, isOpen' := this : OpenSubgroup G } : Set G) ⊆ W
Tactic: use ⟨S, this⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
n : ℕ
ih : W * V ^ (n + 1) ⊆ W
⊢ W * V ^ (n + 1 + 1) ⊆ W
have succ : W * V ^ (n + 1) * V ⊆ W
Tactic: rw [pow_succ, ← mul_assoc]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
n : ℕ
⊢ W * V ^ (n + 1) ⊆ W
have zero : W * V ^ HAdd.hAdd (α := ℕ) 0 1 ⊆ W
Tactic: induction n with
    | zero => simp [hV.mul]
    | succ n ih =>
      rw [pow_succ, ← mul_assoc]
      exact (Set.mul_subset_mul_right ih).trans hV.mul
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
n : ℕ
zero : W * V ^ HAdd.hAdd (α := ℕ) 0 1 ⊆ W
⊢ W * V ^ (n + 1) ⊆ W
have succ : ∀ (n : ℕ) (ih : W * V ^ (n + 1) ⊆ W), W * V ^ (n + 1 + 1) ⊆ W
Tactic: induction n with
    | zero => simp [hV.mul]
    | succ n ih =>
      rw [pow_succ, ← mul_assoc]
      exact (Set.mul_subset_mul_right ih).trans hV.mul
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (hypothesis)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
⊢ (↑{ toSubgroup := S, isOpen' := this : OpenSubgroup G } : Set G) ⊆ W
have mulVpow : ∀ (n : ℕ), W * V ^ (n + 1) ⊆ W
Tactic: have mulVpow (n : ℕ) : W * V ^ (n + 1) ⊆ W := by
    induction n with
    | zero => simp [hV.mul]
    | succ n ih =>
      rw [pow_succ, ← mul_assoc]
      exact (Set.mul_subset_mul_right ih).trans hV.mul
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
mulVpow : ∀ (n : ℕ), W * V ^ (n + 1) ⊆ W
n : ℕ
x : G
xin : x ∈ V ^ (n + 1)
⊢ ∃ x_1 ∈ W, ∃ y ∈ V ^ (n + 1), x_1 * y = x
have right : 1 * x = x
Tactic: use 1, einW, x, xin
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (goal)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
mulVpow : ∀ (n : ℕ), W * V ^ (n + 1) ⊆ W
n : ℕ
x : G
xin : x ∈ V ^ (n + 1)
⊢ 1 * x = x
have right : x = x
Tactic: rw [one_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.OpenSubgroup (hypothesis)
Goal:
G : Type u_2
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ (n : ℕ), V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst_1] (⋃ (n : ℕ), V ^ (n + 1))
mulVpow : ∀ (n : ℕ), W * V ^ (n + 1) ⊆ W
⊢ (↑{ toSubgroup := S, isOpen' := this : OpenSubgroup G } : Set G) ⊆ W
have this₁ : ∀ (n : ℕ), V ^ (n + 1) ⊆ W * V ^ (n + 1)
Tactic: have (n : ℕ) : V ^ (n + 1) ⊆ W * V ^ (n + 1) := by
    intro x xin
    rw [Set.mem_mul]
    use 1, einW, x, xin
    rw [one_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
⊢ Dense (↑S : Set G)
have inl : Subsingleton G → Dense (↑S : Set G)
Tactic: cases subsingleton_or_nontrivial G
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
inl : Subsingleton G → Dense (↑S : Set G)
⊢ Dense (↑S : Set G)
have inr : Nontrivial G → Dense (↑S : Set G)
Tactic: cases subsingleton_or_nontrivial G
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h : Subsingleton G
⊢ Dense (↑S : Set G)
have inl : ∀ (x : G), x ∈ (↑S : Set G)
Tactic: refine fun x => _root_.subset_closure ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h : Subsingleton G
x : G
⊢ x ∈ (↑S : Set G)
have inl : 1 ∈ (↑S : Set G)
Tactic: rw [Subsingleton.elim x 1]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h : Nontrivial G
⊢ Dense (↑S : Set G)
have inr : ∀ (a b : G) (hlt : a < b), ∃ c ∈ (↑S : Set G), c ∈ Ioo a b
Tactic: refine dense_of_exists_between fun a b hlt => ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h : Nontrivial G
a b : G
hlt : a < b
⊢ ∃ c ∈ (↑S : Set G), c ∈ Ioo a b
have inr : ∀ (g : G) (hgS : g ∈ S) (hg0 : 1 < g) (hg : g < b / a), ∃ c ∈ (↑S : Set G), c ∈ Ioo a b
Tactic: rcases hS (b / a) (one_lt_div'.2 hlt) with ⟨g, hgS, hg0, hg⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h : Nontrivial G
a b : G
hlt : a < b
g : G
hgS : g ∈ S
hg0 : 1 < g
hg : g < b / a
⊢ ∃ c ∈ (↑S : Set G), c ∈ Ioo a b
have inr : ∀ (m : ℤ) (hm : 1 * g ^ m ∈ Ioc a (a * g)), ∃ c ∈ (↑S : Set G), c ∈ Ioo a b
Tactic: rcases (existsUnique_add_zpow_mem_Ioc hg0 1 a).exists with ⟨m, hm⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (hypothesis)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h : Nontrivial G
a b : G
hlt : a < b
g : G
hgS : g ∈ S
hg0 : 1 < g
hg : g < b / a
m : ℤ
hm : 1 * g ^ m ∈ Ioc a (a * g)
⊢ ∃ c ∈ (↑S : Set G), c ∈ Ioo a b
have hm₁ : g ^ m ∈ Ioc a (a * g)
Tactic: rw [one_mul] at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.Topology.Algebra.Order.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : TopologicalSpace G
inst_4 : OrderTopology G
inst_5 : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h : Nontrivial G
a b : G
hlt : a < b
g : G
hgS : g ∈ S
hg0 : 1 < g
hg : g < b / a
m : ℤ
hm : g ^ m ∈ Ioc a (a * g)
⊢ ∃ c ∈ (↑S : Set G), c ∈ Ioo a b
have inr : a * g < b
Tactic: refine ⟨g ^ m, zpow_mem hgS _, hm.1, hm.2.trans_lt ?_⟩