ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearIsometryEquiv.fderivWithin
protected theorem fderivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 iso s x = iso 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} (iso : E ≃ₗᵢ[𝕜] F) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (⇑iso) s x = ↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }
-- LinearIsometryEquiv.differentiable
@[fun_prop]
protected theorem differentiable : Differentiable 𝕜 iso 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (iso : E ≃ₗᵢ[𝕜] F)
Goal: Differentiable 𝕜 ⇑iso
-- LinearIsometryEquiv.differentiableOn
@[fun_prop]
protected theorem differentiableOn : DifferentiableOn 𝕜 iso s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} (iso : E ≃ₗᵢ[𝕜] F)
Goal: DifferentiableOn 𝕜 (⇑iso) s
-- LinearIsometryEquiv.comp_differentiableWithinAt_iff
theorem comp_differentiableWithinAt_iff {f : G → E} {s : Set G} {x : G} :
    DifferentiableWithinAt 𝕜 (iso ∘ f) s x ↔ DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {s : Set G} {x : G}
Goal: DifferentiableWithinAt 𝕜 (⇑iso ∘ f) s x ↔ DifferentiableWithinAt 𝕜 f s x
-- LinearIsometryEquiv.comp_differentiableAt_iff
theorem comp_differentiableAt_iff {f : G → E} {x : G} :
    DifferentiableAt 𝕜 (iso ∘ f) x ↔ DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {x : G}
Goal: DifferentiableAt 𝕜 (⇑iso ∘ f) x ↔ DifferentiableAt 𝕜 f x
-- LinearIsometryEquiv.comp_differentiableOn_iff
theorem comp_differentiableOn_iff {f : G → E} {s : Set G} :
    DifferentiableOn 𝕜 (iso ∘ f) s ↔ DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {s : Set G}
Goal: DifferentiableOn 𝕜 (⇑iso ∘ f) s ↔ DifferentiableOn 𝕜 f s
-- LinearIsometryEquiv.comp_differentiable_iff
theorem comp_differentiable_iff {f : G → E} : Differentiable 𝕜 (iso ∘ f) ↔ Differentiable 𝕜 f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E}
Goal: Differentiable 𝕜 (⇑iso ∘ f) ↔ Differentiable 𝕜 f
-- LinearIsometryEquiv.comp_hasFDerivWithinAt_iff
theorem comp_hasFDerivWithinAt_iff {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] E} :
    HasFDerivWithinAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') s x ↔ HasFDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] E}
Goal: HasFDerivWithinAt (⇑iso ∘ f)
    ((↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp f') s x ↔
  HasFDerivWithinAt f f' s x
-- LinearIsometryEquiv.comp_hasStrictFDerivAt_iff
theorem comp_hasStrictFDerivAt_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
    HasStrictFDerivAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ HasStrictFDerivAt f f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {x : G} {f' : G →L[𝕜] E}
Goal: HasStrictFDerivAt (⇑iso ∘ f)
    ((↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp f') x ↔
  HasStrictFDerivAt f f' x
-- LinearIsometryEquiv.comp_hasFDerivAt_iff
theorem comp_hasFDerivAt_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
    HasFDerivAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ HasFDerivAt f f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {x : G} {f' : G →L[𝕜] E}
Goal: HasFDerivAt (⇑iso ∘ f)
    ((↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp f') x ↔
  HasFDerivAt f f' x
-- LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'
theorem comp_hasFDerivWithinAt_iff' {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] F} :
    HasFDerivWithinAt (iso ∘ f) f' s x ↔ HasFDerivWithinAt f ((iso.symm : F →L[𝕜] E).comp f') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] F}
Goal: HasFDerivWithinAt (⇑iso ∘ f) f' s x ↔
  HasFDerivWithinAt f
    ((↑{ toLinearEquiv := iso.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp f') s x
-- LinearIsometryEquiv.comp_hasFDerivAt_iff'
theorem comp_hasFDerivAt_iff' {f : G → E} {x : G} {f' : G →L[𝕜] F} :
    HasFDerivAt (iso ∘ f) f' x ↔ HasFDerivAt f ((iso.symm : F →L[𝕜] E).comp f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {x : G} {f' : G →L[𝕜] F}
Goal: HasFDerivAt (⇑iso ∘ f) f' x ↔
  HasFDerivAt f ((↑{ toLinearEquiv := iso.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp f')
    x
-- LinearIsometryEquiv.comp_fderivWithin
theorem comp_fderivWithin {f : G → E} {s : Set G} {x : G} (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (iso ∘ f) s x = (iso : E →L[𝕜] F).comp (fderivWithin 𝕜 f s x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {s : Set G} {x : G} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (⇑iso ∘ f) s x =
  (↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp (fderivWithin 𝕜 f s x)
-- LinearIsometryEquiv.comp_fderiv
theorem comp_fderiv {f : G → E} {x : G} :
    fderiv 𝕜 (iso ∘ f) x = (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E} {x : G}
Goal: fderiv 𝕜 (⇑iso ∘ f) x =
  (↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp (fderiv 𝕜 f x)
-- LinearIsometryEquiv.comp_fderiv'
theorem comp_fderiv' {f : G → E} :
    fderiv 𝕜 (iso ∘ f) = fun x ↦ (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃ₗᵢ[𝕜] F) {f : G → E}
Goal: fderiv 𝕜 (⇑iso ∘ f) = fun x =>
  (↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp (fderiv 𝕜 f x)
-- HasStrictFDerivAt.of_local_left_inverse
theorem HasStrictFDerivAt.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
    (hg : ContinuousAt g a) (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) (g a))
    (hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F} (hg : ContinuousAt g a) (hf : HasStrictFDerivAt f (↑f') (g a)) (hfg : ∀ᶠ (y : F) in 𝓝 a, f (g y) = y)
Goal: HasStrictFDerivAt g (↑f'.symm) a
-- HasFDerivAt.of_local_left_inverse
theorem HasFDerivAt.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
    (hg : ContinuousAt g a) (hf : HasFDerivAt f (f' : E →L[𝕜] F) (g a))
    (hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) : HasFDerivAt g (f'.symm : F →L[𝕜] E) a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F} (hg : ContinuousAt g a) (hf : HasFDerivAt f (↑f') (g a)) (hfg : ∀ᶠ (y : F) in 𝓝 a, f (g y) = y)
Goal: HasFDerivAt g (↑f'.symm) a
-- PartialHomeomorph.hasStrictFDerivAt_symm
theorem PartialHomeomorph.hasStrictFDerivAt_symm (f : PartialHomeomorph E F) {f' : E ≃L[𝕜] F}
    {a : F} (ha : a ∈ f.target) (htff' : HasStrictFDerivAt f (f' : E →L[𝕜] F) (f.symm a)) :
    HasStrictFDerivAt f.symm (f'.symm : F →L[𝕜] E) a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) {f' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target) (htff' : HasStrictFDerivAt (↑f) (↑f') (↑f.symm a))
Goal: HasStrictFDerivAt (↑f.symm) (↑f'.symm) a
-- PartialHomeomorph.hasFDerivAt_symm
theorem PartialHomeomorph.hasFDerivAt_symm (f : PartialHomeomorph E F) {f' : E ≃L[𝕜] F} {a : F}
    (ha : a ∈ f.target) (htff' : HasFDerivAt f (f' : E →L[𝕜] F) (f.symm a)) :
    HasFDerivAt f.symm (f'.symm : F →L[𝕜] E) a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) {f' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target) (htff' : HasFDerivAt (↑f) (↑f') (↑f.symm a))
Goal: HasFDerivAt (↑f.symm) (↑f'.symm) a
-- HasFDerivWithinAt.eventually_ne
theorem HasFDerivWithinAt.eventually_ne (h : HasFDerivWithinAt f f' s x)
    (hf' : ∃ C, ∀ z, ‖z‖ ≤ C * ‖f' z‖) : ∀ᶠ z in 𝓝[s \ {x}] x, f z ≠ f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (hf' : ∃ C, ∀ (z : E), ‖z‖ ≤ C * ‖f' z‖)
Goal: ∀ᶠ (z : E) in 𝓝[s \ {x}] x, f z ≠ f x
-- HasFDerivAt.eventually_ne
theorem HasFDerivAt.eventually_ne (h : HasFDerivAt f f' x) (hf' : ∃ C, ∀ z, ‖z‖ ≤ C * ‖f' z‖) :
    ∀ᶠ z in 𝓝[≠] x, f z ≠ f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x) (hf' : ∃ C, ∀ (z : E), ‖z‖ ≤ C * ‖f' z‖)
Goal: ∀ᶠ (z : E) in 𝓝[≠] x, f z ≠ f x
-- has_fderiv_at_filter_real_equiv
theorem has_fderiv_at_filter_real_equiv {L : Filter E} :
    Tendsto (fun x' : E => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) ↔
      Tendsto (fun x' : E => ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {f' : E →L[ℝ] F} {x : E} {L : Filter E}
Goal: Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) ↔
  Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0)
-- HasFDerivAt.lim_real
theorem HasFDerivAt.lim_real (hf : HasFDerivAt f f' x) (v : E) :
    Tendsto (fun c : ℝ => c • (f (x + c⁻¹ • v) - f x)) atTop (𝓝 (f' v)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {f' : E →L[ℝ] F} {x : E} (hf : HasFDerivAt f f' x) (v : E)
Goal: Filter.Tendsto (fun c => c • (f (x + c⁻¹ • v) - f x)) Filter.atTop (𝓝 (f' v))
-- HasFDerivWithinAt.mapsTo_tangent_cone
theorem HasFDerivWithinAt.mapsTo_tangent_cone {x : E} (h : HasFDerivWithinAt f f' s x) :
    MapsTo f' (tangentConeAt 𝕜 s x) (tangentConeAt 𝕜 (f '' s) (f x)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivWithinAt f f' s x)
Goal: Set.MapsTo (⇑f') (tangentConeAt 𝕜 s x) (tangentConeAt 𝕜 (f '' s) (f x))
-- HasFDerivWithinAt.uniqueDiffWithinAt
theorem HasFDerivWithinAt.uniqueDiffWithinAt {x : E} (h : HasFDerivWithinAt f f' s x)
    (hs : UniqueDiffWithinAt 𝕜 s x) (h' : DenseRange f') : UniqueDiffWithinAt 𝕜 (f '' s) (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivWithinAt f f' s x) (hs : UniqueDiffWithinAt 𝕜 s x) (h' : DenseRange ⇑f')
Goal: UniqueDiffWithinAt 𝕜 (f '' s) (f x)
-- UniqueDiffOn.image
theorem UniqueDiffOn.image {f' : E → E →L[𝕜] F} (hs : UniqueDiffOn 𝕜 s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hd : ∀ x ∈ s, DenseRange (f' x)) :
    UniqueDiffOn 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {f' : E → E →L[𝕜] F} (hs : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hd : ∀ x ∈ s, DenseRange ⇑(f' x))
Goal: UniqueDiffOn 𝕜 (f '' s)
-- HasFDerivWithinAt.uniqueDiffWithinAt_of_continuousLinearEquiv
theorem HasFDerivWithinAt.uniqueDiffWithinAt_of_continuousLinearEquiv {x : E} (e' : E ≃L[𝕜] F)
    (h : HasFDerivWithinAt f (e' : E →L[𝕜] F) s x) (hs : UniqueDiffWithinAt 𝕜 s x) :
    UniqueDiffWithinAt 𝕜 (f '' s) (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {x : E} (e' : E ≃L[𝕜] F) (h : HasFDerivWithinAt f (↑e') s x) (hs : UniqueDiffWithinAt 𝕜 s x)
Goal: UniqueDiffWithinAt 𝕜 (f '' s) (f x)
-- ContinuousLinearEquiv.uniqueDiffOn_image
theorem ContinuousLinearEquiv.uniqueDiffOn_image (e : E ≃L[𝕜] F) (h : UniqueDiffOn 𝕜 s) :
    UniqueDiffOn 𝕜 (e '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} (e : E ≃L[𝕜] F) (h : UniqueDiffOn 𝕜 s)
Goal: UniqueDiffOn 𝕜 (⇑e '' s)
-- ContinuousLinearEquiv.uniqueDiffOn_image_iff
@[simp]
theorem ContinuousLinearEquiv.uniqueDiffOn_image_iff (e : E ≃L[𝕜] F) :
    UniqueDiffOn 𝕜 (e '' s) ↔ UniqueDiffOn 𝕜 s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} (e : E ≃L[𝕜] F)
Goal: UniqueDiffOn 𝕜 (⇑e '' s) ↔ UniqueDiffOn 𝕜 s
-- ContinuousLinearEquiv.uniqueDiffOn_preimage_iff
@[simp]
theorem ContinuousLinearEquiv.uniqueDiffOn_preimage_iff (e : F ≃L[𝕜] E) :
    UniqueDiffOn 𝕜 (e ⁻¹' s) ↔ UniqueDiffOn 𝕜 s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} (e : F ≃L[𝕜] E)
Goal: UniqueDiffOn 𝕜 (⇑e ⁻¹' s) ↔ UniqueDiffOn 𝕜 s
-- UpperHalfPlane.atImInfty_basis
theorem atImInfty_basis : atImInfty.HasBasis (fun _ => True) fun i : ℝ => im ⁻¹' Set.Ici i 
Goal: UpperHalfPlane.atImInfty.HasBasis (fun x => True) fun i => UpperHalfPlane.im ⁻¹' Set.Ici i
-- UpperHalfPlane.atImInfty_mem
theorem atImInfty_mem (S : Set ℍ) : S ∈ atImInfty ↔ ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → z ∈ S 
Arguments: (S : Set ℍ)
Goal: S ∈ UpperHalfPlane.atImInfty ↔ ∃ A, ∀ (z : ℍ), A ≤ z.im → z ∈ S
-- UpperHalfPlane.zero_form_isBoundedAtImInfty
theorem zero_form_isBoundedAtImInfty {α : Type*} [NormedField α] :
    IsBoundedAtImInfty (0 : ℍ → α) 
Arguments: {α : Type u_1} [NormedField α]
Goal: UpperHalfPlane.IsBoundedAtImInfty 0
-- UpperHalfPlane.IsBoundedAtImInfty.mul
nonrec theorem IsBoundedAtImInfty.mul {f g : ℍ → ℂ} (hf : IsBoundedAtImInfty f)
    (hg : IsBoundedAtImInfty g) : IsBoundedAtImInfty (f * g) 
Arguments: {f g : ℍ → ℂ} (hf : UpperHalfPlane.IsBoundedAtImInfty f) (hg : UpperHalfPlane.IsBoundedAtImInfty g)
Goal: UpperHalfPlane.IsBoundedAtImInfty (f * g)
-- UpperHalfPlane.bounded_mem
theorem bounded_mem (f : ℍ → ℂ) :
    IsBoundedAtImInfty f ↔ ∃ M A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ M 
Arguments: (f : ℍ → ℂ)
Goal: UpperHalfPlane.IsBoundedAtImInfty f ↔ ∃ M A, ∀ (z : ℍ), A ≤ z.im → Complex.abs (f z) ≤ M
-- UpperHalfPlane.zero_at_im_infty
theorem zero_at_im_infty (f : ℍ → ℂ) :
    IsZeroAtImInfty f ↔ ∀ ε : ℝ, 0 < ε → ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ ε 
Arguments: (f : ℍ → ℂ)
Goal: UpperHalfPlane.IsZeroAtImInfty f ↔ ∀ (ε : ℝ), 0 < ε → ∃ A, ∀ (z : ℍ), A ≤ z.im → Complex.abs (f z) ≤ ε
-- UpperHalfPlane.openEmbedding_coe
theorem openEmbedding_coe : OpenEmbedding ((↑) : ℍ → ℂ) 
Goal: OpenEmbedding UpperHalfPlane.coe
-- UpperHalfPlane.embedding_coe
theorem embedding_coe : Embedding ((↑) : ℍ → ℂ) 
Goal: Embedding UpperHalfPlane.coe
-- UpperHalfPlane.continuous_coe
theorem continuous_coe : Continuous ((↑) : ℍ → ℂ) 
Goal: Continuous UpperHalfPlane.coe
-- UpperHalfPlane.continuous_re
theorem continuous_re : Continuous re 
Goal: Continuous UpperHalfPlane.re
-- UpperHalfPlane.continuous_im
theorem continuous_im : Continuous im 
Goal: Continuous UpperHalfPlane.im
-- UpperHalfPlane.mem_verticalStrip_iff
theorem mem_verticalStrip_iff (A B : ℝ) (z : ℍ) : z ∈ verticalStrip A B ↔ |z.re| ≤ A ∧ B ≤ z.im 
Arguments: (A B : ℝ) (z : ℍ)
Goal: z ∈ UpperHalfPlane.verticalStrip A B ↔ |z.re| ≤ A ∧ B ≤ z.im
-- UpperHalfPlane.verticalStrip_mono
@[gcongr]
lemma verticalStrip_mono {A B A' B' : ℝ} (hA : A ≤ A') (hB : B' ≤ B) :
    verticalStrip A B ⊆ verticalStrip A' B' 
Arguments: {A B A' B' : ℝ} (hA : A ≤ A') (hB : B' ≤ B)
Goal: UpperHalfPlane.verticalStrip A B ⊆ UpperHalfPlane.verticalStrip A' B'
-- UpperHalfPlane.verticalStrip_mono_left
@[gcongr]
lemma verticalStrip_mono_left {A A'} (h : A ≤ A') (B) : verticalStrip A B ⊆ verticalStrip A' B 
Arguments: {A A' : ℝ} (h : A ≤ A') (B : ℝ)
Goal: UpperHalfPlane.verticalStrip A B ⊆ UpperHalfPlane.verticalStrip A' B
-- UpperHalfPlane.verticalStrip_anti_right
@[gcongr]
lemma verticalStrip_anti_right (A) {B B'} (h : B' ≤ B) : verticalStrip A B ⊆ verticalStrip A B' 
Arguments: (A : ℝ) {B B' : ℝ} (h : B' ≤ B)
Goal: UpperHalfPlane.verticalStrip A B ⊆ UpperHalfPlane.verticalStrip A B'
-- UpperHalfPlane.subset_verticalStrip_of_isCompact
lemma subset_verticalStrip_of_isCompact {K : Set ℍ} (hK : IsCompact K) :
    ∃ A B : ℝ, 0 < B ∧ K ⊆ verticalStrip A B 
Arguments: {K : Set ℍ} (hK : IsCompact K)
Goal: ∃ A B, 0 < B ∧ K ⊆ UpperHalfPlane.verticalStrip A B
-- UpperHalfPlane.ModularGroup_T_zpow_mem_verticalStrip
theorem ModularGroup_T_zpow_mem_verticalStrip (z : ℍ) {N : ℕ} (hn : 0 < N) :
    ∃ n : ℤ, ModularGroup.T ^ (N * n) • z ∈ verticalStrip N z.im 
Arguments: (z : ℍ) {N : ℕ} (hn : 0 < N)
Goal: ∃ n, ModularGroup.T ^ (↑N * n) • z ∈ UpperHalfPlane.verticalStrip (↑N) z.im
-- UpperHalfPlane.ofComplex_apply
lemma ofComplex_apply (z : ℍ) : ofComplex (z : ℂ) = z 
Arguments: (z : ℍ)
Goal: ↑UpperHalfPlane.ofComplex ↑z = z
-- UpperHalfPlane.comp_ofComplex
lemma comp_ofComplex (f : ℍ → ℂ) (z : ℍ) : (↑ₕ f) z = f z 
Arguments: (f : ℍ → ℂ) (z : ℍ)
Goal: (f ∘ ↑UpperHalfPlane.ofComplex) ↑z = f z
-- rotation_apply
@[simp]
theorem rotation_apply (a : circle) (z : ℂ) : rotation a z = a * z 
Arguments: (a : ↥circle) (z : ℂ)
Goal: (rotation a) z = ↑a * z
-- rotation_symm
@[simp]
theorem rotation_symm (a : circle) : (rotation a).symm = rotation a⁻¹ 
Arguments: (a : ↥circle)
Goal: (rotation a).symm = rotation a⁻¹
-- rotation_trans
@[simp]
theorem rotation_trans (a b : circle) : (rotation a).trans (rotation b) = rotation (b * a) 
Arguments: (a b : ↥circle)
Goal: (rotation a).trans (rotation b) = rotation (b * a)
-- rotation_ne_conjLIE
theorem rotation_ne_conjLIE (a : circle) : rotation a ≠ conjLIE 
Arguments: (a : ↥circle)
Goal: rotation a ≠ Complex.conjLIE
-- rotationOf_coe
@[simps]
def rotationOf (e : ℂ ≃ₗᵢ[ℝ] ℂ) : circle 
Arguments: (e : ℂ ≃ₗᵢ[ℝ] ℂ)
Goal: ↑(rotationOf e) = e 1 / ↑(Complex.abs (e 1))
-- rotationOf_rotation
@[simp]
theorem rotationOf_rotation (a : circle) : rotationOf (rotation a) = a 
Arguments: (a : ↥circle)
Goal: rotationOf (rotation a) = a
-- rotation_injective
theorem rotation_injective : Function.Injective rotation 
Goal: Function.Injective ⇑rotation
-- LinearIsometry.re_apply_eq_re_of_add_conj_eq
theorem LinearIsometry.re_apply_eq_re_of_add_conj_eq (f : ℂ →ₗᵢ[ℝ] ℂ)
    (h₃ : ∀ z, z + conj z = f z + conj (f z)) (z : ℂ) : (f z).re = z.re 
Arguments: (f : ℂ →ₗᵢ[ℝ] ℂ) (h₃ : ∀ (z : ℂ), z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)) (z : ℂ)
Goal: (f z).re = z.re
-- LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re
theorem LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ}
    (h₂ : ∀ z, (f z).re = z.re) (z : ℂ) : (f z).im = z.im ∨ (f z).im = -z.im 
Arguments: {f : ℂ →ₗᵢ[ℝ] ℂ} (h₂ : ∀ (z : ℂ), (f z).re = z.re) (z : ℂ)
Goal: (f z).im = z.im ∨ (f z).im = -z.im
-- LinearIsometry.im_apply_eq_im
theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
    z + conj z = f z + conj (f z) 
Arguments: {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ)
Goal: z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)
-- LinearIsometry.re_apply_eq_re
theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re 
Arguments: {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ)
Goal: (f z).re = z.re
-- linear_isometry_complex_aux
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLIE 
Arguments: {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1)
Goal: f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = Complex.conjLIE
-- linear_isometry_complex
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLIE.trans (rotation a) 
Arguments: (f : ℂ ≃ₗᵢ[ℝ] ℂ)
Goal: ∃ a, f = rotation a ∨ f = Complex.conjLIE.trans (rotation a)
-- toMatrix_rotation
theorem toMatrix_rotation (a : circle) :
    LinearMap.toMatrix basisOneI basisOneI (rotation a).toLinearEquiv =
      Matrix.planeConformalMatrix (re a) (im a) (by simp [pow_two, ← normSq_apply]) 
Arguments: (a : ↥circle)
Goal: (LinearMap.toMatrix Complex.basisOneI Complex.basisOneI) ↑(rotation a).toLinearEquiv =
  ↑(Matrix.planeConformalMatrix (↑a).re (↑a).im ⋯)
-- det_rotation
@[simp]
theorem det_rotation (a : circle) : LinearMap.det ((rotation a).toLinearEquiv : ℂ →ₗ[ℝ] ℂ) = 1 
Arguments: (a : ↥circle)
Goal: LinearMap.det ↑(rotation a).toLinearEquiv = 1
-- linearEquiv_det_rotation
@[simp]
theorem linearEquiv_det_rotation (a : circle) : LinearEquiv.det (rotation a).toLinearEquiv = 1 
Arguments: (a : ↥circle)
Goal: LinearEquiv.det (rotation a).toLinearEquiv = 1
-- Complex.UnitDisc.coe_injective
theorem coe_injective : Injective ((↑) : 𝔻 → ℂ) 
Goal: Function.Injective Subtype.val
-- Complex.UnitDisc.abs_lt_one
theorem abs_lt_one (z : 𝔻) : abs (z : ℂ) < 1 
Arguments: (z : Complex.UnitDisc)
Goal: Complex.abs ↑z < 1
-- Complex.UnitDisc.abs_ne_one
theorem abs_ne_one (z : 𝔻) : abs (z : ℂ) ≠ 1 
Arguments: (z : Complex.UnitDisc)
Goal: Complex.abs ↑z ≠ 1
-- Complex.UnitDisc.normSq_lt_one
theorem normSq_lt_one (z : 𝔻) : normSq z < 1 
Arguments: (z : Complex.UnitDisc)
Goal: Complex.normSq ↑z < 1