ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FirstCountableTopology.properlyDiscontinuousSMul_iff_properSMul
theorem FirstCountableTopology.properlyDiscontinuousSMul_iff_properSMul [T2Space X]
    [FirstCountableTopology X] [DiscreteTopology G] [ContinuousConstSMul G X] :
    ProperlyDiscontinuousSMul G X ↔ ProperSMul G X 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X] [T2Space X] [FirstCountableTopology X] [DiscreteTopology G] [ContinuousConstSMul G X]
Goal: ProperlyDiscontinuousSMul G X ↔ ProperSMul G X
-- Filter.Tendsto.div_const
theorem Filter.Tendsto.div_const {x : G₀} (hf : Tendsto f l (𝓝 x)) (y : G₀) :
    Tendsto (fun a => f a / y) l (𝓝 (x / y)) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [DivInvMonoid G₀] [TopologicalSpace G₀] [ContinuousMul G₀] {f : α → G₀} {l : Filter α} {x : G₀} (hf : Filter.Tendsto f l (𝓝 x)) (y : G₀)
Goal: Filter.Tendsto (fun a => f a / y) l (𝓝 (x / y))
-- ContinuousAt.div_const
nonrec theorem ContinuousAt.div_const {a : α} (hf : ContinuousAt f a) (y : G₀) :
    ContinuousAt (fun x => f x / y) a 
Arguments: {α : Type u_1} {G₀ : Type u_3} [DivInvMonoid G₀] [TopologicalSpace G₀] [ContinuousMul G₀] {f : α → G₀} [TopologicalSpace α] {a : α} (hf : ContinuousAt f a) (y : G₀)
Goal: ContinuousAt (fun x => f x / y) a
-- ContinuousWithinAt.div_const
nonrec theorem ContinuousWithinAt.div_const {a} (hf : ContinuousWithinAt f s a) (y : G₀) :
    ContinuousWithinAt (fun x => f x / y) s a 
Arguments: {α : Type u_1} {G₀ : Type u_3} [DivInvMonoid G₀] [TopologicalSpace G₀] [ContinuousMul G₀] {f : α → G₀} {s : Set α} [TopologicalSpace α] {a : α} (hf : ContinuousWithinAt f s a) (y : G₀)
Goal: ContinuousWithinAt (fun x => f x / y) s a
-- ContinuousOn.div_const
theorem ContinuousOn.div_const (hf : ContinuousOn f s) (y : G₀) :
    ContinuousOn (fun x => f x / y) s 
Arguments: {α : Type u_1} {G₀ : Type u_3} [DivInvMonoid G₀] [TopologicalSpace G₀] [ContinuousMul G₀] {f : α → G₀} {s : Set α} [TopologicalSpace α] (hf : ContinuousOn f s) (y : G₀)
Goal: ContinuousOn (fun x => f x / y) s
-- Continuous.div_const
@[continuity, fun_prop]
theorem Continuous.div_const (hf : Continuous f) (y : G₀) : Continuous fun x => f x / y 
Arguments: {α : Type u_1} {G₀ : Type u_3} [DivInvMonoid G₀] [TopologicalSpace G₀] [ContinuousMul G₀] {f : α → G₀} [TopologicalSpace α] (hf : Continuous f) (y : G₀)
Goal: Continuous fun x => f x / y
-- HasContinuousInv₀.continuousAt_inv₀
class HasContinuousInv₀ (G₀ : Type*) [Zero G₀] [Inv G₀] [TopologicalSpace G₀] : Prop where
    continuousAt_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x
Arguments: {G₀ : Type u_4} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [self : HasContinuousInv₀ G₀] ⦃x : G₀⦄
Goal: x ≠ 0 → ContinuousAt Inv.inv x
-- tendsto_inv₀
theorem tendsto_inv₀ {x : G₀} (hx : x ≠ 0) : Tendsto Inv.inv (𝓝 x) (𝓝 x⁻¹) 
Arguments: {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {x : G₀} (hx : x ≠ 0)
Goal: Filter.Tendsto Inv.inv (𝓝 x) (𝓝 x⁻¹)
-- continuousOn_inv₀
theorem continuousOn_inv₀ : ContinuousOn (Inv.inv : G₀ → G₀) {0}ᶜ 
Arguments: {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀]
Goal: ContinuousOn Inv.inv {0}ᶜ
-- Filter.Tendsto.inv₀
theorem Filter.Tendsto.inv₀ {a : G₀} (hf : Tendsto f l (𝓝 a)) (ha : a ≠ 0) :
    Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {l : Filter α} {f : α → G₀} {a : G₀} (hf : Filter.Tendsto f l (𝓝 a)) (ha : a ≠ 0)
Goal: Filter.Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
-- ContinuousWithinAt.inv₀
nonrec theorem ContinuousWithinAt.inv₀ (hf : ContinuousWithinAt f s a) (ha : f a ≠ 0) :
    ContinuousWithinAt (fun x => (f x)⁻¹) s a 
Arguments: {α : Type u_1} {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {f : α → G₀} {s : Set α} {a : α} [TopologicalSpace α] (hf : ContinuousWithinAt f s a) (ha : f a ≠ 0)
Goal: ContinuousWithinAt (fun x => (f x)⁻¹) s a
-- ContinuousAt.inv₀
@[fun_prop]
nonrec theorem ContinuousAt.inv₀ (hf : ContinuousAt f a) (ha : f a ≠ 0) :
    ContinuousAt (fun x => (f x)⁻¹) a 
Arguments: {α : Type u_1} {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {f : α → G₀} {a : α} [TopologicalSpace α] (hf : ContinuousAt f a) (ha : f a ≠ 0)
Goal: ContinuousAt (fun x => (f x)⁻¹) a
-- Continuous.inv₀
@[continuity, fun_prop]
theorem Continuous.inv₀ (hf : Continuous f) (h0 : ∀ x, f x ≠ 0) : Continuous fun x => (f x)⁻¹ 
Arguments: {α : Type u_1} {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {f : α → G₀} [TopologicalSpace α] (hf : Continuous f) (h0 : ∀ (x : α), f x ≠ 0)
Goal: Continuous fun x => (f x)⁻¹
-- ContinuousOn.inv₀
@[fun_prop]
theorem ContinuousOn.inv₀ (hf : ContinuousOn f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
    ContinuousOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u_1} {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {f : α → G₀} {s : Set α} [TopologicalSpace α] (hf : ContinuousOn f s) (h0 : ∀ x ∈ s, f x ≠ 0)
Goal: ContinuousOn (fun x => (f x)⁻¹) s
-- Units.embedding_val₀
theorem Units.embedding_val₀ [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] :
    Embedding (val : G₀ˣ → G₀) 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀]
Goal: Embedding Units.val
-- nhds_inv₀
lemma nhds_inv₀ (hx : x ≠ 0) : 𝓝 x⁻¹ = (𝓝 x)⁻¹ 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {x : G₀} (hx : x ≠ 0)
Goal: 𝓝 x⁻¹ = (𝓝 x)⁻¹
-- tendsto_inv_iff₀
lemma tendsto_inv_iff₀ {l : Filter α} {f : α → G₀} (hx : x ≠ 0) :
    Tendsto (fun x ↦ (f x)⁻¹) l (𝓝 x⁻¹) ↔ Tendsto f l (𝓝 x) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] {x : G₀} {l : Filter α} {f : α → G₀} (hx : x ≠ 0)
Goal: Filter.Tendsto (fun x => (f x)⁻¹) l (𝓝 x⁻¹) ↔ Filter.Tendsto f l (𝓝 x)
-- Filter.Tendsto.div
theorem Filter.Tendsto.div {l : Filter α} {a b : G₀} (hf : Tendsto f l (𝓝 a))
    (hg : Tendsto g l (𝓝 b)) (hy : b ≠ 0) : Tendsto (f / g) l (𝓝 (a / b)) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} {l : Filter α} {a b : G₀} (hf : Filter.Tendsto f l (𝓝 a)) (hg : Filter.Tendsto g l (𝓝 b)) (hy : b ≠ 0)
Goal: Filter.Tendsto (f / g) l (𝓝 (a / b))
-- Filter.tendsto_mul_iff_of_ne_zero
theorem Filter.tendsto_mul_iff_of_ne_zero [T1Space G₀] {f g : α → G₀} {l : Filter α} {x y : G₀}
    (hg : Tendsto g l (𝓝 y)) (hy : y ≠ 0) :
    Tendsto (fun n => f n * g n) l (𝓝 <| x * y) ↔ Tendsto f l (𝓝 x) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] [T1Space G₀] {f g : α → G₀} {l : Filter α} {x y : G₀} (hg : Filter.Tendsto g l (𝓝 y)) (hy : y ≠ 0)
Goal: Filter.Tendsto (fun n => f n * g n) l (𝓝 (x * y)) ↔ Filter.Tendsto f l (𝓝 x)
-- ContinuousWithinAt.div
nonrec theorem ContinuousWithinAt.div (hf : ContinuousWithinAt f s a)
    (hg : ContinuousWithinAt g s a) (h₀ : g a ≠ 0) : ContinuousWithinAt (f / g) s a 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] {s : Set α} {a : α} (hf : ContinuousWithinAt f s a) (hg : ContinuousWithinAt g s a) (h₀ : g a ≠ 0)
Goal: ContinuousWithinAt (f / g) s a
-- ContinuousOn.div
theorem ContinuousOn.div (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h₀ : ∀ x ∈ s, g x ≠ 0) :
    ContinuousOn (f / g) s 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h₀ : ∀ x ∈ s, g x ≠ 0)
Goal: ContinuousOn (f / g) s
-- ContinuousAt.div
nonrec theorem ContinuousAt.div (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0) :
    ContinuousAt (f / g) a 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] {a : α} (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0)
Goal: ContinuousAt (f / g) a
-- Continuous.div
@[continuity]
theorem Continuous.div (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) :
    Continuous (f / g) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ (x : α), g x ≠ 0)
Goal: Continuous (f / g)
-- continuousOn_div
theorem continuousOn_div : ContinuousOn (fun p : G₀ × G₀ => p.1 / p.2) { p | p.2 ≠ 0 } 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀]
Goal: ContinuousOn (fun p => p.1 / p.2) {p | p.2 ≠ 0}
-- Continuous.div₀
@[fun_prop]
theorem Continuous.div₀ (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) :
    Continuous (fun x => f x / g x) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ (x : α), g x ≠ 0)
Goal: Continuous fun x => f x / g x
-- ContinuousAt.div₀
@[fun_prop]
theorem ContinuousAt.div₀ (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0) :
    ContinuousAt (fun x => f x / g x) a 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] {a : α} (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0)
Goal: ContinuousAt (fun x => f x / g x) a
-- ContinuousOn.div₀
@[fun_prop]
theorem ContinuousOn.div₀ (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h₀ : ∀ x ∈ s, g x ≠ 0) :
    ContinuousOn (fun x => f x / g x) s 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h₀ : ∀ x ∈ s, g x ≠ 0)
Goal: ContinuousOn (fun x => f x / g x) s
-- ContinuousAt.comp_div_cases
theorem ContinuousAt.comp_div_cases {f g : α → G₀} (h : α → G₀ → β) (hf : ContinuousAt f a)
    (hg : ContinuousAt g a) (hh : g a ≠ 0 → ContinuousAt (↿h) (a, f a / g a))
    (h2h : g a = 0 → Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))) :
    ContinuousAt (fun x => h x (f x / g x)) a 
Arguments: {α : Type u_1} {β : Type u_2} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] [TopologicalSpace α] [TopologicalSpace β] {a : α} {f g : α → G₀} (h : α → G₀ → β) (hf : ContinuousAt f a) (hg : ContinuousAt g a) (hh : g a ≠ 0 → ContinuousAt (↿h) (a, f a / g a)) (h2h : g a = 0 → Filter.Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0)))
Goal: ContinuousAt (fun x => h x (f x / g x)) a
-- Continuous.comp_div_cases
theorem Continuous.comp_div_cases {f g : α → G₀} (h : α → G₀ → β) (hf : Continuous f)
    (hg : Continuous g) (hh : ∀ a, g a ≠ 0 → ContinuousAt (↿h) (a, f a / g a))
    (h2h : ∀ a, g a = 0 → Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))) :
    Continuous fun x => h x (f x / g x) 
Arguments: {α : Type u_1} {β : Type u_2} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] [TopologicalSpace α] [TopologicalSpace β] {f g : α → G₀} (h : α → G₀ → β) (hf : Continuous f) (hg : Continuous g) (hh : ∀ (a : α), g a ≠ 0 → ContinuousAt (↿h) (a, f a / g a)) (h2h : ∀ (a : α), g a = 0 → Filter.Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0)))
Goal: Continuous fun x => h x (f x / g x)
-- Homeomorph.coe_mulLeft₀
@[simp]
theorem coe_mulLeft₀ (c : α) (hc : c ≠ 0) : ⇑(Homeomorph.mulLeft₀ c hc) = (c * ·) 
Arguments: {α : Type u_1} [TopologicalSpace α] [GroupWithZero α] [ContinuousMul α] (c : α) (hc : c ≠ 0)
Goal: ⇑(Homeomorph.mulLeft₀ c hc) = fun x => c * x
-- Homeomorph.mulLeft₀_symm_apply
@[simp]
theorem mulLeft₀_symm_apply (c : α) (hc : c ≠ 0) :
    ((Homeomorph.mulLeft₀ c hc).symm : α → α) = (c⁻¹ * ·) 
Arguments: {α : Type u_1} [TopologicalSpace α] [GroupWithZero α] [ContinuousMul α] (c : α) (hc : c ≠ 0)
Goal: ⇑(Homeomorph.mulLeft₀ c hc).symm = fun x => c⁻¹ * x
-- Homeomorph.coe_mulRight₀
@[simp]
theorem coe_mulRight₀ (c : α) (hc : c ≠ 0) : ⇑(Homeomorph.mulRight₀ c hc) = (· * c) 
Arguments: {α : Type u_1} [TopologicalSpace α] [GroupWithZero α] [ContinuousMul α] (c : α) (hc : c ≠ 0)
Goal: ⇑(Homeomorph.mulRight₀ c hc) = fun x => x * c
-- Homeomorph.mulRight₀_symm_apply
@[simp]
theorem mulRight₀_symm_apply (c : α) (hc : c ≠ 0) :
    ((Homeomorph.mulRight₀ c hc).symm : α → α) = (· * c⁻¹) 
Arguments: {α : Type u_1} [TopologicalSpace α] [GroupWithZero α] [ContinuousMul α] (c : α) (hc : c ≠ 0)
Goal: ⇑(Homeomorph.mulRight₀ c hc).symm = fun x => x * c⁻¹
-- map_mul_left_nhds₀
theorem map_mul_left_nhds₀ (ha : a ≠ 0) (b : G₀) : map (a * ·) (𝓝 b) = 𝓝 (a * b) 
Arguments: {G₀ : Type u_3} [TopologicalSpace G₀] [GroupWithZero G₀] [ContinuousMul G₀] {a : G₀} (ha : a ≠ 0) (b : G₀)
Goal: Filter.map (fun x => a * x) (𝓝 b) = 𝓝 (a * b)
-- map_mul_left_nhds_one₀
theorem map_mul_left_nhds_one₀ (ha : a ≠ 0) : map (a * ·) (𝓝 1) = 𝓝 (a) 
Arguments: {G₀ : Type u_3} [TopologicalSpace G₀] [GroupWithZero G₀] [ContinuousMul G₀] {a : G₀} (ha : a ≠ 0)
Goal: Filter.map (fun x => a * x) (𝓝 1) = 𝓝 a
-- map_mul_right_nhds₀
theorem map_mul_right_nhds₀ (ha : a ≠ 0) (b : G₀) : map (· * a) (𝓝 b) = 𝓝 (b * a) 
Arguments: {G₀ : Type u_3} [TopologicalSpace G₀] [GroupWithZero G₀] [ContinuousMul G₀] {a : G₀} (ha : a ≠ 0) (b : G₀)
Goal: Filter.map (fun x => x * a) (𝓝 b) = 𝓝 (b * a)
-- map_mul_right_nhds_one₀
theorem map_mul_right_nhds_one₀ (ha : a ≠ 0) : map (· * a) (𝓝 1) = 𝓝 (a) 
Arguments: {G₀ : Type u_3} [TopologicalSpace G₀] [GroupWithZero G₀] [ContinuousMul G₀] {a : G₀} (ha : a ≠ 0)
Goal: Filter.map (fun x => x * a) (𝓝 1) = 𝓝 a
-- nhds_translation_mul_inv₀
theorem nhds_translation_mul_inv₀ (ha : a ≠ 0) : comap (· * a⁻¹) (𝓝 1) = 𝓝 a 
Arguments: {G₀ : Type u_3} [TopologicalSpace G₀] [GroupWithZero G₀] [ContinuousMul G₀] {a : G₀} (ha : a ≠ 0)
Goal: Filter.comap (fun x => x * a⁻¹) (𝓝 1) = 𝓝 a
-- HasContinuousInv₀.of_nhds_one
theorem HasContinuousInv₀.of_nhds_one (h : Tendsto Inv.inv (𝓝 (1 : G₀)) (𝓝 1)) :
    HasContinuousInv₀ G₀ 
Arguments: {G₀ : Type u_3} [TopologicalSpace G₀] [GroupWithZero G₀] [ContinuousMul G₀] (h : Filter.Tendsto Inv.inv (𝓝 1) (𝓝 1))
Goal: HasContinuousInv₀ G₀
-- continuousAt_zpow₀
theorem continuousAt_zpow₀ (x : G₀) (m : ℤ) (h : x ≠ 0 ∨ 0 ≤ m) :
    ContinuousAt (fun x => x ^ m) x 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] (x : G₀) (m : ℤ) (h : x ≠ 0 ∨ 0 ≤ m)
Goal: ContinuousAt (fun x => x ^ m) x
-- continuousOn_zpow₀
theorem continuousOn_zpow₀ (m : ℤ) : ContinuousOn (fun x : G₀ => x ^ m) {0}ᶜ 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] (m : ℤ)
Goal: ContinuousOn (fun x => x ^ m) {0}ᶜ
-- Filter.Tendsto.zpow₀
theorem Filter.Tendsto.zpow₀ {f : α → G₀} {l : Filter α} {a : G₀} (hf : Tendsto f l (𝓝 a)) (m : ℤ)
    (h : a ≠ 0 ∨ 0 ≤ m) : Tendsto (fun x => f x ^ m) l (𝓝 (a ^ m)) 
Arguments: {α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {f : α → G₀} {l : Filter α} {a : G₀} (hf : Filter.Tendsto f l (𝓝 a)) (m : ℤ) (h : a ≠ 0 ∨ 0 ≤ m)
Goal: Filter.Tendsto (fun x => f x ^ m) l (𝓝 (a ^ m))
-- ContinuousAt.zpow₀
@[fun_prop]
nonrec theorem ContinuousAt.zpow₀ (hf : ContinuousAt f a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m) :
    ContinuousAt (fun x => f x ^ m) a 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {X : Type u_4} [TopologicalSpace X] {a : X} {f : X → G₀} (hf : ContinuousAt f a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m)
Goal: ContinuousAt (fun x => f x ^ m) a
-- ContinuousWithinAt.zpow₀
nonrec theorem ContinuousWithinAt.zpow₀ (hf : ContinuousWithinAt f s a) (m : ℤ)
    (h : f a ≠ 0 ∨ 0 ≤ m) : ContinuousWithinAt (fun x => f x ^ m) s a 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {X : Type u_4} [TopologicalSpace X] {a : X} {s : Set X} {f : X → G₀} (hf : ContinuousWithinAt f s a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m)
Goal: ContinuousWithinAt (fun x => f x ^ m) s a
-- ContinuousOn.zpow₀
@[fun_prop]
theorem ContinuousOn.zpow₀ (hf : ContinuousOn f s) (m : ℤ) (h : ∀ a ∈ s, f a ≠ 0 ∨ 0 ≤ m) :
    ContinuousOn (fun x => f x ^ m) s 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {X : Type u_4} [TopologicalSpace X] {s : Set X} {f : X → G₀} (hf : ContinuousOn f s) (m : ℤ) (h : ∀ a ∈ s, f a ≠ 0 ∨ 0 ≤ m)
Goal: ContinuousOn (fun x => f x ^ m) s
-- Continuous.zpow₀
@[continuity, fun_prop]
theorem Continuous.zpow₀ (hf : Continuous f) (m : ℤ) (h0 : ∀ a, f a ≠ 0 ∨ 0 ≤ m) :
    Continuous fun x => f x ^ m 
Arguments: {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [HasContinuousInv₀ G₀] [ContinuousMul G₀] {X : Type u_4} [TopologicalSpace X] {f : X → G₀} (hf : Continuous f) (m : ℤ) (h0 : ∀ (a : X), f a ≠ 0 ∨ 0 ≤ m)
Goal: Continuous fun x => f x ^ m
-- AccPt.map
theorem AccPt.map {β : Type*} [TopologicalSpace β] {F : Filter X} {x : X}
    (h : AccPt x F) {f : X → β} (hf1 : ContinuousAt f x) (hf2 : Function.Injective f) :
    AccPt (f x) (map f F) 
Arguments: {X : Type u_1} [TopologicalSpace X] {β : Type u_2} [TopologicalSpace β] {F : Filter X} {x : X} (h : AccPt x F) {f : X → β} (hf1 : ContinuousAt f x) (hf2 : Function.Injective f)
Goal: AccPt (f x) (Filter.map f F)
-- mem_derivedSet
@[simp]
lemma mem_derivedSet {A : Set X} {x : X} : x ∈ derivedSet A ↔ AccPt x (𝓟 A) 
Arguments: {X : Type u_1} [TopologicalSpace X] {A : Set X} {x : X}
Goal: x ∈ derivedSet A ↔ AccPt x (𝓟 A)
-- derivedSet_union
lemma derivedSet_union (A B : Set X) : derivedSet (A ∪ B) = derivedSet A ∪ derivedSet B 
Arguments: {X : Type u_1} [TopologicalSpace X] (A B : Set X)
Goal: derivedSet (A ∪ B) = derivedSet A ∪ derivedSet B
-- derivedSet_mono
lemma derivedSet_mono (A B : Set X) (h : A ⊆ B) : derivedSet A ⊆ derivedSet B 
Arguments: {X : Type u_1} [TopologicalSpace X] (A B : Set X) (h : A ⊆ B)
Goal: derivedSet A ⊆ derivedSet B
-- Continuous.image_derivedSet
theorem Continuous.image_derivedSet {β : Type*} [TopologicalSpace β] {A : Set X} {f : X → β}
    (hf1 : Continuous f) (hf2 : Function.Injective f) :
    f '' derivedSet A ⊆ derivedSet (f '' A) 
Arguments: {X : Type u_1} [TopologicalSpace X] {β : Type u_2} [TopologicalSpace β] {A : Set X} {f : X → β} (hf1 : Continuous f) (hf2 : Function.Injective f)
Goal: f '' derivedSet A ⊆ derivedSet (f '' A)
-- derivedSet_subset_closure
lemma derivedSet_subset_closure (A : Set X) : derivedSet A ⊆ closure A 
Arguments: {X : Type u_1} [TopologicalSpace X] (A : Set X)
Goal: derivedSet A ⊆ closure A
-- isClosed_iff_derivedSet_subset
lemma isClosed_iff_derivedSet_subset (A : Set X) : IsClosed A ↔ derivedSet A ⊆ A 
Arguments: {X : Type u_1} [TopologicalSpace X] (A : Set X)
Goal: IsClosed A ↔ derivedSet A ⊆ A
-- derivedSet_closure
lemma derivedSet_closure [T1Space X] (A : Set X) : derivedSet (closure A) = derivedSet A 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (A : Set X)
Goal: derivedSet (closure A) = derivedSet A
-- isClosed_derivedSet
@[simp]
lemma isClosed_derivedSet [T1Space X] (A : Set X) : IsClosed (derivedSet A) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (A : Set X)
Goal: IsClosed (derivedSet A)
-- preperfect_iff_subset_derivedSet
lemma preperfect_iff_subset_derivedSet {U : Set X} : Preperfect U ↔ U ⊆ derivedSet U 
Arguments: {X : Type u_1} [TopologicalSpace X] {U : Set X}
Goal: Preperfect U ↔ U ⊆ derivedSet U
-- perfect_iff_eq_derivedSet
lemma perfect_iff_eq_derivedSet {U : Set X} : Perfect U ↔ U = derivedSet U 
Arguments: {X : Type u_1} [TopologicalSpace X] {U : Set X}
Goal: Perfect U ↔ U = derivedSet U
-- IsPreconnected.inter_derivedSet_nonempty
lemma IsPreconnected.inter_derivedSet_nonempty [T1Space X] {U : Set X} (hs : IsPreconnected U)
    (a b : Set X) (h : U ⊆ a ∪ b) (ha : (U ∩ derivedSet a).Nonempty)
    (hb : (U ∩ derivedSet b).Nonempty) : (U ∩ (derivedSet a ∩ derivedSet b)).Nonempty 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {U : Set X} (hs : IsPreconnected U) (a b : Set X) (h : U ⊆ a ∪ b) (ha : (U ∩ derivedSet a).Nonempty) (hb : (U ∩ derivedSet b).Nonempty)
Goal: (U ∩ (derivedSet a ∩ derivedSet b)).Nonempty
-- vadd_singleton_mem_nhds_of_sigmaCompact
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x 
Arguments: {G : Type u_1} {X : Type u_2} [TopologicalSpace G] [TopologicalSpace X] [AddGroup G] [TopologicalAddGroup G] [AddAction G X] [SigmaCompactSpace G] [BaireSpace X] [T2Space X] [ContinuousVAdd G X] [AddAction.IsPretransitive G X] {U : Set G} (hU : U ∈ 𝓝 0) (x : X)
Goal: U +ᵥ {x} ∈ 𝓝 x
-- smul_singleton_mem_nhds_of_sigmaCompact
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x 
Arguments: {G : Type u_1} {X : Type u_2} [TopologicalSpace G] [TopologicalSpace X] [Group G] [TopologicalGroup G] [MulAction G X] [SigmaCompactSpace G] [BaireSpace X] [T2Space X] [ContinuousSMul G X] [MulAction.IsPretransitive G X] {U : Set G} (hU : U ∈ 𝓝 1) (x : X)
Goal: U • {x} ∈ 𝓝 x
-- isOpenMap_vadd_of_sigmaCompact
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open. This is a version of the open mapping theorem, valid
notably for the action of a sigma-compact locally compact group on a locally compact space."]
theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) 
Arguments: {G : Type u_1} {X : Type u_2} [TopologicalSpace G] [TopologicalSpace X] [AddGroup G] [TopologicalAddGroup G] [AddAction G X] [SigmaCompactSpace G] [BaireSpace X] [T2Space X] [ContinuousVAdd G X] [AddAction.IsPretransitive G X] (x : X)
Goal: IsOpenMap fun g => g +ᵥ x
-- isOpenMap_smul_of_sigmaCompact
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open. This is a version of the open mapping theorem, valid
notably for the action of a sigma-compact locally compact group on a locally compact space."]
theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) 
Arguments: {G : Type u_1} {X : Type u_2} [TopologicalSpace G] [TopologicalSpace X] [Group G] [TopologicalGroup G] [MulAction G X] [SigmaCompactSpace G] [BaireSpace X] [T2Space X] [ContinuousSMul G X] [MulAction.IsPretransitive G X] (x : X)
Goal: IsOpenMap fun g => g • x
-- MonoidHom.isOpenMap_of_sigmaCompact
@[to_additive]
theorem MonoidHom.isOpenMap_of_sigmaCompact
    {H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
    (f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
    IsOpenMap f 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [SigmaCompactSpace G] {H : Type u_3} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H] (f : G →* H) (hf : Function.Surjective ⇑f) (h'f : Continuous ⇑f)
Goal: IsOpenMap ⇑f
-- AddMonoidHom.isOpenMap_of_sigmaCompact
@[to_additive]
theorem MonoidHom.isOpenMap_of_sigmaCompact
    {H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
    (f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
    IsOpenMap f 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [SigmaCompactSpace G] {H : Type u_3} [AddGroup H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousAdd H] (f : G →+ H) (hf : Function.Surjective ⇑f) (h'f : Continuous ⇑f)
Goal: IsOpenMap ⇑f