ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.tendsto_const_div_iff
@[to_additive]
lemma Filter.tendsto_const_div_iff {G : Type*} [CommGroup G] [TopologicalSpace G] [ContinuousDiv G]
    (b : G) {c : G} {f : α → G} {l : Filter α} :
    Tendsto (fun k : α ↦ b / f k) l (𝓝 (b / c)) ↔ Tendsto f l (𝓝 c) 
Arguments: {α : Type u} {G : Type u_1} [CommGroup G] [TopologicalSpace G] [ContinuousDiv G] (b : G) {c : G} {f : α → G} {l : Filter α}
Goal: Filter.Tendsto (fun k => b / f k) l (𝓝 (b / c)) ↔ Filter.Tendsto f l (𝓝 c)
-- Filter.tendsto_const_sub_iff
@[to_additive]
lemma Filter.tendsto_const_div_iff {G : Type*} [CommGroup G] [TopologicalSpace G] [ContinuousDiv G]
    (b : G) {c : G} {f : α → G} {l : Filter α} :
    Tendsto (fun k : α ↦ b / f k) l (𝓝 (b / c)) ↔ Tendsto f l (𝓝 c) 
Arguments: {α : Type u} {G : Type u_1} [AddCommGroup G] [TopologicalSpace G] [ContinuousSub G] (b : G) {c : G} {f : α → G} {l : Filter α}
Goal: Filter.Tendsto (fun k => b - f k) l (𝓝 (b - c)) ↔ Filter.Tendsto f l (𝓝 c)
-- Filter.Tendsto.div_const'
@[to_additive sub_const]
theorem Filter.Tendsto.div_const' {c : G} {f : α → G} {l : Filter α} (h : Tendsto f l (𝓝 c))
    (b : G) : Tendsto (f · / b) l (𝓝 (c / b)) 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Div G] [ContinuousDiv G] {c : G} {f : α → G} {l : Filter α} (h : Filter.Tendsto f l (𝓝 c)) (b : G)
Goal: Filter.Tendsto (fun x => f x / b) l (𝓝 (c / b))
-- Filter.Tendsto.sub_const
@[to_additive sub_const]
theorem Filter.Tendsto.div_const' {c : G} {f : α → G} {l : Filter α} (h : Tendsto f l (𝓝 c))
    (b : G) : Tendsto (f · / b) l (𝓝 (c / b)) 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Sub G] [ContinuousSub G] {c : G} {f : α → G} {l : Filter α} (h : Filter.Tendsto f l (𝓝 c)) (b : G)
Goal: Filter.Tendsto (fun x => f x - b) l (𝓝 (c - b))
-- Filter.tendsto_div_const_iff
lemma Filter.tendsto_div_const_iff {G : Type*}
    [CommGroupWithZero G] [TopologicalSpace G] [ContinuousDiv G]
    {b : G} (hb : b ≠ 0) {c : G} {f : α → G} {l : Filter α} :
    Tendsto (f · / b) l (𝓝 (c / b)) ↔ Tendsto f l (𝓝 c) 
Arguments: {α : Type u} {G : Type u_1} [CommGroupWithZero G] [TopologicalSpace G] [ContinuousDiv G] {b : G} (hb : b ≠ 0) {c : G} {f : α → G} {l : Filter α}
Goal: Filter.Tendsto (fun x => f x / b) l (𝓝 (c / b)) ↔ Filter.Tendsto f l (𝓝 c)
-- Filter.tendsto_sub_const_iff
lemma Filter.tendsto_sub_const_iff {G : Type*}
    [AddCommGroup G] [TopologicalSpace G] [ContinuousSub G]
    (b : G) {c : G} {f : α → G} {l : Filter α} :
    Tendsto (f · - b) l (𝓝 (c - b)) ↔ Tendsto f l (𝓝 c) 
Arguments: {α : Type u} {G : Type u_1} [AddCommGroup G] [TopologicalSpace G] [ContinuousSub G] (b : G) {c : G} {f : α → G} {l : Filter α}
Goal: Filter.Tendsto (fun x => f x - b) l (𝓝 (c - b)) ↔ Filter.Tendsto f l (𝓝 c)
-- Continuous.div'
@[to_additive (attr := continuity, fun_prop) sub]
theorem Continuous.div' (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x / g x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Div G] [ContinuousDiv G] [TopologicalSpace α] {f g : α → G} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x / g x
-- Continuous.sub
@[to_additive (attr := continuity, fun_prop) sub]
theorem Continuous.div' (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x / g x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Sub G] [ContinuousSub G] [TopologicalSpace α] {f g : α → G} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x - g x
-- continuous_sub_left
@[to_additive (attr := continuity) continuous_sub_left]
lemma continuous_div_left' (a : G) : Continuous (a / ·) 
Arguments: {G : Type w} [TopologicalSpace G] [Sub G] [ContinuousSub G] (a : G)
Goal: Continuous fun x => a - x
-- continuous_div_left'
@[to_additive (attr := continuity) continuous_sub_left]
lemma continuous_div_left' (a : G) : Continuous (a / ·) 
Arguments: {G : Type w} [TopologicalSpace G] [Div G] [ContinuousDiv G] (a : G)
Goal: Continuous fun x => a / x
-- continuous_div_right'
@[to_additive (attr := continuity) continuous_sub_right]
lemma continuous_div_right' (a : G) : Continuous (· / a) 
Arguments: {G : Type w} [TopologicalSpace G] [Div G] [ContinuousDiv G] (a : G)
Goal: Continuous fun x => x / a
-- continuous_sub_right
@[to_additive (attr := continuity) continuous_sub_right]
lemma continuous_div_right' (a : G) : Continuous (· / a) 
Arguments: {G : Type w} [TopologicalSpace G] [Sub G] [ContinuousSub G] (a : G)
Goal: Continuous fun x => x - a
-- ContinuousAt.sub
@[to_additive (attr := fun_prop) sub]
theorem ContinuousAt.div' {f g : α → G} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (fun x => f x / g x) x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Sub G] [ContinuousSub G] [TopologicalSpace α] {f g : α → G} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (fun x => f x - g x) x
-- ContinuousAt.div'
@[to_additive (attr := fun_prop) sub]
theorem ContinuousAt.div' {f g : α → G} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (fun x => f x / g x) x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Div G] [ContinuousDiv G] [TopologicalSpace α] {f g : α → G} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (fun x => f x / g x) x
-- ContinuousWithinAt.sub
@[to_additive sub]
theorem ContinuousWithinAt.div' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun x => f x / g x) s x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Sub G] [ContinuousSub G] [TopologicalSpace α] {f g : α → G} {s : Set α} {x : α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (fun x => f x - g x) s x
-- ContinuousWithinAt.div'
@[to_additive sub]
theorem ContinuousWithinAt.div' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun x => f x / g x) s x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Div G] [ContinuousDiv G] [TopologicalSpace α] {f g : α → G} {s : Set α} {x : α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (fun x => f x / g x) s x
-- ContinuousOn.sub
@[to_additive (attr := fun_prop) sub]
theorem ContinuousOn.div' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => f x / g x) s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Sub G] [ContinuousSub G] [TopologicalSpace α] {f g : α → G} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => f x - g x) s
-- ContinuousOn.div'
@[to_additive (attr := fun_prop) sub]
theorem ContinuousOn.div' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => f x / g x) s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Div G] [ContinuousDiv G] [TopologicalSpace α] {f g : α → G} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => f x / g x) s
-- Homeomorph.subLeft_symm_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  " A version of `Homeomorph.addLeft a (-b)` that is defeq to `a - b`. "]
def Homeomorph.divLeft (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (x b : G)
Goal: (Homeomorph.subLeft x).symm b = -b + x
-- Homeomorph.subLeft_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  " A version of `Homeomorph.addLeft a (-b)` that is defeq to `a - b`. "]
def Homeomorph.divLeft (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (x b : G)
Goal: (Homeomorph.subLeft x) b = x - b
-- Homeomorph.divLeft_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  " A version of `Homeomorph.addLeft a (-b)` that is defeq to `a - b`. "]
def Homeomorph.divLeft (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (x b : G)
Goal: (Homeomorph.divLeft x) b = x / b
-- Homeomorph.divLeft_symm_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  " A version of `Homeomorph.addLeft a (-b)` that is defeq to `a - b`. "]
def Homeomorph.divLeft (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (x b : G)
Goal: (Homeomorph.divLeft x).symm b = b⁻¹ * x
-- isOpenMap_sub_left
@[to_additive]
theorem isOpenMap_div_left (a : G) : IsOpenMap (a / ·) 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (a : G)
Goal: IsOpenMap fun x => a - x
-- isOpenMap_div_left
@[to_additive]
theorem isOpenMap_div_left (a : G) : IsOpenMap (a / ·) 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (a : G)
Goal: IsOpenMap fun x => a / x
-- isClosedMap_sub_left
@[to_additive]
theorem isClosedMap_div_left (a : G) : IsClosedMap (a / ·) 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (a : G)
Goal: IsClosedMap fun x => a - x
-- isClosedMap_div_left
@[to_additive]
theorem isClosedMap_div_left (a : G) : IsClosedMap (a / ·) 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (a : G)
Goal: IsClosedMap fun x => a / x
-- Homeomorph.divRight_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  "A version of `Homeomorph.addRight (-a) b` that is defeq to `b - a`. "]
def Homeomorph.divRight (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (x b : G)
Goal: (Homeomorph.divRight x) b = b / x
-- Homeomorph.divRight_symm_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  "A version of `Homeomorph.addRight (-a) b` that is defeq to `b - a`. "]
def Homeomorph.divRight (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (x b : G)
Goal: (Homeomorph.divRight x).symm b = b * x
-- Homeomorph.subRight_symm_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  "A version of `Homeomorph.addRight (-a) b` that is defeq to `b - a`. "]
def Homeomorph.divRight (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (x b : G)
Goal: (Homeomorph.subRight x).symm b = b + x
-- Homeomorph.subRight_apply
@[to_additive (attr := simps! (config := { simpRhs := true }))
  "A version of `Homeomorph.addRight (-a) b` that is defeq to `b - a`. "]
def Homeomorph.divRight (x : G) : G ≃ₜ G 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (x b : G)
Goal: (Homeomorph.subRight x) b = b - x
-- isOpenMap_div_right
@[to_additive]
lemma isOpenMap_div_right (a : G) : IsOpenMap (· / a) 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (a : G)
Goal: IsOpenMap fun x => x / a
-- isOpenMap_sub_right
@[to_additive]
lemma isOpenMap_div_right (a : G) : IsOpenMap (· / a) 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (a : G)
Goal: IsOpenMap fun x => x - a
-- isClosedMap_sub_right
@[to_additive]
lemma isClosedMap_div_right (a : G) : IsClosedMap (· / a) 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (a : G)
Goal: IsClosedMap fun x => x - a
-- isClosedMap_div_right
@[to_additive]
lemma isClosedMap_div_right (a : G) : IsClosedMap (· / a) 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (a : G)
Goal: IsClosedMap fun x => x / a
-- tendsto_sub_nhds_zero_iff
@[to_additive]
theorem tendsto_div_nhds_one_iff {α : Type*} {l : Filter α} {x : G} {u : α → G} :
    Tendsto (u · / x) l (𝓝 1) ↔ Tendsto u l (𝓝 x) 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {α : Type u_1} {l : Filter α} {x : G} {u : α → G}
Goal: Filter.Tendsto (fun x_1 => u x_1 - x) l (𝓝 0) ↔ Filter.Tendsto u l (𝓝 x)
-- tendsto_div_nhds_one_iff
@[to_additive]
theorem tendsto_div_nhds_one_iff {α : Type*} {l : Filter α} {x : G} {u : α → G} :
    Tendsto (u · / x) l (𝓝 1) ↔ Tendsto u l (𝓝 x) 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] {α : Type u_1} {l : Filter α} {x : G} {u : α → G}
Goal: Filter.Tendsto (fun x_1 => u x_1 / x) l (𝓝 1) ↔ Filter.Tendsto u l (𝓝 x)
-- nhds_translation_div
@[to_additive]
theorem nhds_translation_div (x : G) : comap (· / x) (𝓝 1) = 𝓝 x 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [TopologicalGroup G] (x : G)
Goal: Filter.comap (fun x_1 => x_1 / x) (𝓝 1) = 𝓝 x
-- nhds_translation_sub
@[to_additive]
theorem nhds_translation_div (x : G) : comap (· / x) (𝓝 1) = 𝓝 x 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (x : G)
Goal: Filter.comap (fun x_1 => x_1 - x) (𝓝 0) = 𝓝 x
-- IsOpen.vadd_left
@[to_additive]
theorem IsOpen.smul_left (ht : IsOpen t) : IsOpen (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [AddGroup α] [AddAction α β] [ContinuousConstVAdd α β] {s : Set α} {t : Set β} (ht : IsOpen t)
Goal: IsOpen (s +ᵥ t)
-- IsOpen.smul_left
@[to_additive]
theorem IsOpen.smul_left (ht : IsOpen t) : IsOpen (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousConstSMul α β] {s : Set α} {t : Set β} (ht : IsOpen t)
Goal: IsOpen (s • t)
-- subset_interior_vadd_right
@[to_additive]
theorem subset_interior_smul_right : s • interior t ⊆ interior (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [AddGroup α] [AddAction α β] [ContinuousConstVAdd α β] {s : Set α} {t : Set β}
Goal: s +ᵥ interior t ⊆ interior (s +ᵥ t)
-- subset_interior_smul_right
@[to_additive]
theorem subset_interior_smul_right : s • interior t ⊆ interior (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousConstSMul α β] {s : Set α} {t : Set β}
Goal: s • interior t ⊆ interior (s • t)
-- vadd_mem_nhds
@[to_additive]
theorem smul_mem_nhds (a : α) {x : β} (ht : t ∈ 𝓝 x) : a • t ∈ 𝓝 (a • x) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [AddGroup α] [AddAction α β] [ContinuousConstVAdd α β] {t : Set β} (a : α) {x : β} (ht : t ∈ 𝓝 x)
Goal: a +ᵥ t ∈ 𝓝 (a +ᵥ x)
-- smul_mem_nhds
@[to_additive]
theorem smul_mem_nhds (a : α) {x : β} (ht : t ∈ 𝓝 x) : a • t ∈ 𝓝 (a • x) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousConstSMul α β] {t : Set β} (a : α) {x : β} (ht : t ∈ 𝓝 x)
Goal: a • t ∈ 𝓝 (a • x)
-- subset_interior_vadd
@[to_additive]
theorem subset_interior_smul : interior s • interior t ⊆ interior (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [AddGroup α] [AddAction α β] [ContinuousConstVAdd α β] {s : Set α} {t : Set β} [TopologicalSpace α]
Goal: interior s +ᵥ interior t ⊆ interior (s +ᵥ t)
-- subset_interior_smul
@[to_additive]
theorem subset_interior_smul : interior s • interior t ⊆ interior (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousConstSMul α β] {s : Set α} {t : Set β} [TopologicalSpace α]
Goal: interior s • interior t ⊆ interior (s • t)
-- IsClosed.vadd_left_of_isCompact
@[to_additive]
theorem IsClosed.smul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
    IsClosed (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddGroup α] [AddAction α β] [ContinuousNeg α] [ContinuousVAdd α β] {s : Set α} {t : Set β} (ht : IsClosed t) (hs : IsCompact s)
Goal: IsClosed (s +ᵥ t)
-- IsClosed.smul_left_of_isCompact
@[to_additive]
theorem IsClosed.smul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
    IsClosed (s • t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousInv α] [ContinuousSMul α β] {s : Set α} {t : Set β} (ht : IsClosed t) (hs : IsCompact s)
Goal: IsClosed (s • t)
-- AddAction.isClosedMap_quotient
@[to_additive]
theorem MulAction.isClosedMap_quotient [CompactSpace α] :
    letI := orbitRel α β
    IsClosedMap (Quotient.mk' : β → Quotient (orbitRel α β)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddGroup α] [AddAction α β] [ContinuousNeg α] [ContinuousVAdd α β] [CompactSpace α]
Goal: IsClosedMap Quotient.mk'
-- MulAction.isClosedMap_quotient
@[to_additive]
theorem MulAction.isClosedMap_quotient [CompactSpace α] :
    letI := orbitRel α β
    IsClosedMap (Quotient.mk' : β → Quotient (orbitRel α β)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousInv α] [ContinuousSMul α β] [CompactSpace α]
Goal: IsClosedMap Quotient.mk'
-- IsOpen.add_left
@[to_additive]
theorem IsOpen.mul_left : IsOpen t → IsOpen (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd α α] {s t : Set α}
Goal: IsOpen t → IsOpen (s + t)
-- IsOpen.mul_left
@[to_additive]
theorem IsOpen.mul_left : IsOpen t → IsOpen (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul α α] {s t : Set α}
Goal: IsOpen t → IsOpen (s * t)
-- subset_interior_add_right
@[to_additive]
theorem subset_interior_mul_right : s * interior t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd α α] {s t : Set α}
Goal: s + interior t ⊆ interior (s + t)
-- subset_interior_mul_right
@[to_additive]
theorem subset_interior_mul_right : s * interior t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul α α] {s t : Set α}
Goal: s * interior t ⊆ interior (s * t)
-- subset_interior_add
@[to_additive]
theorem subset_interior_mul : interior s * interior t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd α α] {s t : Set α}
Goal: interior s + interior t ⊆ interior (s + t)
-- subset_interior_mul
@[to_additive]
theorem subset_interior_mul : interior s * interior t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul α α] {s t : Set α}
Goal: interior s * interior t ⊆ interior (s * t)
-- singleton_mul_mem_nhds
@[to_additive]
theorem singleton_mul_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : {a} * s ∈ 𝓝 (a * b) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul α α] {s : Set α} (a : α) {b : α} (h : s ∈ 𝓝 b)
Goal: {a} * s ∈ 𝓝 (a * b)
-- singleton_add_mem_nhds
@[to_additive]
theorem singleton_mul_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : {a} * s ∈ 𝓝 (a * b) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd α α] {s : Set α} (a : α) {b : α} (h : s ∈ 𝓝 b)
Goal: {a} + s ∈ 𝓝 (a + b)
-- singleton_mul_mem_nhds_of_nhds_one
@[to_additive]
theorem singleton_mul_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : {a} * s ∈ 𝓝 a 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul α α] {s : Set α} (a : α) (h : s ∈ 𝓝 1)
Goal: {a} * s ∈ 𝓝 a
-- singleton_add_mem_nhds_of_nhds_zero
@[to_additive]
theorem singleton_mul_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : {a} * s ∈ 𝓝 a 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd α α] {s : Set α} (a : α) (h : s ∈ 𝓝 0)
Goal: {a} + s ∈ 𝓝 a
-- IsOpen.add_right
@[to_additive]
theorem IsOpen.mul_right (hs : IsOpen s) : IsOpen (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd αᵃᵒᵖ α] {s t : Set α} (hs : IsOpen s)
Goal: IsOpen (s + t)
-- IsOpen.mul_right
@[to_additive]
theorem IsOpen.mul_right (hs : IsOpen s) : IsOpen (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul αᵐᵒᵖ α] {s t : Set α} (hs : IsOpen s)
Goal: IsOpen (s * t)
-- subset_interior_add_left
@[to_additive]
theorem subset_interior_mul_left : interior s * t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd αᵃᵒᵖ α] {s t : Set α}
Goal: interior s + t ⊆ interior (s + t)
-- subset_interior_mul_left
@[to_additive]
theorem subset_interior_mul_left : interior s * t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul αᵐᵒᵖ α] {s t : Set α}
Goal: interior s * t ⊆ interior (s * t)
-- subset_interior_add'
@[to_additive]
theorem subset_interior_mul' : interior s * interior t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd αᵃᵒᵖ α] {s t : Set α}
Goal: interior s + interior t ⊆ interior (s + t)
-- subset_interior_mul'
@[to_additive]
theorem subset_interior_mul' : interior s * interior t ⊆ interior (s * t) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul αᵐᵒᵖ α] {s t : Set α}
Goal: interior s * interior t ⊆ interior (s * t)
-- mul_singleton_mem_nhds
@[to_additive]
theorem mul_singleton_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : s * {a} ∈ 𝓝 (b * a) 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul αᵐᵒᵖ α] {s : Set α} (a : α) {b : α} (h : s ∈ 𝓝 b)
Goal: s * {a} ∈ 𝓝 (b * a)
-- add_singleton_mem_nhds
@[to_additive]
theorem mul_singleton_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : s * {a} ∈ 𝓝 (b * a) 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd αᵃᵒᵖ α] {s : Set α} (a : α) {b : α} (h : s ∈ 𝓝 b)
Goal: s + {a} ∈ 𝓝 (b + a)
-- add_singleton_mem_nhds_of_nhds_zero
@[to_additive]
theorem mul_singleton_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : s * {a} ∈ 𝓝 a 
Arguments: {α : Type u} [TopologicalSpace α] [AddGroup α] [ContinuousConstVAdd αᵃᵒᵖ α] {s : Set α} (a : α) (h : s ∈ 𝓝 0)
Goal: s + {a} ∈ 𝓝 a
-- mul_singleton_mem_nhds_of_nhds_one
@[to_additive]
theorem mul_singleton_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : s * {a} ∈ 𝓝 a 
Arguments: {α : Type u} [TopologicalSpace α] [Group α] [ContinuousConstSMul αᵐᵒᵖ α] {s : Set α} (a : α) (h : s ∈ 𝓝 1)
Goal: s * {a} ∈ 𝓝 a
-- IsOpen.sub_left
@[to_additive]
theorem IsOpen.div_left (ht : IsOpen t) : IsOpen (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s t : Set G} (ht : IsOpen t)
Goal: IsOpen (s - t)
-- IsOpen.div_left
@[to_additive]
theorem IsOpen.div_left (ht : IsOpen t) : IsOpen (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G} (ht : IsOpen t)
Goal: IsOpen (s / t)
-- IsOpen.sub_right
@[to_additive]
theorem IsOpen.div_right (hs : IsOpen s) : IsOpen (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s t : Set G} (hs : IsOpen s)
Goal: IsOpen (s - t)
-- IsOpen.div_right
@[to_additive]
theorem IsOpen.div_right (hs : IsOpen s) : IsOpen (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G} (hs : IsOpen s)
Goal: IsOpen (s / t)
-- subset_interior_sub_left
@[to_additive]
theorem subset_interior_div_left : interior s / t ⊆ interior (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s t : Set G}
Goal: interior s - t ⊆ interior (s - t)
-- subset_interior_div_left
@[to_additive]
theorem subset_interior_div_left : interior s / t ⊆ interior (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G}
Goal: interior s / t ⊆ interior (s / t)
-- subset_interior_sub_right
@[to_additive]
theorem subset_interior_div_right : s / interior t ⊆ interior (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s t : Set G}
Goal: s - interior t ⊆ interior (s - t)
-- subset_interior_div_right
@[to_additive]
theorem subset_interior_div_right : s / interior t ⊆ interior (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G}
Goal: s / interior t ⊆ interior (s / t)
-- subset_interior_sub
@[to_additive]
theorem subset_interior_div : interior s / interior t ⊆ interior (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] {s t : Set G}
Goal: interior s - interior t ⊆ interior (s - t)
-- subset_interior_div
@[to_additive]
theorem subset_interior_div : interior s / interior t ⊆ interior (s / t) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G}
Goal: interior s / interior t ⊆ interior (s / t)
-- IsOpen.mul_closure
@[to_additive]
theorem IsOpen.mul_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