ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsClosed.neg
@[to_additive]
theorem IsClosed.inv (hs : IsClosed s) : IsClosed s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] {s : Set G} (hs : IsClosed s)
Goal: IsClosed (-s)
-- neg_closure
@[to_additive]
theorem inv_closure : ∀ s : Set G, (closure s)⁻¹ = closure s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] (s : Set G)
Goal: -closure s = closure (-s)
-- inv_closure
@[to_additive]
theorem inv_closure : ∀ s : Set G, (closure s)⁻¹ = closure s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] (s : Set G)
Goal: (closure s)⁻¹ = closure s⁻¹
-- continuous_neg_iff
@[to_additive (attr := simp)]
lemma continuous_inv_iff : Continuous f⁻¹ ↔ Continuous f 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G}
Goal: Continuous (-f) ↔ Continuous f
-- continuous_inv_iff
@[to_additive (attr := simp)]
lemma continuous_inv_iff : Continuous f⁻¹ ↔ Continuous f 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G}
Goal: Continuous f⁻¹ ↔ Continuous f
-- continuousAt_inv_iff
@[to_additive (attr := simp)]
lemma continuousAt_inv_iff : ContinuousAt f⁻¹ x ↔ ContinuousAt f x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} {x : α}
Goal: ContinuousAt f⁻¹ x ↔ ContinuousAt f x
-- continuousAt_neg_iff
@[to_additive (attr := simp)]
lemma continuousAt_inv_iff : ContinuousAt f⁻¹ x ↔ ContinuousAt f x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} {x : α}
Goal: ContinuousAt (-f) x ↔ ContinuousAt f x
-- continuousOn_inv_iff
@[to_additive (attr := simp)]
lemma continuousOn_inv_iff : ContinuousOn f⁻¹ s ↔ ContinuousOn f s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} {s : Set α}
Goal: ContinuousOn f⁻¹ s ↔ ContinuousOn f s
-- continuousOn_neg_iff
@[to_additive (attr := simp)]
lemma continuousOn_inv_iff : ContinuousOn f⁻¹ s ↔ ContinuousOn f s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} {s : Set α}
Goal: ContinuousOn (-f) s ↔ ContinuousOn f s
-- continuousNeg_sInf
@[to_additive]
theorem continuousInv_sInf {ts : Set (TopologicalSpace G)}
    (h : ∀ t ∈ ts, @ContinuousInv G t _) : @ContinuousInv G (sInf ts) _ 
Arguments: {G : Type w} [Neg G] {ts : Set (TopologicalSpace G)} (h : ∀ t ∈ ts, ContinuousNeg G)
Goal: ContinuousNeg G
-- continuousInv_sInf
@[to_additive]
theorem continuousInv_sInf {ts : Set (TopologicalSpace G)}
    (h : ∀ t ∈ ts, @ContinuousInv G t _) : @ContinuousInv G (sInf ts) _ 
Arguments: {G : Type w} [Inv G] {ts : Set (TopologicalSpace G)} (h : ∀ t ∈ ts, ContinuousInv G)
Goal: ContinuousInv G
-- continuousNeg_iInf
@[to_additive]
theorem continuousInv_iInf {ts' : ι' → TopologicalSpace G}
    (h' : ∀ i, @ContinuousInv G (ts' i) _) : @ContinuousInv G (⨅ i, ts' i) _ 
Arguments: {G : Type w} {ι' : Sort u_1} [Neg G] {ts' : ι' → TopologicalSpace G} (h' : ∀ (i : ι'), ContinuousNeg G)
Goal: ContinuousNeg G
-- continuousInv_iInf
@[to_additive]
theorem continuousInv_iInf {ts' : ι' → TopologicalSpace G}
    (h' : ∀ i, @ContinuousInv G (ts' i) _) : @ContinuousInv G (⨅ i, ts' i) _ 
Arguments: {G : Type w} {ι' : Sort u_1} [Inv G] {ts' : ι' → TopologicalSpace G} (h' : ∀ (i : ι'), ContinuousInv G)
Goal: ContinuousInv G
-- continuousInv_inf
@[to_additive]
theorem continuousInv_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @ContinuousInv G t₁ _)
    (h₂ : @ContinuousInv G t₂ _) : @ContinuousInv G (t₁ ⊓ t₂) _ 
Arguments: {G : Type w} [Inv G] {t₁ t₂ : TopologicalSpace G} (h₁ : ContinuousInv G) (h₂ : ContinuousInv G)
Goal: ContinuousInv G
-- continuousNeg_inf
@[to_additive]
theorem continuousInv_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @ContinuousInv G t₁ _)
    (h₂ : @ContinuousInv G t₂ _) : @ContinuousInv G (t₁ ⊓ t₂) _ 
Arguments: {G : Type w} [Neg G] {t₁ t₂ : TopologicalSpace G} (h₁ : ContinuousNeg G) (h₂ : ContinuousNeg G)
Goal: ContinuousNeg G
-- Inducing.continuousNeg
@[to_additive]
theorem Inducing.continuousInv {G H : Type*} [Inv G] [Inv H] [TopologicalSpace G]
    [TopologicalSpace H] [ContinuousInv H] {f : G → H} (hf : Inducing f)
    (hf_inv : ∀ x, f x⁻¹ = (f x)⁻¹) : ContinuousInv G 
Arguments: {G : Type u_1} {H : Type u_2} [Neg G] [Neg H] [TopologicalSpace G] [TopologicalSpace H] [ContinuousNeg H] {f : G → H} (hf : Inducing f) (hf_inv : ∀ (x : G), f (-x) = -f x)
Goal: ContinuousNeg G
-- Inducing.continuousInv
@[to_additive]
theorem Inducing.continuousInv {G H : Type*} [Inv G] [Inv H] [TopologicalSpace G]
    [TopologicalSpace H] [ContinuousInv H] {f : G → H} (hf : Inducing f)
    (hf_inv : ∀ x, f x⁻¹ = (f x)⁻¹) : ContinuousInv G 
Arguments: {G : Type u_1} {H : Type u_2} [Inv G] [Inv H] [TopologicalSpace G] [TopologicalSpace H] [ContinuousInv H] {f : G → H} (hf : Inducing f) (hf_inv : ∀ (x : G), f x⁻¹ = (f x)⁻¹)
Goal: ContinuousInv G
-- TopologicalGroup.continuous_conj_prod
@[to_additive
  "Conjugation is jointly continuous on `G × G` when both `add` and `neg` are continuous."]
theorem TopologicalGroup.continuous_conj_prod [ContinuousInv G] :
    Continuous fun g : G × G => g.fst * g.snd * g.fst⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [Mul G] [ContinuousMul G] [ContinuousInv G]
Goal: Continuous fun g => g.1 * g.2 * g.1⁻¹
-- TopologicalAddGroup.continuous_conj_sum
@[to_additive
  "Conjugation is jointly continuous on `G × G` when both `add` and `neg` are continuous."]
theorem TopologicalGroup.continuous_conj_prod [ContinuousInv G] :
    Continuous fun g : G × G => g.fst * g.snd * g.fst⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [Add G] [ContinuousAdd G] [ContinuousNeg G]
Goal: Continuous fun g => g.1 + g.2 + -g.1
-- TopologicalAddGroup.continuous_conj
@[to_additive (attr := continuity)
  "Conjugation by a fixed element is continuous when `add` is continuous."]
theorem TopologicalGroup.continuous_conj (g : G) : Continuous fun h : G => g * h * g⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [Add G] [ContinuousAdd G] (g : G)
Goal: Continuous fun h => g + h + -g
-- TopologicalGroup.continuous_conj
@[to_additive (attr := continuity)
  "Conjugation by a fixed element is continuous when `add` is continuous."]
theorem TopologicalGroup.continuous_conj (g : G) : Continuous fun h : G => g * h * g⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [Mul G] [ContinuousMul G] (g : G)
Goal: Continuous fun h => g * h * g⁻¹
-- TopologicalGroup.continuous_conj'
@[to_additive (attr := continuity)
  "Conjugation acting on fixed element of the additive group is continuous when both
    `add` and `neg` are continuous."]
theorem TopologicalGroup.continuous_conj' [ContinuousInv G] (h : G) :
    Continuous fun g : G => g * h * g⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [Mul G] [ContinuousMul G] [ContinuousInv G] (h : G)
Goal: Continuous fun g => g * h * g⁻¹
-- TopologicalAddGroup.continuous_conj'
@[to_additive (attr := continuity)
  "Conjugation acting on fixed element of the additive group is continuous when both
    `add` and `neg` are continuous."]
theorem TopologicalGroup.continuous_conj' [ContinuousInv G] (h : G) :
    Continuous fun g : G => g * h * g⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [Add G] [ContinuousAdd G] [ContinuousNeg G] (h : G)
Goal: Continuous fun g => g + h + -g
-- continuous_zsmul
@[to_additive (attr := continuity)]
theorem continuous_zpow : ∀ z : ℤ, Continuous fun a : G => a ^ z
  
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (z : ℤ)
Goal: Continuous fun a => z • a
-- continuous_zpow
@[to_additive (attr := continuity)]
theorem continuous_zpow : ∀ z : ℤ, Continuous fun a : G => a ^ z
  
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (z : ℤ)
Goal: Continuous fun a => a ^ z
-- Continuous.zsmul
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.zpow {f : α → G} (h : Continuous f) (z : ℤ) : Continuous fun b => f b ^ z 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [TopologicalSpace α] {f : α → G} (h : Continuous f) (z : ℤ)
Goal: Continuous fun b => z • f b
-- Continuous.zpow
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.zpow {f : α → G} (h : Continuous f) (z : ℤ) : Continuous fun b => f b ^ z 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G] [TopologicalSpace α] {f : α → G} (h : Continuous f) (z : ℤ)
Goal: Continuous fun b => f b ^ z
-- continuousOn_zpow
@[to_additive]
theorem continuousOn_zpow {s : Set G} (z : ℤ) : ContinuousOn (fun x => x ^ z) s 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s : Set G} (z : ℤ)
Goal: ContinuousOn (fun x => x ^ z) s
-- continuousOn_zsmul
@[to_additive]
theorem continuousOn_zpow {s : Set G} (z : ℤ) : ContinuousOn (fun x => x ^ z) s 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s : Set G} (z : ℤ)
Goal: ContinuousOn (fun x => z • x) s
-- continuousAt_zsmul
@[to_additive]
theorem continuousAt_zpow (x : G) (z : ℤ) : ContinuousAt (fun x => x ^ z) x 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (x : G) (z : ℤ)
Goal: ContinuousAt (fun x => z • x) x
-- continuousAt_zpow
@[to_additive]
theorem continuousAt_zpow (x : G) (z : ℤ) : ContinuousAt (fun x => x ^ z) x 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (x : G) (z : ℤ)
Goal: ContinuousAt (fun x => x ^ z) x
-- Filter.Tendsto.zsmul
@[to_additive]
theorem Filter.Tendsto.zpow {α} {l : Filter α} {f : α → G} {x : G} (hf : Tendsto f l (𝓝 x))
    (z : ℤ) : Tendsto (fun x => f x ^ z) l (𝓝 (x ^ z)) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {α : Type u_1} {l : Filter α} {f : α → G} {x : G} (hf : Filter.Tendsto f l (𝓝 x)) (z : ℤ)
Goal: Filter.Tendsto (fun x => z • f x) l (𝓝 (z • x))
-- Filter.Tendsto.zpow
@[to_additive]
theorem Filter.Tendsto.zpow {α} {l : Filter α} {f : α → G} {x : G} (hf : Tendsto f l (𝓝 x))
    (z : ℤ) : Tendsto (fun x => f x ^ z) l (𝓝 (x ^ z)) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {α : Type u_1} {l : Filter α} {f : α → G} {x : G} (hf : Filter.Tendsto f l (𝓝 x)) (z : ℤ)
Goal: Filter.Tendsto (fun x => f x ^ z) l (𝓝 (x ^ z))
-- ContinuousWithinAt.zsmul
@[to_additive]
theorem ContinuousWithinAt.zpow {f : α → G} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x)
    (z : ℤ) : ContinuousWithinAt (fun x => f x ^ z) s x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [TopologicalSpace α] {f : α → G} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (z : ℤ)
Goal: ContinuousWithinAt (fun x => z • f x) s x
-- ContinuousWithinAt.zpow
@[to_additive]
theorem ContinuousWithinAt.zpow {f : α → G} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x)
    (z : ℤ) : ContinuousWithinAt (fun x => f x ^ z) s x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G] [TopologicalSpace α] {f : α → G} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (z : ℤ)
Goal: ContinuousWithinAt (fun x => f x ^ z) s x
-- ContinuousAt.zpow
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.zpow {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) :
    ContinuousAt (fun x => f x ^ z) x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ)
Goal: ContinuousAt (fun x => f x ^ z) x
-- ContinuousAt.zsmul
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.zpow {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) :
    ContinuousAt (fun x => f x ^ z) x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ)
Goal: ContinuousAt (fun x => z • f x) x
-- ContinuousOn.zsmul
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.zpow {f : α → G} {s : Set α} (hf : ContinuousOn f s) (z : ℤ) :
    ContinuousOn (fun x => f x ^ z) s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [TopologicalSpace α] {f : α → G} {s : Set α} (hf : ContinuousOn f s) (z : ℤ)
Goal: ContinuousOn (fun x => z • f x) s
-- ContinuousOn.zpow
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.zpow {f : α → G} {s : Set α} (hf : ContinuousOn f s) (z : ℤ) :
    ContinuousOn (fun x => f x ^ z) s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G] [TopologicalSpace α] {f : α → G} {s : Set α} (hf : ContinuousOn f s) (z : ℤ)
Goal: ContinuousOn (fun x => f x ^ z) s
-- tendsto_inv_nhdsWithin_Ioi
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ioi {a : H} : Tendsto Inv.inv (𝓝[>] a) (𝓝[<] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[>] a) (𝓝[<] a⁻¹)
-- tendsto_neg_nhdsWithin_Ioi
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ioi {a : H} : Tendsto Inv.inv (𝓝[>] a) (𝓝[<] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[>] a) (𝓝[<] (-a))
-- tendsto_neg_nhdsWithin_Iio
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iio {a : H} : Tendsto Inv.inv (𝓝[<] a) (𝓝[>] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[<] a) (𝓝[>] (-a))
-- tendsto_inv_nhdsWithin_Iio
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iio {a : H} : Tendsto Inv.inv (𝓝[<] a) (𝓝[>] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[<] a) (𝓝[>] a⁻¹)
-- tendsto_neg_nhdsWithin_Ioi_neg
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ioi_inv {a : H} : Tendsto Inv.inv (𝓝[>] a⁻¹) (𝓝[<] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[>] (-a)) (𝓝[<] a)
-- tendsto_inv_nhdsWithin_Ioi_inv
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ioi_inv {a : H} : Tendsto Inv.inv (𝓝[>] a⁻¹) (𝓝[<] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[>] a⁻¹) (𝓝[<] a)
-- tendsto_neg_nhdsWithin_Iio_neg
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iio_inv {a : H} : Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[<] (-a)) (𝓝[>] a)
-- tendsto_inv_nhdsWithin_Iio_inv
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iio_inv {a : H} : Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a)
-- tendsto_neg_nhdsWithin_Ici
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ici {a : H} : Tendsto Inv.inv (𝓝[≥] a) (𝓝[≤] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[≥] a) (𝓝[≤] (-a))
-- tendsto_inv_nhdsWithin_Ici
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ici {a : H} : Tendsto Inv.inv (𝓝[≥] a) (𝓝[≤] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[≥] a) (𝓝[≤] a⁻¹)
-- tendsto_neg_nhdsWithin_Iic
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iic {a : H} : Tendsto Inv.inv (𝓝[≤] a) (𝓝[≥] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[≤] a) (𝓝[≥] (-a))
-- tendsto_inv_nhdsWithin_Iic
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iic {a : H} : Tendsto Inv.inv (𝓝[≤] a) (𝓝[≥] a⁻¹) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[≤] a) (𝓝[≥] a⁻¹)
-- tendsto_neg_nhdsWithin_Ici_neg
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ici_inv {a : H} : Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[≥] (-a)) (𝓝[≤] a)
-- tendsto_inv_nhdsWithin_Ici_inv
@[to_additive]
theorem tendsto_inv_nhdsWithin_Ici_inv {a : H} : Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a)
-- tendsto_inv_nhdsWithin_Iic_inv
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iic_inv {a : H} : Tendsto Inv.inv (𝓝[≤] a⁻¹) (𝓝[≥] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] {a : H}
Goal: Filter.Tendsto Inv.inv (𝓝[≤] a⁻¹) (𝓝[≥] a)
-- tendsto_neg_nhdsWithin_Iic_neg
@[to_additive]
theorem tendsto_inv_nhdsWithin_Iic_inv {a : H} : Tendsto Inv.inv (𝓝[≤] a⁻¹) (𝓝[≥] a) 
Arguments: {H : Type x} [TopologicalSpace H] [OrderedAddCommGroup H] [ContinuousNeg H] {a : H}
Goal: Filter.Tendsto Neg.neg (𝓝[≤] (-a)) (𝓝[≥] a)
-- nhds_zero_symm
@[to_additive]
theorem nhds_one_symm : comap Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) 
Arguments: (G : Type w) [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G]
Goal: Filter.comap Neg.neg (𝓝 0) = 𝓝 0
-- nhds_one_symm
@[to_additive]
theorem nhds_one_symm : comap Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) 
Arguments: (G : Type w) [TopologicalSpace G] [Group G] [TopologicalGroup G]
Goal: Filter.comap Inv.inv (𝓝 1) = 𝓝 1
-- nhds_one_symm'
@[to_additive]
theorem nhds_one_symm' : map Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) 
Arguments: (G : Type w) [TopologicalSpace G] [Group G] [TopologicalGroup G]
Goal: Filter.map Inv.inv (𝓝 1) = 𝓝 1
-- nhds_zero_symm'
@[to_additive]
theorem nhds_one_symm' : map Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) 
Arguments: (G : Type w) [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G]
Goal: Filter.map Neg.neg (𝓝 0) = 𝓝 0
-- inv_mem_nhds_one
@[to_additive]
theorem inv_mem_nhds_one {S : Set G} (hS : S ∈ (𝓝 1 : Filter G)) : S⁻¹ ∈ 𝓝 (1 : G) 
Arguments: (G : Type w) [TopologicalSpace G] [Group G] [TopologicalGroup G] {S : Set G} (hS : S ∈ 𝓝 1)
Goal: S⁻¹ ∈ 𝓝 1
-- neg_mem_nhds_zero
@[to_additive]
theorem inv_mem_nhds_one {S : Set G} (hS : S ∈ (𝓝 1 : Filter G)) : S⁻¹ ∈ 𝓝 (1 : G) 
Arguments: (G : Type w) [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {S : Set G} (hS : S ∈ 𝓝 0)
Goal: -S ∈ 𝓝 0
-- Homeomorph.shearMulRight_coe
@[to_additive (attr := simp)]
theorem Homeomorph.shearMulRight_coe :
    ⇑(Homeomorph.shearMulRight G) = fun z : G × G => (z.1, z.1 * z.2) 
Arguments: (G : Type w) [TopologicalSpace G] [Group G] [TopologicalGroup G]
Goal: ⇑(Homeomorph.shearMulRight G) = fun z => (z.1, z.1 * z.2)
-- Homeomorph.shearAddRight_coe
@[to_additive (attr := simp)]
theorem Homeomorph.shearMulRight_coe :
    ⇑(Homeomorph.shearMulRight G) = fun z : G × G => (z.1, z.1 * z.2) 
Arguments: (G : Type w) [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G]
Goal: ⇑(Homeomorph.shearAddRight G) = fun z => (z.1, z.1 + z.2)
-- Homeomorph.shearMulRight_symm_coe
@[to_additive (attr := simp)]
theorem Homeomorph.shearMulRight_symm_coe :
    ⇑(Homeomorph.shearMulRight G).symm = fun z : G × G => (z.1, z.1⁻¹ * z.2) 
Arguments: (G : Type w) [TopologicalSpace G] [Group G] [TopologicalGroup G]
Goal: ⇑(Homeomorph.shearMulRight G).symm = fun z => (z.1, z.1⁻¹ * z.2)
-- Homeomorph.shearAddRight_symm_coe
@[to_additive (attr := simp)]
theorem Homeomorph.shearMulRight_symm_coe :
    ⇑(Homeomorph.shearMulRight G).symm = fun z : G × G => (z.1, z.1⁻¹ * z.2) 
Arguments: (G : Type w) [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G]
Goal: ⇑(Homeomorph.shearAddRight G).symm = fun z => (z.1, -z.1 + z.2)
-- Inducing.topologicalGroup
@[to_additive]
protected theorem Inducing.topologicalGroup {F : Type*} [Group H] [TopologicalSpace H]
    [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Inducing f) : TopologicalGroup H 
Arguments: {G : Type w} {H : Type x} [TopologicalSpace G] [Group G] [TopologicalGroup G] {F : Type u_1} [Group H] [TopologicalSpace H] [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Inducing ⇑f)
Goal: TopologicalGroup H
-- Inducing.topologicalAddGroup
@[to_additive]
protected theorem Inducing.topologicalGroup {F : Type*} [Group H] [TopologicalSpace H]
    [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Inducing f) : TopologicalGroup H 
Arguments: {G : Type w} {H : Type x} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {F : Type u_1} [AddGroup H] [TopologicalSpace H] [FunLike F H G] [AddMonoidHomClass F H G] (f : F) (hf : Inducing ⇑f)
Goal: TopologicalAddGroup H
-- topologicalGroup_induced
@[to_additive]
-- Porting note: removed `protected` (needs to be in namespace)
theorem topologicalGroup_induced {F : Type*} [Group H] [FunLike F H G] [MonoidHomClass F H G]
    (f : F) :
    @TopologicalGroup H (induced f ‹_›) _ 
Arguments: {G : Type w} {H : Type x} [TopologicalSpace G] [Group G] [TopologicalGroup G] {F : Type u_1} [Group H] [FunLike F H G] [MonoidHomClass F H G] (f : F)
Goal: TopologicalGroup H
-- topologicalAddGroup_induced
@[to_additive]
-- Porting note: removed `protected` (needs to be in namespace)
theorem topologicalGroup_induced {F : Type*} [Group H] [FunLike F H G] [MonoidHomClass F H G]
    (f : F) :
    @TopologicalGroup H (induced f ‹_›) _ 
Arguments: {G : Type w} {H : Type x} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {F : Type u_1} [AddGroup H] [FunLike F H G] [AddMonoidHomClass F H G] (f : F)
Goal: TopologicalAddGroup H
-- AddSubgroup.topologicalClosure_coe
@[to_additive (attr := simp)]
theorem Subgroup.topologicalClosure_coe {s : Subgroup G} :
    (s.topologicalClosure : Set G) = _root_.closure s 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s : AddSubgroup G}
Goal: ↑s.topologicalClosure = closure ↑s
-- Subgroup.topologicalClosure_coe
@[to_additive (attr := simp)]
theorem Subgroup.topologicalClosure_coe {s : Subgroup G} :
    (s.topologicalClosure : Set G) = _root_.closure s 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s : Subgroup G}
Goal: ↑s.topologicalClosure = closure ↑s
-- Subgroup.le_topologicalClosure
@[to_additive]
theorem Subgroup.le_topologicalClosure (s : Subgroup G) : s ≤ s.topologicalClosure 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (s : Subgroup G)
Goal: s ≤ s.topologicalClosure
-- AddSubgroup.le_topologicalClosure
@[to_additive]
theorem Subgroup.le_topologicalClosure (s : Subgroup G) : s ≤ s.topologicalClosure 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (s : AddSubgroup G)
Goal: s ≤ s.topologicalClosure
-- Subgroup.isClosed_topologicalClosure
@[to_additive]
theorem Subgroup.isClosed_topologicalClosure (s : Subgroup G) :
    IsClosed (s.topologicalClosure : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (s : Subgroup G)
Goal: IsClosed ↑s.topologicalClosure
-- AddSubgroup.isClosed_topologicalClosure
@[to_additive]
theorem Subgroup.isClosed_topologicalClosure (s : Subgroup G) :
    IsClosed (s.topologicalClosure : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (s : AddSubgroup G)
Goal: IsClosed ↑s.topologicalClosure
-- Subgroup.topologicalClosure_minimal
@[to_additive]
theorem Subgroup.topologicalClosure_minimal (s : Subgroup G) {t : Subgroup G} (h : s ≤ t)
    (ht : IsClosed (t : Set G)) : s.topologicalClosure ≤ t 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (s : Subgroup G) {t : Subgroup G} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- AddSubgroup.topologicalClosure_minimal
@[to_additive]
theorem Subgroup.topologicalClosure_minimal (s : Subgroup G) {t : Subgroup G} (h : s ≤ t)
    (ht : IsClosed (t : Set G)) : s.topologicalClosure ≤ t 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (s : AddSubgroup G) {t : AddSubgroup G} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t