awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x

Tactic:
4

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x

Tactic:
3 → 4

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x

Tactic:
fun h ↦ closure_mono (range_subset_iff.2 fun n ↦ ?_) h

Premises used: Iff.mpr, closure_mono, Set.range_subset_iff
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
refine :
  ∀ (h : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x) (n : ℕ),
    Membership.mem (γ := Set G) (range fun (x : ℤ) ↦ y ^ x) (y ^ n)
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x

Tactic:
refine fun h ↦ closure_mono (range_subset_iff.2 fun n ↦ ?_) h

Premises used: Iff.mpr, closure_mono, Set.range_subset_iff
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
h : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x
n : ℕ
⊢ Membership.mem (γ := Set G) (range fun (x : ℤ) ↦ y ^ x) (y ^ n)

Tactic:
⟨n, zpow_natCast _ _⟩

Premises used: zpow_natCast, Set.Mem, setOf, Nat.cast, Exists.intro, Set.range
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
h : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x
n : ℕ
⊢ Membership.mem (γ := Set G) (range fun (x : ℤ) ↦ y ^ x) (y ^ n)

Tactic:
exact ⟨n, zpow_natCast _ _⟩
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x

Tactic:
tfae_have 3 → 4 := by
    refine fun h ↦ closure_mono (range_subset_iff.2 fun n ↦ ?_) h
    exact ⟨n, zpow_natCast _ _⟩
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
⊢ Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x

Tactic:
4

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
⊢ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x

Tactic:
1

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x

Tactic:
4 → 1

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x

Tactic:
fun h ↦ closure_minimal ?_ isClosed_setOf_clusterPt h

Premises used: Set.Mem, closure_minimal, setOf, isClosed_setOf_clusterPt, ClusterPt, MapClusterPt
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
refine :
  ∀ (h : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    Subset.{u_1} (α := Set G) (range fun (x : ℤ) ↦ y ^ x) {x : G | ClusterPt x (map (fun (x : ℕ) ↦ y ^ x) atTop)}
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x

Tactic:
refine fun h ↦ closure_minimal ?_ isClosed_setOf_clusterPt h

Premises used: closure_minimal, isClosed_setOf_clusterPt
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
h : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
⊢ Subset.{u_1} (α := Set G) (range fun (x : ℤ) ↦ y ^ x) {x : G | ClusterPt x (map (fun (x : ℕ) ↦ y ^ x) atTop)}

Tactic:
range_subset_iff.2 (mapClusterPt_self_zpow_atTop_pow _)

Premises used: Iff.mpr, mapClusterPt_self_zpow_atTop_pow, Set.range_subset_iff
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
h : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
⊢ Subset.{u_1} (α := Set G) (range fun (x : ℤ) ↦ y ^ x) {x : G | ClusterPt x (map (fun (x : ℕ) ↦ y ^ x) atTop)}

Tactic:
exact range_subset_iff.2 (mapClusterPt_self_zpow_atTop_pow _)

Premises used: Iff.mpr, mapClusterPt_self_zpow_atTop_pow, Set.range_subset_iff
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
⊢ ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x

Tactic:
tfae_have 4 → 1 := by
    refine fun h ↦ closure_minimal ?_ isClosed_setOf_clusterPt h
    exact range_subset_iff.2 (mapClusterPt_self_zpow_atTop_pow _)
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x

Tactic:
1

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x

Tactic:
3

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ ∀ (a : MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x

Tactic:
1 → 3

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
rw : ∀ (a : MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x), ClusterPt x (𝓟 (range fun (x : ℕ) ↦ y ^ x))
⊢ ∀ (a : MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x

Tactic:
rw [mem_closure_iff_clusterPt]

Premises used: mem_closure_iff_clusterPt
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ ∀ (a : MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x), ClusterPt x (𝓟 (range fun (x : ℕ) ↦ y ^ x))

Tactic:
(ClusterPt.mono · (le_principal_iff.2 range_mem_map))

Premises used: Iff.mpr, ClusterPt.mono, Filter.le_principal_iff, Filter.range_mem_map
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ ∀ (a : MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x), ClusterPt x (𝓟 (range fun (x : ℕ) ↦ y ^ x))

Tactic:
exact (ClusterPt.mono · (le_principal_iff.2 range_mem_map))

Premises used: Iff.mpr, ClusterPt.mono, Filter.le_principal_iff, Filter.range_mem_map
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
⊢ ∀ (a : MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x

Tactic:
tfae_have 1 → 3 := by
    rw [mem_closure_iff_clusterPt]
    exact (ClusterPt.mono · (le_principal_iff.2 range_mem_map))
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
tfae_2_iff_1 : (MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_3_to_4 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x
tfae_4_to_1 :
  ∀ (a : Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x),
    MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x
tfae_1_to_3 :
  ∀ (a : MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x
⊢ [MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x, MapClusterPt x atTop fun (x : ℤ) ↦ y ^ x,
      Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℕ) ↦ y ^ x)) x,
      Membership.mem (γ := Set G) (closure[inst_1] (range fun (x : ℤ) ↦ y ^ x)) x].TFAE

Tactic:
tfae_finish
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
⊢ (MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x) ↔
    Membership.mem (γ := Subgroup G) (Subgroup.topologicalClosure (Subgroup.zpowers y)) x

Tactic:
(mapClusterPt_atTop_pow_tfae x y).out 0 3

Premises used: mapClusterPt_atTop_pow_tfae, List.TFAE.out, OfNat.ofNat
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
⊢ (MapClusterPt x⁻¹ atTop fun (x : ℕ) ↦ y ^ x) ↔ MapClusterPt x atTop fun (x : ℕ) ↦ y ^ x

Tactic:
simp only [mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers, inv_mem_iff]

Premises used: inv_mem_iff, mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x : G
h :
  ∀ (y : G),
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x_1 : ℤ) ↦ x ^ x_1)) y ↔
      Membership.mem (γ := Set G) (closure[inst_1] (range fun (x_1 : ℕ) ↦ x ^ x_1)) y
⊢ Eq.{u_1 + 1} (α := Set G) (closure[inst_1] (range fun (x_1 : ℤ) ↦ x ^ x_1))
    (closure[inst_1] (range fun (x_1 : ℕ) ↦ x ^ x_1))

Tactic:
ext y
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
⊢ Membership.mem (γ := Set G) (closure[inst_1] (range fun (x_1 : ℤ) ↦ x ^ x_1)) y ↔
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x_1 : ℕ) ↦ x ^ x_1)) y

Tactic:
(mapClusterPt_atTop_pow_tfae y x).out 3 2

Premises used: mapClusterPt_atTop_pow_tfae, List.TFAE.out, OfNat.ofNat
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x y : G
⊢ Membership.mem (γ := Set G) (closure[inst_1] (range fun (x_1 : ℤ) ↦ x ^ x_1)) y ↔
    Membership.mem (γ := Set G) (closure[inst_1] (range fun (x_1 : ℕ) ↦ x ^ x_1)) y

Tactic:
exact (mapClusterPt_atTop_pow_tfae y x).out 3 2

Premises used: mapClusterPt_atTop_pow_tfae, List.TFAE.out, OfNat.ofNat
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
x : G
⊢ (DenseRange.{u_1, 0} (X := G) fun (x_1 : ℤ) ↦ x ^ x_1) ↔ DenseRange.{u_1, 0} (X := G) fun (x_1 : ℕ) ↦ x ^ x_1

Tactic:
simp only [DenseRange, dense_iff_closure_eq, closure_range_zpow_eq_pow]

Premises used: dense_iff_closure_eq, DenseRange, closure_range_zpow_eq_pow
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
⊢ Eq (α := Submonoid G) (Submonoid.topologicalClosure (Subgroup.toSubmonoid (Subgroup.closure s)))
    (Submonoid.topologicalClosure (Submonoid.closure s))

Tactic:
le_antisymm ?_ (closure_mono <| Subgroup.le_closure_toSubmonoid _)

Premises used: closure_mono, Subgroup.le_closure_toSubmonoid, le_antisymm
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
refine :
  LE.le (α := Submonoid G) (Submonoid.topologicalClosure (Subgroup.toSubmonoid (Subgroup.closure s)))
    (Submonoid.topologicalClosure (Submonoid.closure s))
⊢ Eq (α := Submonoid G) (Submonoid.topologicalClosure (Subgroup.toSubmonoid (Subgroup.closure s)))
    (Submonoid.topologicalClosure (Submonoid.closure s))

Tactic:
refine le_antisymm ?_ (closure_mono <| Subgroup.le_closure_toSubmonoid _)

Premises used: closure_mono, Subgroup.le_closure_toSubmonoid, le_antisymm
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
⊢ LE.le (α := Submonoid G) (Submonoid.topologicalClosure (Subgroup.toSubmonoid (Subgroup.closure s)))
    (Submonoid.topologicalClosure (Submonoid.closure s))

Tactic:
Submonoid.topologicalClosure_minimal _ ?_ isClosed_closure

Premises used: Submonoid.topologicalClosure_minimal, isClosed_closure
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
refine :
  LE.le (α := Submonoid G) (Subgroup.toSubmonoid (Subgroup.closure s))
    (Submonoid.topologicalClosure (Submonoid.closure s))
⊢ LE.le (α := Submonoid G) (Submonoid.topologicalClosure (Subgroup.toSubmonoid (Subgroup.closure s)))
    (Submonoid.topologicalClosure (Submonoid.closure s))

Tactic:
refine Submonoid.topologicalClosure_minimal _ ?_ isClosed_closure

Premises used: Submonoid.topologicalClosure_minimal
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
rw : s ∪ s⁻¹ ⊆ (↑(Submonoid.topologicalClosure (Submonoid.closure s)) : Set G)
⊢ LE.le (α := Submonoid G) (Subgroup.toSubmonoid (Subgroup.closure s))
    (Submonoid.topologicalClosure (Submonoid.closure s))

Tactic:
rw [Subgroup.closure_toSubmonoid, Submonoid.closure_le]

Premises used: Subgroup.closure_toSubmonoid, Submonoid.closure_le
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
⊢ s ∪ s⁻¹ ⊆ (↑(Submonoid.topologicalClosure (Submonoid.closure s)) : Set G)

Tactic:
union_subset (Submonoid.subset_closure.trans subset_closure) fun x hx ↦ ?_

Premises used: subset_closure, HasSubset.Subset.trans, Set.union_subset, Submonoid.subset_closure
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
refine : ∀ (x : G) (hx : x ∈ s⁻¹), x ∈ (↑(Submonoid.topologicalClosure (Submonoid.closure s)) : Set G)
⊢ s ∪ s⁻¹ ⊆ (↑(Submonoid.topologicalClosure (Submonoid.closure s)) : Set G)

Tactic:
refine union_subset (Submonoid.subset_closure.trans subset_closure) fun x hx ↦ ?_

Premises used: subset_closure, HasSubset.Subset.trans, Set.union_subset, Submonoid.subset_closure
-- Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure
Goal:
G : Type u_1
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : CompactSpace G
inst_3 : IsTopologicalGroup G
s : Set G
x : G
hx : x ∈ s⁻¹
⊢ x ∈ (↑(Submonoid.topologicalClosure (Submonoid.closure s)) : Set G)

Tactic:
closure_mono (Submonoid.powers_le.2 (Submonoid.subset_closure <| Set.mem_inv.1 hx)) ?_

Premises used: Iff.mpr, Set.Mem, setOf, closure_mono, closure, Iff.mp, Submonoid.subset_closure, Set.sInter, Submonoid.powers_le, Set.mem_inv