ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PartialHomeomorph.MDifferentiable.comp_symm_deriv
theorem comp_symm_deriv {x : M'} (hx : x ∈ e.target) :
    (mfderiv I I' e (e.symm x)).comp (mfderiv I' I e.symm x) =
      ContinuousLinearMap.id 𝕜 (TangentSpace I' 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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M'} (hx : x ∈ e.target)
Goal: (mfderiv I I' (↑e) (↑e.symm x)).comp (mfderiv I' I (↑e.symm) x) = ContinuousLinearMap.id 𝕜 (TangentSpace I' x)
-- PartialHomeomorph.MDifferentiable.mfderiv_bijective
theorem mfderiv_bijective {x : M} (hx : x ∈ e.source) : Function.Bijective (mfderiv I I' e 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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M} (hx : x ∈ e.source)
Goal: Function.Bijective ⇑(mfderiv I I' (↑e) x)
-- PartialHomeomorph.MDifferentiable.mfderiv_injective
theorem mfderiv_injective {x : M} (hx : x ∈ e.source) : Function.Injective (mfderiv I I' e 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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M} (hx : x ∈ e.source)
Goal: Function.Injective ⇑(mfderiv I I' (↑e) x)
-- PartialHomeomorph.MDifferentiable.mfderiv_surjective
theorem mfderiv_surjective {x : M} (hx : x ∈ e.source) : Function.Surjective (mfderiv I I' e 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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M} (hx : x ∈ e.source)
Goal: Function.Surjective ⇑(mfderiv I I' (↑e) x)
-- PartialHomeomorph.MDifferentiable.ker_mfderiv_eq_bot
theorem ker_mfderiv_eq_bot {x : M} (hx : x ∈ e.source) : LinearMap.ker (mfderiv I I' e 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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M} (hx : x ∈ e.source)
Goal: LinearMap.ker (mfderiv I I' (↑e) x) = ⊥
-- PartialHomeomorph.MDifferentiable.range_mfderiv_eq_top
theorem range_mfderiv_eq_top {x : M} (hx : x ∈ e.source) : LinearMap.range (mfderiv I I' e 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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M} (hx : x ∈ e.source)
Goal: LinearMap.range (mfderiv I I' (↑e) x) = ⊤
-- PartialHomeomorph.MDifferentiable.range_mfderiv_eq_univ
theorem range_mfderiv_eq_univ {x : M} (hx : x ∈ e.source) : range (mfderiv I I' e x) = univ 
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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M} (hx : x ∈ e.source)
Goal: Set.range ⇑(mfderiv I I' (↑e) x) = Set.univ
-- PartialHomeomorph.MDifferentiable.trans
theorem trans (he' : e'.MDifferentiable I' I'') : (e.trans e').MDifferentiable I I'' 
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''] {e : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) {e' : PartialHomeomorph M' M''} [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] [SmoothManifoldWithCorners I'' M''] (he' : PartialHomeomorph.MDifferentiable I' I'' e')
Goal: PartialHomeomorph.MDifferentiable I I'' (e ≫ₕ e')
-- hasMFDerivAt_extChartAt
theorem hasMFDerivAt_extChartAt (h : y ∈ (chartAt H x).source) :
    HasMFDerivAt I 𝓘(𝕜, E) (extChartAt I x) y (mfderiv I I (chartAt H x) y : _) 
Arguments: {𝕜 : Type u_11} [NontriviallyNormedField 𝕜] {E : Type u_12} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_13} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_14} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {x y : M} (h : y ∈ (chartAt H x).source)
Goal: HasMFDerivAt I 𝓘(𝕜, E) (↑(extChartAt I x)) y (mfderiv I I (↑(chartAt H x)) y)
-- hasMFDerivWithinAt_extChartAt
theorem hasMFDerivWithinAt_extChartAt (h : y ∈ (chartAt H x).source) :
    HasMFDerivWithinAt I 𝓘(𝕜, E) (extChartAt I x) s y (mfderiv I I (chartAt H x) y : _) 
Arguments: {𝕜 : Type u_11} [NontriviallyNormedField 𝕜] {E : Type u_12} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_13} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_14} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {s : Set M} {x y : M} (h : y ∈ (chartAt H x).source)
Goal: HasMFDerivWithinAt I 𝓘(𝕜, E) (↑(extChartAt I x)) s y (mfderiv I I (↑(chartAt H x)) y)
-- mdifferentiableAt_extChartAt
theorem mdifferentiableAt_extChartAt (h : y ∈ (chartAt H x).source) :
    MDifferentiableAt I 𝓘(𝕜, E) (extChartAt I x) y 
Arguments: {𝕜 : Type u_11} [NontriviallyNormedField 𝕜] {E : Type u_12} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_13} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_14} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {x y : M} (h : y ∈ (chartAt H x).source)
Goal: MDifferentiableAt I 𝓘(𝕜, E) (↑(extChartAt I x)) y
-- mdifferentiableOn_extChartAt
theorem mdifferentiableOn_extChartAt :
    MDifferentiableOn I 𝓘(𝕜, E) (extChartAt I x) (chartAt H x).source 
Arguments: {𝕜 : Type u_11} [NontriviallyNormedField 𝕜] {E : Type u_12} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_13} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_14} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {x : M}
Goal: MDifferentiableOn I 𝓘(𝕜, E) (↑(extChartAt I x)) (chartAt H x).source
-- ofSet_mem_analyticGroupoid
theorem ofSet_mem_analyticGroupoid {s : Set H} (hs : IsOpen s) :
    PartialHomeomorph.ofSet s hs ∈ analyticGroupoid I 
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) {s : Set H} (hs : IsOpen s)
Goal: PartialHomeomorph.ofSet s hs ∈ analyticGroupoid I
-- symm_trans_mem_analyticGroupoid
theorem symm_trans_mem_analyticGroupoid (e : PartialHomeomorph M H) :
    e.symm.trans e ∈ analyticGroupoid I 
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] (e : PartialHomeomorph M H)
Goal: e.symm ≫ₕ e ∈ analyticGroupoid I
-- mem_analyticGroupoid_of_boundaryless
theorem mem_analyticGroupoid_of_boundaryless [CompleteSpace E] [I.Boundaryless]
    (e : PartialHomeomorph H H) :
    e ∈ analyticGroupoid I ↔ AnalyticOn 𝕜 (I ∘ e ∘ I.symm) (I '' e.source) ∧
    AnalyticOn 𝕜 (I ∘ e.symm ∘ I.symm) (I '' e.target) 
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) [CompleteSpace E] [I.Boundaryless] (e : PartialHomeomorph H H)
Goal: e ∈ analyticGroupoid I ↔
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
-- SmoothRing.smooth_mul
class SmoothRing (I : ModelWithCorners 𝕜 E H) (R : Type*) [Semiring R] [TopologicalSpace R]
    [ChartedSpace H R] extends SmoothAdd I R : Prop where
  smooth_mul : Smooth (I.prod I) I fun p : R × R => p.1 * p.2
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {R : Type u_4} [Semiring R] [TopologicalSpace R] [ChartedSpace H R] [self : SmoothRing I R]
Goal: Smooth (I.prod I) I fun p => p.1 * p.2
-- topologicalSemiring_of_smooth
theorem topologicalSemiring_of_smooth [Semiring R] [SmoothRing I R] : TopologicalSemiring R 
Arguments: {𝕜 : Type u_1} {R : Type u_2} {E : Type u_3} {H : Type u_4} [TopologicalSpace R] [TopologicalSpace H] [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [ChartedSpace H R] (I : ModelWithCorners 𝕜 E H) [Semiring R] [SmoothRing I R]
Goal: TopologicalSemiring R
-- UniformGroup.uniformContinuous_div
class UniformGroup (α : Type*) [UniformSpace α] [Group α] : Prop where
  uniformContinuous_div : UniformContinuous fun p : α × α => p.1 / p.2
Arguments: {α : Type u_3} [UniformSpace α] [Group α] [self : UniformGroup α]
Goal: UniformContinuous fun p => p.1 / p.2
-- UniformAddGroup.uniformContinuous_sub
class UniformAddGroup (α : Type*) [UniformSpace α] [AddGroup α] : Prop where
  uniformContinuous_sub : UniformContinuous fun p : α × α => p.1 - p.2
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] [self : UniformAddGroup α]
Goal: UniformContinuous fun p => p.1 - p.2
-- UniformGroup.mk'
@[to_additive]
theorem UniformGroup.mk' {α} [UniformSpace α] [Group α]
    (h₁ : UniformContinuous fun p : α × α => p.1 * p.2) (h₂ : UniformContinuous fun p : α => p⁻¹) :
    UniformGroup α 
Arguments: {α : Type u_3} [UniformSpace α] [Group α] (h₁ : UniformContinuous fun p => p.1 * p.2) (h₂ : UniformContinuous fun p => p⁻¹)
Goal: UniformGroup α
-- UniformAddGroup.mk'
@[to_additive]
theorem UniformGroup.mk' {α} [UniformSpace α] [Group α]
    (h₁ : UniformContinuous fun p : α × α => p.1 * p.2) (h₂ : UniformContinuous fun p : α => p⁻¹) :
    UniformGroup α 
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] (h₁ : UniformContinuous fun p => p.1 + p.2) (h₂ : UniformContinuous fun p => -p)
Goal: UniformAddGroup α
-- uniformContinuous_div
@[to_additive]
theorem uniformContinuous_div : UniformContinuous fun p : α × α => p.1 / p.2 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α]
Goal: UniformContinuous fun p => p.1 / p.2
-- uniformContinuous_sub
@[to_additive]
theorem uniformContinuous_div : UniformContinuous fun p : α × α => p.1 / p.2 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: UniformContinuous fun p => p.1 - p.2
-- UniformContinuous.div
@[to_additive]
theorem UniformContinuous.div [UniformSpace β] {f : β → α} {g : β → α} (hf : UniformContinuous f)
    (hg : UniformContinuous g) : UniformContinuous fun x => f x / g x 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g)
Goal: UniformContinuous fun x => f x / g x
-- UniformContinuous.sub
@[to_additive]
theorem UniformContinuous.div [UniformSpace β] {f : β → α} {g : β → α} (hf : UniformContinuous f)
    (hg : UniformContinuous g) : UniformContinuous fun x => f x / g x 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g)
Goal: UniformContinuous fun x => f x - g x
-- UniformContinuous.inv
@[to_additive]
theorem UniformContinuous.inv [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
    UniformContinuous fun x => (f x)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
Goal: UniformContinuous fun x => (f x)⁻¹
-- UniformContinuous.neg
@[to_additive]
theorem UniformContinuous.inv [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
    UniformContinuous fun x => (f x)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
Goal: UniformContinuous fun x => -f x
-- uniformContinuous_inv
@[to_additive]
theorem uniformContinuous_inv : UniformContinuous fun x : α => x⁻¹ 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α]
Goal: UniformContinuous fun x => x⁻¹
-- uniformContinuous_neg
@[to_additive]
theorem uniformContinuous_inv : UniformContinuous fun x : α => x⁻¹ 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: UniformContinuous fun x => -x
-- UniformContinuous.add
@[to_additive]
theorem UniformContinuous.mul [UniformSpace β] {f : β → α} {g : β → α} (hf : UniformContinuous f)
    (hg : UniformContinuous g) : UniformContinuous fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g)
Goal: UniformContinuous fun x => f x + g x
-- UniformContinuous.mul
@[to_additive]
theorem UniformContinuous.mul [UniformSpace β] {f : β → α} {g : β → α} (hf : UniformContinuous f)
    (hg : UniformContinuous g) : UniformContinuous fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g)
Goal: UniformContinuous fun x => f x * g x
-- uniformContinuous_add
@[to_additive]
theorem uniformContinuous_mul : UniformContinuous fun p : α × α => p.1 * p.2 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: UniformContinuous fun p => p.1 + p.2
-- uniformContinuous_mul
@[to_additive]
theorem uniformContinuous_mul : UniformContinuous fun p : α × α => p.1 * p.2 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α]
Goal: UniformContinuous fun p => p.1 * p.2
-- UniformContinuous.const_nsmul
@[to_additive UniformContinuous.const_nsmul]
theorem UniformContinuous.pow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
    ∀ n : ℕ, UniformContinuous fun x => f x ^ n
  
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℕ)
Goal: UniformContinuous fun x => n • f x
-- UniformContinuous.pow_const
@[to_additive UniformContinuous.const_nsmul]
theorem UniformContinuous.pow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
    ∀ n : ℕ, UniformContinuous fun x => f x ^ n
  
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℕ)
Goal: UniformContinuous fun x => f x ^ n
-- uniformContinuous_const_nsmul
@[to_additive uniformContinuous_const_nsmul]
theorem uniformContinuous_pow_const (n : ℕ) : UniformContinuous fun x : α => x ^ n 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] (n : ℕ)
Goal: UniformContinuous fun x => n • x
-- uniformContinuous_pow_const
@[to_additive uniformContinuous_const_nsmul]
theorem uniformContinuous_pow_const (n : ℕ) : UniformContinuous fun x : α => x ^ n 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] (n : ℕ)
Goal: UniformContinuous fun x => x ^ n
-- UniformContinuous.const_zsmul
@[to_additive UniformContinuous.const_zsmul]
theorem UniformContinuous.zpow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
    ∀ n : ℤ, UniformContinuous fun x => f x ^ n
  
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℤ)
Goal: UniformContinuous fun x => n • f x
-- UniformContinuous.zpow_const
@[to_additive UniformContinuous.const_zsmul]
theorem UniformContinuous.zpow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
    ∀ n : ℤ, UniformContinuous fun x => f x ^ n
  
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℤ)
Goal: UniformContinuous fun x => f x ^ n
-- uniformContinuous_zpow_const
@[to_additive uniformContinuous_const_zsmul]
theorem uniformContinuous_zpow_const (n : ℤ) : UniformContinuous fun x : α => x ^ n 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] (n : ℤ)
Goal: UniformContinuous fun x => x ^ n
-- uniformContinuous_const_zsmul
@[to_additive uniformContinuous_const_zsmul]
theorem uniformContinuous_zpow_const (n : ℤ) : UniformContinuous fun x : α => x ^ n 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] (n : ℤ)
Goal: UniformContinuous fun x => n • x
-- uniformity_translate_add
@[to_additive]
theorem uniformity_translate_mul (a : α) : ((𝓤 α).map fun x : α × α => (x.1 * a, x.2 * a)) = 𝓤 α 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] (a : α)
Goal: Filter.map (fun x => (x.1 + a, x.2 + a)) (𝓤 α) = 𝓤 α
-- uniformity_translate_mul
@[to_additive]
theorem uniformity_translate_mul (a : α) : ((𝓤 α).map fun x : α × α => (x.1 * a, x.2 * a)) = 𝓤 α 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] (a : α)
Goal: Filter.map (fun x => (x.1 * a, x.2 * a)) (𝓤 α) = 𝓤 α
-- uniformEmbedding_translate_add
@[to_additive]
theorem uniformEmbedding_translate_mul (a : α) : UniformEmbedding fun x : α => x * a 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] (a : α)
Goal: UniformEmbedding fun x => x + a
-- uniformEmbedding_translate_mul
@[to_additive]
theorem uniformEmbedding_translate_mul (a : α) : UniformEmbedding fun x : α => x * a 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] (a : α)
Goal: UniformEmbedding fun x => x * a
-- uniformAddGroup_sInf
@[to_additive]
theorem uniformGroup_sInf {us : Set (UniformSpace β)} (h : ∀ u ∈ us, @UniformGroup β u _) :
    @UniformGroup β (sInf us) _ 
Arguments: {β : Type u_2} [AddGroup β] {us : Set (UniformSpace β)} (h : ∀ u ∈ us, UniformAddGroup β)
Goal: UniformAddGroup β
-- uniformGroup_sInf
@[to_additive]
theorem uniformGroup_sInf {us : Set (UniformSpace β)} (h : ∀ u ∈ us, @UniformGroup β u _) :
    @UniformGroup β (sInf us) _ 
Arguments: {β : Type u_2} [Group β] {us : Set (UniformSpace β)} (h : ∀ u ∈ us, UniformGroup β)
Goal: UniformGroup β
-- uniformGroup_iInf
@[to_additive]
theorem uniformGroup_iInf {ι : Sort*} {us' : ι → UniformSpace β}
    (h' : ∀ i, @UniformGroup β (us' i) _) : @UniformGroup β (⨅ i, us' i) _ 
Arguments: {β : Type u_2} [Group β] {ι : Sort u_3} {us' : ι → UniformSpace β} (h' : ∀ (i : ι), UniformGroup β)
Goal: UniformGroup β
-- uniformAddGroup_iInf
@[to_additive]
theorem uniformGroup_iInf {ι : Sort*} {us' : ι → UniformSpace β}
    (h' : ∀ i, @UniformGroup β (us' i) _) : @UniformGroup β (⨅ i, us' i) _ 
Arguments: {β : Type u_2} [AddGroup β] {ι : Sort u_3} {us' : ι → UniformSpace β} (h' : ∀ (i : ι), UniformAddGroup β)
Goal: UniformAddGroup β
-- uniformGroup_inf
@[to_additive]
theorem uniformGroup_inf {u₁ u₂ : UniformSpace β} (h₁ : @UniformGroup β u₁ _)
    (h₂ : @UniformGroup β u₂ _) : @UniformGroup β (u₁ ⊓ u₂) _ 
Arguments: {β : Type u_2} [Group β] {u₁ u₂ : UniformSpace β} (h₁ : UniformGroup β) (h₂ : UniformGroup β)
Goal: UniformGroup β
-- uniformAddGroup_inf
@[to_additive]
theorem uniformGroup_inf {u₁ u₂ : UniformSpace β} (h₁ : @UniformGroup β u₁ _)
    (h₂ : @UniformGroup β u₂ _) : @UniformGroup β (u₁ ⊓ u₂) _ 
Arguments: {β : Type u_2} [AddGroup β] {u₁ u₂ : UniformSpace β} (h₁ : UniformAddGroup β) (h₂ : UniformAddGroup β)
Goal: UniformAddGroup β
-- UniformInducing.uniformAddGroup
@[to_additive]
lemma UniformInducing.uniformGroup {γ : Type*} [Group γ] [UniformSpace γ] [UniformGroup γ]
    [UniformSpace β] {F : Type*} [FunLike F β γ] [MonoidHomClass F β γ]
    (f : F) (hf : UniformInducing f) :
    UniformGroup β 
Arguments: {β : Type u_2} [AddGroup β] {γ : Type u_3} [AddGroup γ] [UniformSpace γ] [UniformAddGroup γ] [UniformSpace β] {F : Type u_4} [FunLike F β γ] [AddMonoidHomClass F β γ] (f : F) (hf : UniformInducing ⇑f)
Goal: UniformAddGroup β
-- UniformInducing.uniformGroup
@[to_additive]
lemma UniformInducing.uniformGroup {γ : Type*} [Group γ] [UniformSpace γ] [UniformGroup γ]
    [UniformSpace β] {F : Type*} [FunLike F β γ] [MonoidHomClass F β γ]
    (f : F) (hf : UniformInducing f) :
    UniformGroup β 
Arguments: {β : Type u_2} [Group β] {γ : Type u_3} [Group γ] [UniformSpace γ] [UniformGroup γ] [UniformSpace β] {F : Type u_4} [FunLike F β γ] [MonoidHomClass F β γ] (f : F) (hf : UniformInducing ⇑f)
Goal: UniformGroup β
-- UniformGroup.comap
@[to_additive]
protected theorem UniformGroup.comap {γ : Type*} [Group γ] {u : UniformSpace γ} [UniformGroup γ]
    {F : Type*} [FunLike F β γ] [MonoidHomClass F β γ] (f : F) : @UniformGroup β (u.comap f) _ 
Arguments: {β : Type u_2} [Group β] {γ : Type u_3} [Group γ] {u : UniformSpace γ} [UniformGroup γ] {F : Type u_4} [FunLike F β γ] [MonoidHomClass F β γ] (f : F)
Goal: UniformGroup β
-- UniformAddGroup.comap
@[to_additive]
protected theorem UniformGroup.comap {γ : Type*} [Group γ] {u : UniformSpace γ} [UniformGroup γ]
    {F : Type*} [FunLike F β γ] [MonoidHomClass F β γ] (f : F) : @UniformGroup β (u.comap f) _ 
Arguments: {β : Type u_2} [AddGroup β] {γ : Type u_3} [AddGroup γ] {u : UniformSpace γ} [UniformAddGroup γ] {F : Type u_4} [FunLike F β γ] [AddMonoidHomClass F β γ] (f : F)
Goal: UniformAddGroup β
-- uniformity_eq_comap_nhds_zero
@[to_additive]
theorem uniformity_eq_comap_nhds_one : 𝓤 α = comap (fun x : α × α => x.2 / x.1) (𝓝 (1 : α)) 
Arguments: (α : Type u_1) [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: 𝓤 α = Filter.comap (fun x => x.2 - x.1) (𝓝 0)