ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- nhdsSet_prod_le
theorem nhdsSet_prod_le (s : Set X) (t : Set Y) : 𝓝ˢ (s ×ˢ t) ≤ 𝓝ˢ s ×ˢ 𝓝ˢ t 
Arguments: {X : Type u_5} {Y : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] (s : Set X) (t : Set Y)
Goal: 𝓝ˢ (s ×ˢ t) ≤ 𝓝ˢ s ×ˢ 𝓝ˢ t
-- Filter.eventually_nhdsSet_prod_iff
theorem Filter.eventually_nhdsSet_prod_iff {p : X × Y → Prop} :
    (∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q) ↔
      ∀ x ∈ s, ∀ y ∈ t,
          ∃ px : X → Prop, (∀ᶠ x' in 𝓝 x, px x') ∧ ∃ py : Y → Prop, (∀ᶠ y' in 𝓝 y, py y') ∧
            ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y) 
Arguments: {X : Type u_5} {Y : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {t : Set Y} {p : X × Y → Prop}
Goal: (∀ᶠ (q : X × Y) in 𝓝ˢ (s ×ˢ t), p q) ↔
  ∀ x ∈ s,
    ∀ y ∈ t,
      ∃ px,
        (∀ᶠ (x' : X) in 𝓝 x, px x') ∧ ∃ py, (∀ᶠ (y' : Y) in 𝓝 y, py y') ∧ ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y)
-- Filter.Eventually.prod_nhdsSet
theorem Filter.Eventually.prod_nhdsSet {p : X × Y → Prop} {px : X → Prop} {py : Y → Prop}
    (hp : ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y)) (hs : ∀ᶠ x in 𝓝ˢ s, px x)
    (ht : ∀ᶠ y in 𝓝ˢ t, py y) : ∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q 
Arguments: {X : Type u_5} {Y : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {t : Set Y} {p : X × Y → Prop} {px : X → Prop} {py : Y → Prop} (hp : ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y)) (hs : ∀ᶠ (x : X) in 𝓝ˢ s, px x) (ht : ∀ᶠ (y : Y) in 𝓝ˢ t, py y)
Goal: ∀ᶠ (q : X × Y) in 𝓝ˢ (s ×ˢ t), p q
-- PointedSmoothMap.smul_def
theorem smul_def (x : M) (f : C^∞⟮I, M; 𝕜⟯⟨x⟩) (k : 𝕜) : f • k = f x * k 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (x : M) (f : C^⊤⟮I, M; 𝕜⟯⟨x⟩) (k : 𝕜)
Goal: f • k = f x * k
-- Derivation.evalAt_apply
theorem evalAt_apply (x : M) : evalAt x X f = (X f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (X : Derivation 𝕜 C^⊤⟮I, M; 𝓘(𝕜, 𝕜), 𝕜⟯ C^⊤⟮I, M; 𝓘(𝕜, 𝕜), 𝕜⟯) (f : C^⊤⟮I, M; 𝓘(𝕜, 𝕜), 𝕜⟯) (x : M)
Goal: ((Derivation.evalAt x) X) f = (X f) x
-- apply_fdifferential
@[simp]
theorem apply_fdifferential (f : C^∞⟮I, M; I', M'⟯) {x : M} (v : PointDerivation I x)
    (g : C^∞⟮I', M'; 𝕜⟯) : 𝒅 f x v g = v (g.comp f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : C^⊤⟮I, M; I', M'⟯) {x : M} (v : PointDerivation I x) (g : C^⊤⟮I', M'; 𝓘(𝕜, 𝕜), 𝕜⟯)
Goal: ((𝒅 f x) v) g = v (g.comp f)
-- apply_hfdifferential
@[simp]
theorem apply_hfdifferential {f : C^∞⟮I, M; I', M'⟯} {x : M} {y : M'} (h : f x = y)
    (v : PointDerivation I x) (g : C^∞⟮I', M'; 𝕜⟯) : 𝒅ₕ h v g = 𝒅 f x v g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : C^⊤⟮I, M; I', M'⟯} {x : M} {y : M'} (h : f x = y) (v : PointDerivation I x) (g : C^⊤⟮I', M'; 𝓘(𝕜, 𝕜), 𝕜⟯)
Goal: ((𝒅ₕ h) v) g = ((𝒅 f x) v) g
-- fdifferential_comp
@[simp]
theorem fdifferential_comp (g : C^∞⟮I', M'; I'', M''⟯) (f : C^∞⟮I, M; I', M'⟯) (x : M) :
    𝒅 (g.comp f) x = (𝒅 g (f x)).comp (𝒅 f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] (g : C^⊤⟮I', M'; I'', M''⟯) (f : C^⊤⟮I, M; I', M'⟯) (x : M)
Goal: 𝒅 (g.comp f) x = 𝒅 g (f x) ∘ₗ 𝒅 f x
-- Homeomorph.coe_mulLeft
@[to_additive (attr := simp)]
theorem Homeomorph.coe_mulLeft (a : G) : ⇑(Homeomorph.mulLeft a) = (a * ·) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: ⇑(Homeomorph.mulLeft a) = fun x => a * x
-- Homeomorph.coe_addLeft
@[to_additive (attr := simp)]
theorem Homeomorph.coe_mulLeft (a : G) : ⇑(Homeomorph.mulLeft a) = (a * ·) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: ⇑(Homeomorph.addLeft a) = fun x => a + x
-- Homeomorph.mulLeft_symm
@[to_additive]
theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹
-- Homeomorph.addLeft_symm
@[to_additive]
theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: (Homeomorph.addLeft a).symm = Homeomorph.addLeft (-a)
-- isOpenMap_mul_left
@[to_additive]
lemma isOpenMap_mul_left (a : G) : IsOpenMap (a * ·) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: IsOpenMap fun x => a * x
-- isOpenMap_add_left
@[to_additive]
lemma isOpenMap_mul_left (a : G) : IsOpenMap (a * ·) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: IsOpenMap fun x => a + x
-- IsOpen.left_addCoset
@[to_additive IsOpen.left_addCoset]
theorem IsOpen.leftCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] {U : Set G} (h : IsOpen U) (x : G)
Goal: IsOpen (x +ᵥ U)
-- IsOpen.leftCoset
@[to_additive IsOpen.left_addCoset]
theorem IsOpen.leftCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] {U : Set G} (h : IsOpen U) (x : G)
Goal: IsOpen (x • U)
-- isClosedMap_mul_left
@[to_additive]
lemma isClosedMap_mul_left (a : G) : IsClosedMap (a * ·) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: IsClosedMap fun x => a * x
-- isClosedMap_add_left
@[to_additive]
lemma isClosedMap_mul_left (a : G) : IsClosedMap (a * ·) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: IsClosedMap fun x => a + x
-- IsClosed.left_addCoset
@[to_additive IsClosed.left_addCoset]
theorem IsClosed.leftCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] {U : Set G} (h : IsClosed U) (x : G)
Goal: IsClosed (x +ᵥ U)
-- IsClosed.leftCoset
@[to_additive IsClosed.left_addCoset]
theorem IsClosed.leftCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] {U : Set G} (h : IsClosed U) (x : G)
Goal: IsClosed (x • U)
-- Homeomorph.coe_mulRight
@[to_additive (attr := simp)]
lemma Homeomorph.coe_mulRight (a : G) : ⇑(Homeomorph.mulRight a) = (· * a) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: ⇑(Homeomorph.mulRight a) = fun x => x * a
-- Homeomorph.coe_addRight
@[to_additive (attr := simp)]
lemma Homeomorph.coe_mulRight (a : G) : ⇑(Homeomorph.mulRight a) = (· * a) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: ⇑(Homeomorph.addRight a) = fun x => x + a
-- Homeomorph.mulRight_symm
@[to_additive]
theorem Homeomorph.mulRight_symm (a : G) :
    (Homeomorph.mulRight a).symm = Homeomorph.mulRight a⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: (Homeomorph.mulRight a).symm = Homeomorph.mulRight a⁻¹
-- Homeomorph.addRight_symm
@[to_additive]
theorem Homeomorph.mulRight_symm (a : G) :
    (Homeomorph.mulRight a).symm = Homeomorph.mulRight a⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: (Homeomorph.addRight a).symm = Homeomorph.addRight (-a)
-- isOpenMap_add_right
@[to_additive]
theorem isOpenMap_mul_right (a : G) : IsOpenMap (· * a) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: IsOpenMap fun x => x + a
-- isOpenMap_mul_right
@[to_additive]
theorem isOpenMap_mul_right (a : G) : IsOpenMap (· * a) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: IsOpenMap fun x => x * a
-- IsOpen.rightCoset
@[to_additive IsOpen.right_addCoset]
theorem IsOpen.rightCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (op x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] {U : Set G} (h : IsOpen U) (x : G)
Goal: IsOpen (MulOpposite.op x • U)
-- IsOpen.right_addCoset
@[to_additive IsOpen.right_addCoset]
theorem IsOpen.rightCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (op x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] {U : Set G} (h : IsOpen U) (x : G)
Goal: IsOpen (AddOpposite.op x +ᵥ U)
-- isClosedMap_add_right
@[to_additive]
theorem isClosedMap_mul_right (a : G) : IsClosedMap (· * a) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (a : G)
Goal: IsClosedMap fun x => x + a
-- isClosedMap_mul_right
@[to_additive]
theorem isClosedMap_mul_right (a : G) : IsClosedMap (· * a) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (a : G)
Goal: IsClosedMap fun x => x * a
-- IsClosed.rightCoset
@[to_additive IsClosed.right_addCoset]
theorem IsClosed.rightCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (op x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] {U : Set G} (h : IsClosed U) (x : G)
Goal: IsClosed (MulOpposite.op x • U)
-- IsClosed.right_addCoset
@[to_additive IsClosed.right_addCoset]
theorem IsClosed.rightCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (op x • U) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] {U : Set G} (h : IsClosed U) (x : G)
Goal: IsClosed (AddOpposite.op x +ᵥ U)
-- discreteTopology_of_isOpen_singleton_zero
@[to_additive]
theorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :
    DiscreteTopology G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (h : IsOpen {0})
Goal: DiscreteTopology G
-- discreteTopology_of_isOpen_singleton_one
@[to_additive]
theorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :
    DiscreteTopology G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G] (h : IsOpen {1})
Goal: DiscreteTopology G
-- discreteTopology_iff_isOpen_singleton_one
@[to_additive]
theorem discreteTopology_iff_isOpen_singleton_one : DiscreteTopology G ↔ IsOpen ({1} : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [ContinuousMul G]
Goal: DiscreteTopology G ↔ IsOpen {1}
-- discreteTopology_iff_isOpen_singleton_zero
@[to_additive]
theorem discreteTopology_iff_isOpen_singleton_one : DiscreteTopology G ↔ IsOpen ({1} : Set G) 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [ContinuousAdd G]
Goal: DiscreteTopology G ↔ IsOpen {0}
-- ContinuousNeg.continuous_neg
class ContinuousNeg (G : Type u) [TopologicalSpace G] [Neg G] : Prop where
  continuous_neg : Continuous fun a : G => -a
Arguments: {G : Type u} [TopologicalSpace G] [Neg G] [self : ContinuousNeg G]
Goal: Continuous fun a => -a
-- ContinuousInv.continuous_inv
@[to_additive (attr := continuity)]
class ContinuousInv (G : Type u) [TopologicalSpace G] [Inv G] : Prop where
  continuous_inv : Continuous fun a : G => a⁻¹
Arguments: {G : Type u} [TopologicalSpace G] [Inv G] [self : ContinuousInv G]
Goal: Continuous fun a => a⁻¹
-- ContinuousInv.induced
@[to_additive]
theorem ContinuousInv.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F α β] [Group α]
    [Group β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousInv β] (f : F) :
    @ContinuousInv α (tβ.induced f) _ 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [FunLike F α β] [Group α] [Group β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousInv β] (f : F)
Goal: ContinuousInv α
-- ContinuousNeg.induced
@[to_additive]
theorem ContinuousInv.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F α β] [Group α]
    [Group β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousInv β] (f : F) :
    @ContinuousInv α (tβ.induced f) _ 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [FunLike F α β] [AddGroup α] [AddGroup β] [AddMonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousNeg β] (f : F)
Goal: ContinuousNeg α
-- Specializes.neg
@[to_additive]
protected theorem Specializes.inv {x y : G} (h : x ⤳ y) : (x⁻¹) ⤳ (y⁻¹) 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [ContinuousNeg G] {x y : G} (h : x ⤳ y)
Goal: (-x) ⤳ (-y)
-- Specializes.inv
@[to_additive]
protected theorem Specializes.inv {x y : G} (h : x ⤳ y) : (x⁻¹) ⤳ (y⁻¹) 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [ContinuousInv G] {x y : G} (h : x ⤳ y)
Goal: x⁻¹ ⤳ y⁻¹
-- Inseparable.inv
@[to_additive]
protected theorem Inseparable.inv {x y : G} (h : Inseparable x y) : Inseparable (x⁻¹) (y⁻¹) 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [ContinuousInv G] {x y : G} (h : Inseparable x y)
Goal: Inseparable x⁻¹ y⁻¹
-- Inseparable.neg
@[to_additive]
protected theorem Inseparable.inv {x y : G} (h : Inseparable x y) : Inseparable (x⁻¹) (y⁻¹) 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [ContinuousNeg G] {x y : G} (h : Inseparable x y)
Goal: Inseparable (-x) (-y)
-- Specializes.zpow
@[to_additive]
protected theorem Specializes.zpow {G : Type*} [DivInvMonoid G] [TopologicalSpace G]
    [ContinuousMul G] [ContinuousInv G] {x y : G} (h : x ⤳ y) : ∀ m : ℤ, (x ^ m) ⤳ (y ^ m)
  
Arguments: {G : Type u_1} [DivInvMonoid G] [TopologicalSpace G] [ContinuousMul G] [ContinuousInv G] {x y : G} (h : x ⤳ y) (m : ℤ)
Goal: (x ^ m) ⤳ (y ^ m)
-- Specializes.zsmul
@[to_additive]
protected theorem Specializes.zpow {G : Type*} [DivInvMonoid G] [TopologicalSpace G]
    [ContinuousMul G] [ContinuousInv G] {x y : G} (h : x ⤳ y) : ∀ m : ℤ, (x ^ m) ⤳ (y ^ m)
  
Arguments: {G : Type u_1} [SubNegMonoid G] [TopologicalSpace G] [ContinuousAdd G] [ContinuousNeg G] {x y : G} (h : x ⤳ y) (m : ℤ)
Goal: (m • x) ⤳ (m • y)
-- Inseparable.zsmul
@[to_additive]
protected theorem Inseparable.zpow {G : Type*} [DivInvMonoid G] [TopologicalSpace G]
    [ContinuousMul G] [ContinuousInv G] {x y : G} (h : Inseparable x y) (m : ℤ) :
    Inseparable (x ^ m) (y ^ m) 
Arguments: {G : Type u_1} [SubNegMonoid G] [TopologicalSpace G] [ContinuousAdd G] [ContinuousNeg G] {x y : G} (h : Inseparable x y) (m : ℤ)
Goal: Inseparable (m • x) (m • y)
-- Inseparable.zpow
@[to_additive]
protected theorem Inseparable.zpow {G : Type*} [DivInvMonoid G] [TopologicalSpace G]
    [ContinuousMul G] [ContinuousInv G] {x y : G} (h : Inseparable x y) (m : ℤ) :
    Inseparable (x ^ m) (y ^ m) 
Arguments: {G : Type u_1} [DivInvMonoid G] [TopologicalSpace G] [ContinuousMul G] [ContinuousInv G] {x y : G} (h : Inseparable x y) (m : ℤ)
Goal: Inseparable (x ^ m) (y ^ m)
-- continuousOn_inv
@[to_additive]
theorem continuousOn_inv {s : Set G} : ContinuousOn Inv.inv s 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [ContinuousInv G] {s : Set G}
Goal: ContinuousOn Inv.inv s
-- continuousOn_neg
@[to_additive]
theorem continuousOn_inv {s : Set G} : ContinuousOn Inv.inv s 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [ContinuousNeg G] {s : Set G}
Goal: ContinuousOn Neg.neg s
-- continuousWithinAt_neg
@[to_additive]
theorem continuousWithinAt_inv {s : Set G} {x : G} : ContinuousWithinAt Inv.inv s x 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [ContinuousNeg G] {s : Set G} {x : G}
Goal: ContinuousWithinAt Neg.neg s x
-- continuousWithinAt_inv
@[to_additive]
theorem continuousWithinAt_inv {s : Set G} {x : G} : ContinuousWithinAt Inv.inv s x 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [ContinuousInv G] {s : Set G} {x : G}
Goal: ContinuousWithinAt Inv.inv s x
-- continuousAt_neg
@[to_additive]
theorem continuousAt_inv {x : G} : ContinuousAt Inv.inv x 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [ContinuousNeg G] {x : G}
Goal: ContinuousAt Neg.neg x
-- continuousAt_inv
@[to_additive]
theorem continuousAt_inv {x : G} : ContinuousAt Inv.inv x 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [ContinuousInv G] {x : G}
Goal: ContinuousAt Inv.inv x
-- tendsto_neg
@[to_additive]
theorem tendsto_inv (a : G) : Tendsto Inv.inv (𝓝 a) (𝓝 a⁻¹) 
Arguments: {G : Type w} [TopologicalSpace G] [Neg G] [ContinuousNeg G] (a : G)
Goal: Filter.Tendsto Neg.neg (𝓝 a) (𝓝 (-a))
-- tendsto_inv
@[to_additive]
theorem tendsto_inv (a : G) : Tendsto Inv.inv (𝓝 a) (𝓝 a⁻¹) 
Arguments: {G : Type w} [TopologicalSpace G] [Inv G] [ContinuousInv G] (a : G)
Goal: Filter.Tendsto Inv.inv (𝓝 a) (𝓝 a⁻¹)
-- Filter.Tendsto.inv
@[to_additive
  "If a function converges to a value in an additive topological group, then its
  negation converges to the negation of this value."]
theorem Filter.Tendsto.inv {f : α → G} {l : Filter α} {y : G} (h : Tendsto f l (𝓝 y)) :
    Tendsto (fun x => (f x)⁻¹) l (𝓝 y⁻¹) 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Inv G] [ContinuousInv G] {f : α → G} {l : Filter α} {y : G} (h : Filter.Tendsto f l (𝓝 y))
Goal: Filter.Tendsto (fun x => (f x)⁻¹) l (𝓝 y⁻¹)
-- Filter.Tendsto.neg
@[to_additive
  "If a function converges to a value in an additive topological group, then its
  negation converges to the negation of this value."]
theorem Filter.Tendsto.inv {f : α → G} {l : Filter α} {y : G} (h : Tendsto f l (𝓝 y)) :
    Tendsto (fun x => (f x)⁻¹) l (𝓝 y⁻¹) 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Neg G] [ContinuousNeg G] {f : α → G} {l : Filter α} {y : G} (h : Filter.Tendsto f l (𝓝 y))
Goal: Filter.Tendsto (fun x => -f x) l (𝓝 (-y))
-- Continuous.inv
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.inv (hf : Continuous f) : Continuous fun x => (f x)⁻¹ 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Inv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} (hf : Continuous f)
Goal: Continuous fun x => (f x)⁻¹
-- Continuous.neg
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.inv (hf : Continuous f) : Continuous fun x => (f x)⁻¹ 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Neg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} (hf : Continuous f)
Goal: Continuous fun x => -f x
-- ContinuousAt.inv
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.inv (hf : ContinuousAt f x) : ContinuousAt (fun x => (f x)⁻¹) x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Inv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x)
Goal: ContinuousAt (fun x => (f x)⁻¹) x
-- ContinuousAt.neg
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.inv (hf : ContinuousAt f x) : ContinuousAt (fun x => (f x)⁻¹) x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Neg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x)
Goal: ContinuousAt (fun x => -f x) x
-- ContinuousOn.inv
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.inv (hf : ContinuousOn f s) : ContinuousOn (fun x => (f x)⁻¹) s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Inv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} {s : Set α} (hf : ContinuousOn f s)
Goal: ContinuousOn (fun x => (f x)⁻¹) s
-- ContinuousOn.neg
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.inv (hf : ContinuousOn f s) : ContinuousOn (fun x => (f x)⁻¹) s 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Neg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} {s : Set α} (hf : ContinuousOn f s)
Goal: ContinuousOn (fun x => -f x) s
-- ContinuousWithinAt.inv
@[to_additive]
theorem ContinuousWithinAt.inv (hf : ContinuousWithinAt f s x) :
    ContinuousWithinAt (fun x => (f x)⁻¹) s x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Inv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} {s : Set α} {x : α} (hf : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (fun x => (f x)⁻¹) s x
-- ContinuousWithinAt.neg
@[to_additive]
theorem ContinuousWithinAt.inv (hf : ContinuousWithinAt f s x) :
    ContinuousWithinAt (fun x => (f x)⁻¹) s x 
Arguments: {G : Type w} {α : Type u} [TopologicalSpace G] [Neg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} {s : Set α} {x : α} (hf : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (fun x => -f x) s x
-- isClosed_setOf_map_neg
@[to_additive]
theorem isClosed_setOf_map_inv [Inv G₁] [Inv G₂] [ContinuousInv G₂] :
    IsClosed { f : G₁ → G₂ | ∀ x, f x⁻¹ = (f x)⁻¹ } 
Arguments: (G₁ : Type u_2) (G₂ : Type u_3) [TopologicalSpace G₂] [T2Space G₂] [Neg G₁] [Neg G₂] [ContinuousNeg G₂]
Goal: IsClosed {f | ∀ (x : G₁), f (-x) = -f x}
-- isClosed_setOf_map_inv
@[to_additive]
theorem isClosed_setOf_map_inv [Inv G₁] [Inv G₂] [ContinuousInv G₂] :
    IsClosed { f : G₁ → G₂ | ∀ x, f x⁻¹ = (f x)⁻¹ } 
Arguments: (G₁ : Type u_2) (G₂ : Type u_3) [TopologicalSpace G₂] [T2Space G₂] [Inv G₁] [Inv G₂] [ContinuousInv G₂]
Goal: IsClosed {f | ∀ (x : G₁), f x⁻¹ = (f x)⁻¹}
-- IsCompact.neg
@[to_additive]
theorem IsCompact.inv (hs : IsCompact s) : IsCompact s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] {s : Set G} (hs : IsCompact s)
Goal: IsCompact (-s)
-- IsCompact.inv
@[to_additive]
theorem IsCompact.inv (hs : IsCompact s) : IsCompact s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] {s : Set G} (hs : IsCompact s)
Goal: IsCompact s⁻¹
-- Homeomorph.coe_neg
@[to_additive (attr := simp)]
lemma Homeomorph.coe_inv {G : Type*} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] :
    ⇑(Homeomorph.inv G) = Inv.inv 
Arguments: {G : Type u_1} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G]
Goal: ⇑(Homeomorph.neg G) = Neg.neg
-- Homeomorph.coe_inv
@[to_additive (attr := simp)]
lemma Homeomorph.coe_inv {G : Type*} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] :
    ⇑(Homeomorph.inv G) = Inv.inv 
Arguments: {G : Type u_1} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G]
Goal: ⇑(Homeomorph.inv G) = Inv.inv
-- isOpenMap_inv
@[to_additive]
theorem isOpenMap_inv : IsOpenMap (Inv.inv : G → G) 
Arguments: (G : Type w) [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G]
Goal: IsOpenMap Inv.inv
-- isOpenMap_neg
@[to_additive]
theorem isOpenMap_inv : IsOpenMap (Inv.inv : G → G) 
Arguments: (G : Type w) [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G]
Goal: IsOpenMap Neg.neg
-- isClosedMap_neg
@[to_additive]
theorem isClosedMap_inv : IsClosedMap (Inv.inv : G → G) 
Arguments: (G : Type w) [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G]
Goal: IsClosedMap Neg.neg
-- isClosedMap_inv
@[to_additive]
theorem isClosedMap_inv : IsClosedMap (Inv.inv : G → G) 
Arguments: (G : Type w) [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G]
Goal: IsClosedMap Inv.inv
-- IsOpen.inv
@[to_additive]
theorem IsOpen.inv (hs : IsOpen s) : IsOpen s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] {s : Set G} (hs : IsOpen s)
Goal: IsOpen s⁻¹
-- IsOpen.neg
@[to_additive]
theorem IsOpen.inv (hs : IsOpen s) : IsOpen s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] {s : Set G} (hs : IsOpen s)
Goal: IsOpen (-s)
-- IsClosed.inv
@[to_additive]
theorem IsClosed.inv (hs : IsClosed s) : IsClosed s⁻¹ 
Arguments: {G : Type w} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] {s : Set G} (hs : IsClosed s)
Goal: IsClosed s⁻¹