ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasFDerivAt_integral_of_dominated_loc_of_lip'
theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε)
    (hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
    (hF'_meas : AEStronglyMeasurable F' μ)
    (h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
    (bound_integrable : Integrable (bound : α → ℝ) μ)
    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a) x₀) :
    Integrable F' μ ∧ HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_4} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ} {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε) (hF_meas : ∀ x ∈ Metric.ball x₀ ε, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' μ) (h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀)
Goal: MeasureTheory.Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
-- hasFDerivAt_integral_of_dominated_loc_of_lip
theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F' : α → H →L[𝕜] E}
    (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
    (hF_int : Integrable (F x₀) μ) (hF'_meas : AEStronglyMeasurable F' μ)
    (h_lip : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
    (bound_integrable : Integrable (bound : α → ℝ) μ)
    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a) x₀) :
    Integrable F' μ ∧ HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_4} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ} {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' μ) (h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (Metric.ball x₀ ε)) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀)
Goal: MeasureTheory.Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
-- hasFDerivAt_integral_of_dominated_loc_of_lip_interval
theorem hasFDerivAt_integral_of_dominated_loc_of_lip_interval [NormedSpace ℝ H] {μ : Measure ℝ}
    {F : H → ℝ → E} {F' : ℝ → H →L[ℝ] E} {a b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε)
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) <| μ.restrict (Ι a b))
    (hF_int : IntervalIntegrable (F x₀) μ a b)
    (hF'_meas : AEStronglyMeasurable F' <| μ.restrict (Ι a b))
    (h_lip : ∀ᵐ t ∂μ.restrict (Ι a b),
      LipschitzOnWith (Real.nnabs <| bound t) (F · t) (ball x₀ ε))
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_diff : ∀ᵐ t ∂μ.restrict (Ι a b), HasFDerivAt (F · t) (F' t) x₀) :
    IntervalIntegrable F' μ a b ∧
      HasFDerivAt (fun x ↦ ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' t ∂μ) x₀ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_4} [NormedAddCommGroup H] {x₀ : H} {ε : ℝ} [NormedSpace ℝ H] {μ : MeasureTheory.Measure ℝ} {F : H → ℝ → E} {F' : ℝ → H →L[ℝ] E} {a b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (hF_int : IntervalIntegrable (F x₀) μ a b) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' (μ.restrict (Ι a b))) (h_lip : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound t)) (fun x => F x t) (Metric.ball x₀ ε)) (bound_integrable : IntervalIntegrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun x => F x t) (F' t) x₀)
Goal: IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
-- hasFDerivAt_integral_of_dominated_of_fderiv_le
theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F' : H → α → H →L[𝕜] E} (ε_pos : 0 < ε)
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
    (hF'_meas : AEStronglyMeasurable (F' x₀) μ)
    (h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a)
    (bound_integrable : Integrable (bound : α → ℝ) μ)
    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (F · a) (F' x a) x) :
    HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_4} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ} {F' : H → α → H →L[𝕜] E} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable (F' x₀) μ) (h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, HasFDerivAt (fun x => F x a) (F' x a) x)
Goal: HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀
-- hasFDerivAt_integral_of_dominated_of_fderiv_le''
theorem hasFDerivAt_integral_of_dominated_of_fderiv_le'' [NormedSpace ℝ H] {μ : Measure ℝ}
    {F : H → ℝ → E} {F' : H → ℝ → H →L[ℝ] E} {a b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε)
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) <| μ.restrict (Ι a b))
    (hF_int : IntervalIntegrable (F x₀) μ a b)
    (hF'_meas : AEStronglyMeasurable (F' x₀) <| μ.restrict (Ι a b))
    (h_bound : ∀ᵐ t ∂μ.restrict (Ι a b), ∀ x ∈ ball x₀ ε, ‖F' x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_diff : ∀ᵐ t ∂μ.restrict (Ι a b), ∀ x ∈ ball x₀ ε, HasFDerivAt (F · t) (F' x t) x) :
    HasFDerivAt (fun x ↦ ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' x₀ t ∂μ) x₀ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_4} [NormedAddCommGroup H] {x₀ : H} {ε : ℝ} [NormedSpace ℝ H] {μ : MeasureTheory.Measure ℝ} {F : H → ℝ → E} {F' : H → ℝ → H →L[ℝ] E} {a b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (hF_int : IntervalIntegrable (F x₀) μ a b) (hF'_meas : MeasureTheory.AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))) (h_bound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ Metric.ball x₀ ε, ‖F' x t‖ ≤ bound t) (bound_integrable : IntervalIntegrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ Metric.ball x₀ ε, HasFDerivAt (fun x => F x t) (F' x t) x)
Goal: HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀
-- hasDerivAt_integral_of_dominated_loc_of_lip
theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0 < ε)
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
    (hF'_meas : AEStronglyMeasurable F' μ)
    (h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
    (bound_integrable : Integrable (bound : α → ℝ) μ)
    (h_diff : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' a) x₀) :
    Integrable F' μ ∧ HasDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {bound : α → ℝ} {ε : ℝ} {F : 𝕜 → α → E} {x₀ : 𝕜} {F' : α → E} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' μ) (h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (Metric.ball x₀ ε)) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' a) x₀)
Goal: MeasureTheory.Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
-- hasDerivAt_integral_of_dominated_loc_of_deriv_le
theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le (ε_pos : 0 < ε)
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
    {F' : 𝕜 → α → E} (hF'_meas : AEStronglyMeasurable (F' x₀) μ)
    (h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasDerivAt (F · a) (F' x a) x) :
    Integrable (F' x₀) μ ∧ HasDerivAt (fun n ↦ ∫ a, F n a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {bound : α → ℝ} {ε : ℝ} {F : 𝕜 → α → E} {x₀ : 𝕜} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) {F' : 𝕜 → α → E} (hF'_meas : MeasureTheory.AEStronglyMeasurable (F' x₀) μ) (h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, HasDerivAt (fun x => F x a) (F' x a) x)
Goal: MeasureTheory.Integrable (F' x₀) μ ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀
-- HasStrictFDerivAt.approximates_deriv_on_nhds
theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
    (hf : HasStrictFDerivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c) :
    ∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c 
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} {a : E} (hf : HasStrictFDerivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c)
Goal: ∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c
-- HasStrictFDerivAt.map_nhds_eq_of_surj
theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E}
    (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) (h : LinearMap.range f' = ⊤) :
    map f (𝓝 a) = 𝓝 (f 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] [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (h : LinearMap.range f' = ⊤)
Goal: Filter.map f (𝓝 a) = 𝓝 (f a)
-- HasStrictFDerivAt.approximates_deriv_on_open_nhds
theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    ∃ s : Set E, a ∈ s ∧ IsOpen s ∧
      ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) 
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} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: ∃ s, a ∈ s ∧ IsOpen s ∧ ApproximatesLinearOn f (↑f') s (‖↑f'.symm‖₊⁻¹ / 2)
-- HasStrictFDerivAt.toPartialHomeomorph_coe
@[simp]
theorem toPartialHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    (hf.toPartialHomeomorph f : E → F) = 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: ↑(HasStrictFDerivAt.toPartialHomeomorph f hf) = f
-- HasStrictFDerivAt.mem_toPartialHomeomorph_source
theorem mem_toPartialHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    a ∈ (hf.toPartialHomeomorph f).source 
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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: a ∈ (HasStrictFDerivAt.toPartialHomeomorph f hf).source
-- HasStrictFDerivAt.image_mem_toPartialHomeomorph_target
theorem image_mem_toPartialHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    f a ∈ (hf.toPartialHomeomorph f).target 
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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: f a ∈ (HasStrictFDerivAt.toPartialHomeomorph f hf).target
-- HasStrictFDerivAt.map_nhds_eq_of_equiv
theorem map_nhds_eq_of_equiv (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    map f (𝓝 a) = 𝓝 (f 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: Filter.map f (𝓝 a) = 𝓝 (f a)
-- HasStrictFDerivAt.localInverse_def
theorem localInverse_def (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    hf.localInverse f _ _ = (hf.toPartialHomeomorph f).symm 
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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: HasStrictFDerivAt.localInverse f f' a hf = ↑(HasStrictFDerivAt.toPartialHomeomorph f hf).symm
-- HasStrictFDerivAt.eventually_left_inverse
theorem eventually_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: ∀ᶠ (x : E) in 𝓝 a, HasStrictFDerivAt.localInverse f f' a hf (f x) = x
-- HasStrictFDerivAt.localInverse_apply_image
@[simp]
theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    hf.localInverse f f' a (f a) = 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: HasStrictFDerivAt.localInverse f f' a hf (f a) = a
-- HasStrictFDerivAt.eventually_right_inverse
theorem eventually_right_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    ∀ᶠ y in 𝓝 (f a), f (hf.localInverse f f' a y) = y 
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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: ∀ᶠ (y : F) in 𝓝 (f a), f (HasStrictFDerivAt.localInverse f f' a hf y) = y
-- HasStrictFDerivAt.localInverse_continuousAt
theorem localInverse_continuousAt (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    ContinuousAt (hf.localInverse f f' a) (f 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: ContinuousAt (HasStrictFDerivAt.localInverse f f' a hf) (f a)
-- HasStrictFDerivAt.localInverse_tendsto
theorem localInverse_tendsto (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    Tendsto (hf.localInverse f f' a) (𝓝 <| f a) (𝓝 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: Filter.Tendsto (HasStrictFDerivAt.localInverse f f' a hf) (𝓝 (f a)) (𝓝 a)
-- HasStrictFDerivAt.localInverse_unique
theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
    (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : ∀ᶠ y in 𝓝 (f a), g y = localInverse f f' a hf y 
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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a) {g : F → E} (hg : ∀ᶠ (x : E) in 𝓝 a, g (f x) = x)
Goal: ∀ᶠ (y : F) in 𝓝 (f a), g y = HasStrictFDerivAt.localInverse f f' a hf y
-- HasStrictFDerivAt.to_localInverse
theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    HasStrictFDerivAt (hf.localInverse f f' a) (f'.symm : F →L[𝕜] E) (f 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a)
Goal: HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (↑f'.symm) (f a)
-- HasStrictFDerivAt.to_local_left_inverse
theorem to_local_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
    (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) (f 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] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (↑f') a) {g : F → E} (hg : ∀ᶠ (x : E) in 𝓝 a, g (f x) = x)
Goal: HasStrictFDerivAt g (↑f'.symm) (f a)
-- isOpenMap_of_hasStrictFDerivAt_equiv
theorem isOpenMap_of_hasStrictFDerivAt_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
    (hf : ∀ x, HasStrictFDerivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap 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] [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F} (hf : ∀ (x : E), HasStrictFDerivAt f (↑(f' x)) x)
Goal: IsOpenMap f
-- IsBoundedBilinearMap.hasStrictFDerivAt
@[fun_prop]
theorem IsBoundedBilinearMap.hasStrictFDerivAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    HasStrictFDerivAt b (h.deriv p) p 
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] {b : E × F → G} (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
Goal: HasStrictFDerivAt b (h.deriv p) p
-- IsBoundedBilinearMap.hasFDerivAt
@[fun_prop]
theorem IsBoundedBilinearMap.hasFDerivAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    HasFDerivAt b (h.deriv p) p 
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] {b : E × F → G} (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
Goal: HasFDerivAt b (h.deriv p) p
-- IsBoundedBilinearMap.hasFDerivWithinAt
@[fun_prop]
theorem IsBoundedBilinearMap.hasFDerivWithinAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    HasFDerivWithinAt b (h.deriv p) u p 
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] {b : E × F → G} {u : Set (E × F)} (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
Goal: HasFDerivWithinAt b (h.deriv p) u p
-- IsBoundedBilinearMap.differentiableAt
@[fun_prop]
theorem IsBoundedBilinearMap.differentiableAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    DifferentiableAt 𝕜 b p 
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] {b : E × F → G} (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
Goal: DifferentiableAt 𝕜 b p
-- IsBoundedBilinearMap.differentiableWithinAt
@[fun_prop]
theorem IsBoundedBilinearMap.differentiableWithinAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    DifferentiableWithinAt 𝕜 b u p 
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] {b : E × F → G} {u : Set (E × F)} (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
Goal: DifferentiableWithinAt 𝕜 b u p
-- IsBoundedBilinearMap.fderiv
protected theorem IsBoundedBilinearMap.fderiv (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    fderiv 𝕜 b p = h.deriv p 
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] {b : E × F → G} (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
Goal: fderiv 𝕜 b p = h.deriv p
-- IsBoundedBilinearMap.fderivWithin
protected theorem IsBoundedBilinearMap.fderivWithin (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
    (hxs : UniqueDiffWithinAt 𝕜 u p) : fderivWithin 𝕜 b u p = h.deriv p 
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] {b : E × F → G} {u : Set (E × F)} (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) (hxs : UniqueDiffWithinAt 𝕜 u p)
Goal: fderivWithin 𝕜 b u p = h.deriv p
-- IsBoundedBilinearMap.differentiable
@[fun_prop]
theorem IsBoundedBilinearMap.differentiable (h : IsBoundedBilinearMap 𝕜 b) : Differentiable 𝕜 b 
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] {b : E × F → G} (h : IsBoundedBilinearMap 𝕜 b)
Goal: Differentiable 𝕜 b
-- IsBoundedBilinearMap.differentiableOn
@[fun_prop]
theorem IsBoundedBilinearMap.differentiableOn (h : IsBoundedBilinearMap 𝕜 b) :
    DifferentiableOn 𝕜 b u 
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] {b : E × F → G} {u : Set (E × F)} (h : IsBoundedBilinearMap 𝕜 b)
Goal: DifferentiableOn 𝕜 b u
-- ContinuousLinearMap.hasFDerivWithinAt_of_bilinear
@[fun_prop]
theorem ContinuousLinearMap.hasFDerivWithinAt_of_bilinear {f : G' → E} {g : G' → F}
    {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F} {x : G'} {s : Set G'} (hf : HasFDerivWithinAt f f' s x)
    (hg : HasFDerivWithinAt g g' s x) :
    HasFDerivWithinAt (fun y => B (f y) (g y))
      (B.precompR G' (f x) g' + B.precompL G' f' (g x)) 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] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (B : E →L[𝕜] F →L[𝕜] G) {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F} {x : G'} {s : Set G'} (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x)
Goal: HasFDerivWithinAt (fun y => (B (f y)) (g y))
  (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) s x