ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsOpen.add_closure
@[to_additive]
theorem IsOpen.mul_closure (hs : IsOpen s) (t : Set G) : s * closure t = s * t 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s : Set G} (hs : IsOpen s) (t : Set G)
Goal: s + closure t = s + t
-- IsOpen.closure_add
@[to_additive]
theorem IsOpen.closure_mul (ht : IsOpen t) (s : Set G) : closure s * t = s * t 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {t : Set G} (ht : IsOpen t) (s : Set G)
Goal: closure s + t = s + t
-- IsOpen.closure_mul
@[to_additive]
theorem IsOpen.closure_mul (ht : IsOpen t) (s : Set G) : closure s * t = s * t 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {t : Set G} (ht : IsOpen t) (s : Set G)
Goal: closure s * t = s * t
-- IsOpen.div_closure
@[to_additive]
theorem IsOpen.div_closure (hs : IsOpen s) (t : Set G) : s / closure t = s / t 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s : Set G} (hs : IsOpen s) (t : Set G)
Goal: s / closure t = s / t
-- IsOpen.sub_closure
@[to_additive]
theorem IsOpen.div_closure (hs : IsOpen s) (t : Set G) : s / closure t = s / t 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s : Set G} (hs : IsOpen s) (t : Set G)
Goal: s - closure t = s - t
-- IsOpen.closure_div
@[to_additive]
theorem IsOpen.closure_div (ht : IsOpen t) (s : Set G) : closure s / t = s / t 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {t : Set G} (ht : IsOpen t) (s : Set G)
Goal: closure s / t = s / t
-- IsOpen.closure_sub
@[to_additive]
theorem IsOpen.closure_div (ht : IsOpen t) (s : Set G) : closure s / t = s / t 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {t : Set G} (ht : IsOpen t) (s : Set G)
Goal: closure s - t = s - t
-- IsClosed.add_left_of_isCompact
@[to_additive]
theorem IsClosed.mul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) : IsClosed (s * t) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s t : Set G} (ht : IsClosed t) (hs : IsCompact s)
Goal: IsClosed (s + t)
-- IsClosed.mul_left_of_isCompact
@[to_additive]
theorem IsClosed.mul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) : IsClosed (s * t) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G} (ht : IsClosed t) (hs : IsCompact s)
Goal: IsClosed (s * t)
-- IsClosed.add_right_of_isCompact
@[to_additive]
theorem IsClosed.mul_right_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
    IsClosed (t * s) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s t : Set G} (ht : IsClosed t) (hs : IsCompact s)
Goal: IsClosed (t + s)
-- IsClosed.mul_right_of_isCompact
@[to_additive]
theorem IsClosed.mul_right_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
    IsClosed (t * s) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G} (ht : IsClosed t) (hs : IsCompact s)
Goal: IsClosed (t * s)
-- QuotientGroup.isClosedMap_coe
@[to_additive]
theorem QuotientGroup.isClosedMap_coe {H : Subgroup G} (hH : IsCompact (H : Set G)) :
    IsClosedMap ((↑) : G → G ⧸ H) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {H : Subgroup G} (hH : IsCompact ↑H)
Goal: IsClosedMap QuotientGroup.mk
-- QuotientAddGroup.isClosedMap_coe
@[to_additive]
theorem QuotientGroup.isClosedMap_coe {H : Subgroup G} (hH : IsCompact (H : Set G)) :
    IsClosedMap ((↑) : G → G ⧸ H) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {H : AddSubgroup G} (hH : IsCompact ↑H)
Goal: IsClosedMap QuotientAddGroup.mk
-- subset_add_closure_zero
@[to_additive]
lemma subset_mul_closure_one (s : Set G) : s ⊆ s * (closure {1} : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] (s : Set G)
Goal: s ⊆ s + closure {0}
-- subset_mul_closure_one
@[to_additive]
lemma subset_mul_closure_one (s : Set G) : s ⊆ s * (closure {1} : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] (s : Set G)
Goal: s ⊆ s * closure {1}
-- IsCompact.mul_closure_one_eq_closure
@[to_additive]
lemma IsCompact.mul_closure_one_eq_closure {K : Set G} (hK : IsCompact K) :
    K * (closure {1} : Set G) = closure K 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {K : Set G} (hK : IsCompact K)
Goal: K * closure {1} = closure K
-- IsCompact.add_closure_zero_eq_closure
@[to_additive]
lemma IsCompact.mul_closure_one_eq_closure {K : Set G} (hK : IsCompact K) :
    K * (closure {1} : Set G) = closure K 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {K : Set G} (hK : IsCompact K)
Goal: K + closure {0} = closure K
-- IsClosed.mul_closure_one_eq
@[to_additive]
lemma IsClosed.mul_closure_one_eq {F : Set G} (hF : IsClosed F) :
    F * (closure {1} : Set G) = F 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {F : Set G} (hF : IsClosed F)
Goal: F * closure {1} = F
-- IsClosed.add_closure_zero_eq
@[to_additive]
lemma IsClosed.mul_closure_one_eq {F : Set G} (hF : IsClosed F) :
    F * (closure {1} : Set G) = F 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {F : Set G} (hF : IsClosed F)
Goal: F + closure {0} = F
-- compl_mul_closure_one_eq
@[to_additive]
lemma compl_mul_closure_one_eq {t : Set G} (ht : t * (closure {1} : Set G) = t) :
    tᶜ * (closure {1} : Set G) = tᶜ 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {t : Set G} (ht : t * closure {1} = t)
Goal: tᶜ * closure {1} = tᶜ
-- compl_add_closure_zero_eq
@[to_additive]
lemma compl_mul_closure_one_eq {t : Set G} (ht : t * (closure {1} : Set G) = t) :
    tᶜ * (closure {1} : Set G) = tᶜ 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {t : Set G} (ht : t + closure {0} = t)
Goal: tᶜ + closure {0} = tᶜ
-- compl_add_closure_zero_eq_iff
@[to_additive]
lemma compl_mul_closure_one_eq_iff {t : Set G} :
    tᶜ * (closure {1} : Set G) = tᶜ ↔ t * (closure {1} : Set G) = t 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {t : Set G}
Goal: tᶜ + closure {0} = tᶜ ↔ t + closure {0} = t
-- compl_mul_closure_one_eq_iff
@[to_additive]
lemma compl_mul_closure_one_eq_iff {t : Set G} :
    tᶜ * (closure {1} : Set G) = tᶜ ↔ t * (closure {1} : Set G) = t 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {t : Set G}
Goal: tᶜ * closure {1} = tᶜ ↔ t * closure {1} = t
-- IsOpen.mul_closure_one_eq
@[to_additive]
lemma IsOpen.mul_closure_one_eq {U : Set G} (hU : IsOpen U) :
    U * (closure {1} : Set G) = U 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {U : Set G} (hU : IsOpen U)
Goal: U * closure {1} = U
-- IsOpen.add_closure_zero_eq
@[to_additive]
lemma IsOpen.mul_closure_one_eq {U : Set G} (hU : IsOpen U) :
    U * (closure {1} : Set G) = U 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {U : Set G} (hU : IsOpen U)
Goal: U + closure {0} = U
-- TopologicalAddGroup.t1Space
@[to_additive]
theorem TopologicalGroup.t1Space (h : @IsClosed G _ {1}) : T1Space G 
Arguments: (G : Type w) [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (h : IsClosed {0})
Goal: T1Space G
-- TopologicalGroup.t1Space
@[to_additive]
theorem TopologicalGroup.t1Space (h : @IsClosed G _ {1}) : T1Space G 
Arguments: (G : Type w) [TopologicalSpace G] [Group G] [ContinuousMul G] (h : IsClosed {1})
Goal: T1Space G
-- addGroup_inseparable_iff
@[to_additive]
theorem group_inseparable_iff {x y : G} : Inseparable x y ↔ x / y ∈ closure (1 : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {x y : G}
Goal: Inseparable x y ↔ x - y ∈ closure 0
-- group_inseparable_iff
@[to_additive]
theorem group_inseparable_iff {x y : G} : Inseparable x y ↔ x / y ∈ closure (1 : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {x y : G}
Goal: Inseparable x y ↔ x / y ∈ closure 1
-- TopologicalGroup.t2Space_iff_one_closed
@[to_additive]
theorem TopologicalGroup.t2Space_iff_one_closed : T2Space G ↔ IsClosed ({1} : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G]
Goal: T2Space G ↔ IsClosed {1}
-- TopologicalAddGroup.t2Space_iff_zero_closed
@[to_additive]
theorem TopologicalGroup.t2Space_iff_one_closed : T2Space G ↔ IsClosed ({1} : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G]
Goal: T2Space G ↔ IsClosed {0}
-- TopologicalGroup.t2Space_of_one_sep
@[to_additive]
theorem TopologicalGroup.t2Space_of_one_sep (H : ∀ x : G, x ≠ 1 → ∃ U ∈ 𝓝 (1 : G), x ∉ U) :
    T2Space G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (H : ∀ (x : G), x ≠ 1 → ∃ U ∈ 𝓝 1, x ∉ U)
Goal: T2Space G
-- TopologicalAddGroup.t2Space_of_zero_sep
@[to_additive]
theorem TopologicalGroup.t2Space_of_one_sep (H : ∀ x : G, x ≠ 1 → ∃ U ∈ 𝓝 (1 : G), x ∉ U) :
    T2Space G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (H : ∀ (x : G), x ≠ 0 → ∃ U ∈ 𝓝 0, x ∉ U)
Goal: T2Space G
-- exists_closed_nhds_one_inv_eq_mul_subset
@[to_additive "Given a neighborhood `U` of the identity, one may find a neighborhood `V` of the
identity which is closed, symmetric, and satisfies `V + V ⊆ U`."]
theorem exists_closed_nhds_one_inv_eq_mul_subset {U : Set G} (hU : U ∈ 𝓝 1) :
    ∃ V ∈ 𝓝 1, IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {U : Set G} (hU : U ∈ 𝓝 1)
Goal: ∃ V ∈ 𝓝 1, IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U
-- exists_closed_nhds_zero_neg_eq_add_subset
@[to_additive "Given a neighborhood `U` of the identity, one may find a neighborhood `V` of the
identity which is closed, symmetric, and satisfies `V + V ⊆ U`."]
theorem exists_closed_nhds_one_inv_eq_mul_subset {U : Set G} (hU : U ∈ 𝓝 1) :
    ∃ V ∈ 𝓝 1, IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {U : Set G} (hU : U ∈ 𝓝 0)
Goal: ∃ V ∈ 𝓝 0, IsClosed V ∧ -V = V ∧ V + V ⊆ U
-- Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite
@[to_additive
  "A subgroup `S` of an additive topological group `G` acts on `G` properly
  discontinuously on the left, if it is discrete in the sense that `S ∩ K` is finite for all compact
  `K`. (See also `DiscreteTopology`."]
theorem Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite (S : Subgroup G)
    (hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (S : Subgroup G) (hS : Filter.Tendsto (⇑S.subtype) Filter.cofinite (Filter.cocompact G))
Goal: ProperlyDiscontinuousSMul (↥S) G
-- AddSubgroup.properlyDiscontinuousVAdd_of_tendsto_cofinite
@[to_additive
  "A subgroup `S` of an additive topological group `G` acts on `G` properly
  discontinuously on the left, if it is discrete in the sense that `S ∩ K` is finite for all compact
  `K`. (See also `DiscreteTopology`."]
theorem Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite (S : Subgroup G)
    (hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (S : AddSubgroup G) (hS : Filter.Tendsto (⇑S.subtype) Filter.cofinite (Filter.cocompact G))
Goal: ProperlyDiscontinuousVAdd (↥S) G
-- Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite
@[to_additive
  "A subgroup `S` of an additive topological group `G` acts on `G` properly discontinuously
  on the right, if it is discrete in the sense that `S ∩ K` is finite for all compact `K`.
  (See also `DiscreteTopology`.)

  If `G` is Hausdorff, this can be combined with `t2Space_of_properlyDiscontinuousVAdd_of_t2Space`
  to show that the quotient group `G ⧸ S` is Hausdorff."]
theorem Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite (S : Subgroup G)
    (hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S.op G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (S : Subgroup G) (hS : Filter.Tendsto (⇑S.subtype) Filter.cofinite (Filter.cocompact G))
Goal: ProperlyDiscontinuousSMul (↥S.op) G
-- AddSubgroup.properlyDiscontinuousVAdd_opposite_of_tendsto_cofinite
@[to_additive
  "A subgroup `S` of an additive topological group `G` acts on `G` properly discontinuously
  on the right, if it is discrete in the sense that `S ∩ K` is finite for all compact `K`.
  (See also `DiscreteTopology`.)

  If `G` is Hausdorff, this can be combined with `t2Space_of_properlyDiscontinuousVAdd_of_t2Space`
  to show that the quotient group `G ⧸ S` is Hausdorff."]
theorem Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite (S : Subgroup G)
    (hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S.op G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (S : AddSubgroup G) (hS : Filter.Tendsto (⇑S.subtype) Filter.cofinite (Filter.cocompact G))
Goal: ProperlyDiscontinuousVAdd (↥S.op) G
-- compact_open_separated_add_right
@[to_additive
  "Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
  `0` such that `K + V ⊆ U`."]
theorem compact_open_separated_mul_right {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
    (hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), K * V ⊆ U 
Arguments: {G : Type w} [TopologicalSpace G] [AddZeroClass G] [ContinuousAdd G] {K U : Set G} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
Goal: ∃ V ∈ 𝓝 0, K + V ⊆ U
-- compact_open_separated_mul_right
@[to_additive
  "Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
  `0` such that `K + V ⊆ U`."]
theorem compact_open_separated_mul_right {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
    (hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), K * V ⊆ U 
Arguments: {G : Type w} [TopologicalSpace G] [MulOneClass G] [ContinuousMul G] {K U : Set G} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
Goal: ∃ V ∈ 𝓝 1, K * V ⊆ U
-- compact_open_separated_add_left
@[to_additive
  "Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
  `0` such that `V + K ⊆ U`."]
theorem compact_open_separated_mul_left {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
    (hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), V * K ⊆ U 
Arguments: {G : Type w} [TopologicalSpace G] [AddZeroClass G] [ContinuousAdd G] {K U : Set G} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
Goal: ∃ V ∈ 𝓝 0, V + K ⊆ U
-- compact_open_separated_mul_left
@[to_additive
  "Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
  `0` such that `V + K ⊆ U`."]
theorem compact_open_separated_mul_left {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
    (hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), V * K ⊆ U 
Arguments: {G : Type w} [TopologicalSpace G] [MulOneClass G] [ContinuousMul G] {K U : Set G} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
Goal: ∃ V ∈ 𝓝 1, V * K ⊆ U
-- compact_covered_by_add_left_translates
@[to_additive
  "A compact set is covered by finitely many left additive translates of a set
    with non-empty interior."]
theorem compact_covered_by_mul_left_translates {K V : Set G} (hK : IsCompact K)
    (hV : (interior V).Nonempty) : ∃ t : Finset G, K ⊆ ⋃ g ∈ t, (g * ·) ⁻¹' V 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty)
Goal: ∃ t, K ⊆ ⋃ g ∈ t, (fun x => g + x) ⁻¹' V
-- compact_covered_by_mul_left_translates
@[to_additive
  "A compact set is covered by finitely many left additive translates of a set
    with non-empty interior."]
theorem compact_covered_by_mul_left_translates {K V : Set G} (hK : IsCompact K)
    (hV : (interior V).Nonempty) : ∃ t : Finset G, K ⊆ ⋃ g ∈ t, (g * ·) ⁻¹' V 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty)
Goal: ∃ t, K ⊆ ⋃ g ∈ t, (fun x => g * x) ⁻¹' V
-- exists_disjoint_vadd_of_isCompact
@[to_additive
  "Given two compact sets in a noncompact additive topological group, there is a
  translate of the second one that is disjoint from the first one."]
theorem exists_disjoint_smul_of_isCompact [NoncompactSpace G] {K L : Set G} (hK : IsCompact K)
    (hL : IsCompact L) : ∃ g : G, Disjoint K (g • L) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [NoncompactSpace G] {K L : Set G} (hK : IsCompact K) (hL : IsCompact L)
Goal: ∃ g, Disjoint K (g +ᵥ L)
-- exists_disjoint_smul_of_isCompact
@[to_additive
  "Given two compact sets in a noncompact additive topological group, there is a
  translate of the second one that is disjoint from the first one."]
theorem exists_disjoint_smul_of_isCompact [NoncompactSpace G] {K L : Set G} (hK : IsCompact K)
    (hL : IsCompact L) : ∃ g : G, Disjoint K (g • L) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] [NoncompactSpace G] {K L : Set G} (hK : IsCompact K) (hL : IsCompact L)
Goal: ∃ g, Disjoint K (g • L)
-- exists_isCompact_isClosed_subset_isCompact_nhds_one
@[to_additive (attr := deprecated IsCompact.isCompact_isClosed_basis_nhds (since := "2024-01-28"))
  "A compact neighborhood of `0` in a topological additive group
admits a closed compact subset that is a neighborhood of `0`."]
theorem exists_isCompact_isClosed_subset_isCompact_nhds_one
    {L : Set G} (Lcomp : IsCompact L) (L1 : L ∈ 𝓝 (1 : G)) :
    ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ⊆ L ∧ K ∈ 𝓝 (1 : G) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {L : Set G} (Lcomp : IsCompact L) (L1 : L ∈ 𝓝 1)
Goal: ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ L ∧ K ∈ 𝓝 1
-- exists_isCompact_isClosed_subset_isCompact_nhds_zero
@[to_additive (attr := deprecated IsCompact.isCompact_isClosed_basis_nhds (since := "2024-01-28"))
  "A compact neighborhood of `0` in a topological additive group
admits a closed compact subset that is a neighborhood of `0`."]
theorem exists_isCompact_isClosed_subset_isCompact_nhds_one
    {L : Set G} (Lcomp : IsCompact L) (L1 : L ∈ 𝓝 (1 : G)) :
    ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ⊆ L ∧ K ∈ 𝓝 (1 : G) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {L : Set G} (Lcomp : IsCompact L) (L1 : L ∈ 𝓝 0)
Goal: ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ L ∧ K ∈ 𝓝 0
-- IsCompact.locallyCompactSpace_of_mem_nhds_of_addGroup
@[to_additive]
theorem IsCompact.locallyCompactSpace_of_mem_nhds_of_group {K : Set G} (hK : IsCompact K) {x : G}
    (h : K ∈ 𝓝 x) : LocallyCompactSpace G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {K : Set G} (hK : IsCompact K) {x : G} (h : K ∈ 𝓝 x)
Goal: LocallyCompactSpace G
-- IsCompact.locallyCompactSpace_of_mem_nhds_of_group
@[to_additive]
theorem IsCompact.locallyCompactSpace_of_mem_nhds_of_group {K : Set G} (hK : IsCompact K) {x : G}
    (h : K ∈ 𝓝 x) : LocallyCompactSpace G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {K : Set G} (hK : IsCompact K) {x : G} (h : K ∈ 𝓝 x)
Goal: LocallyCompactSpace G
-- instLocallyCompactSpaceOfWeaklyOfAddGroup
@[to_additive
  (attr := deprecated WeaklyLocallyCompactSpace.locallyCompactSpace (since := "2024-01-28"))]
theorem instLocallyCompactSpaceOfWeaklyOfGroup [WeaklyLocallyCompactSpace G] :
    LocallyCompactSpace G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [WeaklyLocallyCompactSpace G]
Goal: LocallyCompactSpace G
-- instLocallyCompactSpaceOfWeaklyOfGroup
@[to_additive
  (attr := deprecated WeaklyLocallyCompactSpace.locallyCompactSpace (since := "2024-01-28"))]
theorem instLocallyCompactSpaceOfWeaklyOfGroup [WeaklyLocallyCompactSpace G] :
    LocallyCompactSpace G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] [WeaklyLocallyCompactSpace G]
Goal: LocallyCompactSpace G
-- eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group
@[to_additive
      "If a function defined on a topological additive group has a support contained in a compact
      set, then either the function is trivial or the group is locally compact."]
theorem eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group
    [TopologicalSpace α] [Zero α] [T1Space α]
    {f : G → α} {k : Set G} (hk : IsCompact k) (hf : support f ⊆ k) (h'f : Continuous f) :
    f = 0 ∨ LocallyCompactSpace G 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G] [TopologicalSpace α] [Zero α] [T1Space α] {f : G → α} {k : Set G} (hk : IsCompact k) (hf : Function.support f ⊆ k) (h'f : Continuous f)
Goal: f = 0 ∨ LocallyCompactSpace G
-- eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_addGroup
@[to_additive
      "If a function defined on a topological additive group has a support contained in a compact
      set, then either the function is trivial or the group is locally compact."]
theorem eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group
    [TopologicalSpace α] [Zero α] [T1Space α]
    {f : G → α} {k : Set G} (hk : IsCompact k) (hf : support f ⊆ k) (h'f : Continuous f) :
    f = 0 ∨ LocallyCompactSpace G 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [TopologicalSpace α] [Zero α] [T1Space α] {f : G → α} {k : Set G} (hk : IsCompact k) (hf : Function.support f ⊆ k) (h'f : Continuous f)
Goal: f = 0 ∨ LocallyCompactSpace G
-- HasCompactSupport.eq_zero_or_locallyCompactSpace_of_group
@[to_additive
      "If a function defined on a topological additive group has compact support,
      then either the function is trivial or the group is locally compact."]
theorem HasCompactSupport.eq_zero_or_locallyCompactSpace_of_group
    [TopologicalSpace α] [Zero α] [T1Space α]
    {f : G → α} (hf : HasCompactSupport f) (h'f : Continuous f) :
    f = 0 ∨ LocallyCompactSpace G 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G] [TopologicalSpace α] [Zero α] [T1Space α] {f : G → α} (hf : HasCompactSupport f) (h'f : Continuous f)
Goal: f = 0 ∨ LocallyCompactSpace G
-- HasCompactSupport.eq_zero_or_locallyCompactSpace_of_addGroup
@[to_additive
      "If a function defined on a topological additive group has compact support,
      then either the function is trivial or the group is locally compact."]
theorem HasCompactSupport.eq_zero_or_locallyCompactSpace_of_group
    [TopologicalSpace α] [Zero α] [T1Space α]
    {f : G → α} (hf : HasCompactSupport f) (h'f : Continuous f) :
    f = 0 ∨ LocallyCompactSpace G 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [TopologicalSpace α] [Zero α] [T1Space α] {f : G → α} (hf : HasCompactSupport f) (h'f : Continuous f)
Goal: f = 0 ∨ LocallyCompactSpace G
-- local_isCompact_isClosed_nhds_of_addGroup
@[to_additive (attr := deprecated isCompact_isClosed_basis_nhds (since := "2024-01-28"))
  "In a locally compact additive group, any neighborhood of the identity contains a
  compact closed neighborhood of the identity, even without separation assumptions on the space."]
theorem local_isCompact_isClosed_nhds_of_group [LocallyCompactSpace G] {U : Set G}
    (hU : U ∈ 𝓝 (1 : G)) :
    ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ (1 : G) ∈ interior K 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [LocallyCompactSpace G] {U : Set G} (hU : U ∈ 𝓝 0)
Goal: ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 0 ∈ interior K
-- local_isCompact_isClosed_nhds_of_group
@[to_additive (attr := deprecated isCompact_isClosed_basis_nhds (since := "2024-01-28"))
  "In a locally compact additive group, any neighborhood of the identity contains a
  compact closed neighborhood of the identity, even without separation assumptions on the space."]
theorem local_isCompact_isClosed_nhds_of_group [LocallyCompactSpace G] {U : Set G}
    (hU : U ∈ 𝓝 (1 : G)) :
    ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ (1 : G) ∈ interior K 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] [LocallyCompactSpace G] {U : Set G} (hU : U ∈ 𝓝 1)
Goal: ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K