ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply
theorem implicitToPartialHomeomorphOfComplemented_apply (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) (y : E) :
    hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker y =
      (f y, Classical.choose hker (y - a)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented) (y : E)
Goal: ↑(HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented f f' hf hf' hker) y =
  (f y, (Classical.choose hker) (y - a))
-- HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply_ker
@[simp]
theorem implicitToPartialHomeomorphOfComplemented_apply_ker (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) (y : ker f') :
    hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker (y + a) = (f (y + a), y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented) (y : ↥(LinearMap.ker f'))
Goal: ↑(HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented f f' hf hf' hker) (↑y + a) = (f (↑y + a), y)
-- HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_self
@[simp]
theorem implicitToPartialHomeomorphOfComplemented_self (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) :
    hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker a = (f a, 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented)
Goal: ↑(HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented f f' hf hf' hker) a = (f a, 0)
-- HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_source
theorem mem_implicitToPartialHomeomorphOfComplemented_source (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) :
    a ∈ (hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker).source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented)
Goal: a ∈ (HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented f f' hf hf' hker).source
-- HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_target
theorem mem_implicitToPartialHomeomorphOfComplemented_target (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) :
    (f a, (0 : ker f')) ∈ (hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker).target 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented)
Goal: (f a, 0) ∈ (HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented f f' hf hf' hker).target
-- HasStrictFDerivAt.map_implicitFunctionOfComplemented_eq
theorem map_implicitFunctionOfComplemented_eq (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤)
    (hker : (ker f').ClosedComplemented) :
    ∀ᶠ p : F × ker f' in 𝓝 (f a, 0),
      f (hf.implicitFunctionOfComplemented f f' hf' hker p.1 p.2) = p.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented)
Goal: ∀ᶠ (p : F × ↥(LinearMap.ker f')) in 𝓝 (f a, 0),
  f (HasStrictFDerivAt.implicitFunctionOfComplemented f f' hf hf' hker p.1 p.2) = p.1
-- HasStrictFDerivAt.eq_implicitFunctionOfComplemented
theorem eq_implicitFunctionOfComplemented (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤)
    (hker : (ker f').ClosedComplemented) :
    ∀ᶠ x in 𝓝 a, hf.implicitFunctionOfComplemented f f' hf' hker (f x)
      (hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker x).snd = x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented)
Goal: ∀ᶠ (x : E) in 𝓝 a,
  HasStrictFDerivAt.implicitFunctionOfComplemented f f' hf hf' hker (f x)
      (↑(HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented f f' hf hf' hker) x).2 =
    x
-- HasStrictFDerivAt.implicitFunctionOfComplemented_apply_image
@[simp]
theorem implicitFunctionOfComplemented_apply_image (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) :
    hf.implicitFunctionOfComplemented f f' hf' hker (f a) 0 = a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented)
Goal: HasStrictFDerivAt.implicitFunctionOfComplemented f f' hf hf' hker (f a) 0 = a
-- HasStrictFDerivAt.to_implicitFunctionOfComplemented
theorem to_implicitFunctionOfComplemented (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤)
    (hker : (ker f').ClosedComplemented) :
    HasStrictFDerivAt (hf.implicitFunctionOfComplemented f f' hf' hker (f a))
      (ker f').subtypeL 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented)
Goal: HasStrictFDerivAt (HasStrictFDerivAt.implicitFunctionOfComplemented f f' hf hf' hker (f a)) (LinearMap.ker f').subtypeL
  0
-- HasStrictFDerivAt.implicitToPartialHomeomorph_fst
@[simp]
theorem implicitToPartialHomeomorph_fst (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤)
    (x : E) : (hf.implicitToPartialHomeomorph f f' hf' x).fst = f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (x : E)
Goal: (↑(HasStrictFDerivAt.implicitToPartialHomeomorph f f' hf hf') x).1 = f x
-- HasStrictFDerivAt.implicitToPartialHomeomorph_apply_ker
@[simp]
theorem implicitToPartialHomeomorph_apply_ker (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤)
    (y : ker f') : hf.implicitToPartialHomeomorph f f' hf' (y + a) = (f (y + a), y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (y : ↥(LinearMap.ker f'))
Goal: ↑(HasStrictFDerivAt.implicitToPartialHomeomorph f f' hf hf') (↑y + a) = (f (↑y + a), y)
-- HasStrictFDerivAt.implicitToPartialHomeomorph_self
@[simp]
theorem implicitToPartialHomeomorph_self (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤) :
    hf.implicitToPartialHomeomorph f f' hf' a = (f a, 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤)
Goal: ↑(HasStrictFDerivAt.implicitToPartialHomeomorph f f' hf hf') a = (f a, 0)
-- HasStrictFDerivAt.mem_implicitToPartialHomeomorph_source
theorem mem_implicitToPartialHomeomorph_source (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) : a ∈ (hf.implicitToPartialHomeomorph f f' hf').source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤)
Goal: a ∈ (HasStrictFDerivAt.implicitToPartialHomeomorph f f' hf hf').source
-- HasStrictFDerivAt.mem_implicitToPartialHomeomorph_target
theorem mem_implicitToPartialHomeomorph_target (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) : (f a, (0 : ker f')) ∈ (hf.implicitToPartialHomeomorph f f' hf').target 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤)
Goal: (f a, 0) ∈ (HasStrictFDerivAt.implicitToPartialHomeomorph f f' hf hf').target
-- HasStrictFDerivAt.tendsto_implicitFunction
theorem tendsto_implicitFunction (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤) {α : Type*}
    {l : Filter α} {g₁ : α → F} {g₂ : α → ker f'} (h₁ : Tendsto g₁ l (𝓝 <| f a))
    (h₂ : Tendsto g₂ l (𝓝 0)) :
    Tendsto (fun t => hf.implicitFunction f f' hf' (g₁ t) (g₂ t)) l (𝓝 a) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) {α : Type u_4} {l : Filter α} {g₁ : α → F} {g₂ : α → ↥(LinearMap.ker f')} (h₁ : Filter.Tendsto g₁ l (𝓝 (f a))) (h₂ : Filter.Tendsto g₂ l (𝓝 0))
Goal: Filter.Tendsto (fun t => HasStrictFDerivAt.implicitFunction f f' hf hf' (g₁ t) (g₂ t)) l (𝓝 a)
-- HasStrictFDerivAt.map_implicitFunction_eq
theorem map_implicitFunction_eq (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤) :
    ∀ᶠ p : F × ker f' in 𝓝 (f a, 0), f (hf.implicitFunction f f' hf' p.1 p.2) = p.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤)
Goal: ∀ᶠ (p : F × ↥(LinearMap.ker f')) in 𝓝 (f a, 0), f (HasStrictFDerivAt.implicitFunction f f' hf hf' p.1 p.2) = p.1
-- HasStrictFDerivAt.implicitFunction_apply_image
@[simp]
theorem implicitFunction_apply_image (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤) :
    hf.implicitFunction f f' hf' (f a) 0 = a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤)
Goal: HasStrictFDerivAt.implicitFunction f f' hf hf' (f a) 0 = a
-- HasStrictFDerivAt.eq_implicitFunction
theorem eq_implicitFunction (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤) :
    ∀ᶠ x in 𝓝 a,
      hf.implicitFunction f f' hf' (f x) (hf.implicitToPartialHomeomorph f f' hf' x).snd = x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤)
Goal: ∀ᶠ (x : E) in 𝓝 a,
  HasStrictFDerivAt.implicitFunction f f' hf hf' (f x)
      (↑(HasStrictFDerivAt.implicitToPartialHomeomorph f f' hf hf') x).2 =
    x
-- HasStrictFDerivAt.to_implicitFunction
theorem to_implicitFunction (hf : HasStrictFDerivAt f f' a) (hf' : range f' = ⊤) :
    HasStrictFDerivAt (hf.implicitFunction f f' hf' (f a)) (ker f').subtypeL 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤)
Goal: HasStrictFDerivAt (HasStrictFDerivAt.implicitFunction f f' hf hf' (f a)) (LinearMap.ker f').subtypeL 0
-- intervalIntegral.hasFDerivAt_integral_of_dominated_loc_of_lip
nonrec theorem hasFDerivAt_integral_of_dominated_loc_of_lip
    {F : H → ℝ → E} {F' : ℝ → H →L[𝕜] E} {x₀ : H}
    (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, t ∈ Ι a b →
      LipschitzOnWith (Real.nnabs <| bound t) (fun x => F x t) (ball x₀ ε))
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_diff : ∀ᵐ t ∂μ, t ∈ Ι a b → HasFDerivAt (fun x => F x 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: {𝕜 : Type u_1} [RCLike 𝕜] {μ : MeasureTheory.Measure ℝ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_3} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {a b ε : ℝ} {bound : ℝ → ℝ} {F : H → ℝ → E} {F' : ℝ → H →L[𝕜] E} {x₀ : H} (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, t ∈ 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, t ∈ 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
-- intervalIntegral.hasFDerivAt_integral_of_dominated_of_fderiv_le
nonrec theorem hasFDerivAt_integral_of_dominated_of_fderiv_le
    {F : H → ℝ → E} {F' : H → ℝ → H →L[𝕜] E} {x₀ : H} (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, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, ‖F' x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_diff : ∀ᵐ t ∂μ, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x 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: {𝕜 : Type u_1} [RCLike 𝕜] {μ : MeasureTheory.Measure ℝ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_3} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {a b ε : ℝ} {bound : ℝ → ℝ} {F : H → ℝ → E} {F' : H → ℝ → H →L[𝕜] E} {x₀ : H} (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, t ∈ 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, t ∈ 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
-- intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_lip
nonrec theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → ℝ → E} {F' : ℝ → E} {x₀ : 𝕜}
    (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_lipsch : ∀ᵐ t ∂μ, t ∈ Ι a b →
      LipschitzOnWith (Real.nnabs <| bound t) (fun x => F x t) (ball x₀ ε))
    (bound_integrable : IntervalIntegrable (bound : ℝ → ℝ) μ a b)
    (h_diff : ∀ᵐ t ∂μ, t ∈ Ι a b → HasDerivAt (fun x => F x t) (F' t) x₀) :
    IntervalIntegrable F' μ a b ∧
      HasDerivAt (fun x => ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' t ∂μ) x₀ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {μ : MeasureTheory.Measure ℝ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {a b ε : ℝ} {bound : ℝ → ℝ} {F : 𝕜 → ℝ → E} {F' : ℝ → E} {x₀ : 𝕜} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) 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_lipsch : ∀ᵐ (t : ℝ) ∂μ, t ∈ 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, t ∈ Ι a b → HasDerivAt (fun x => F x t) (F' t) x₀)
Goal: IntervalIntegrable F' μ a b ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
-- intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
nonrec theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le
    {F : 𝕜 → ℝ → E} {F' : 𝕜 → ℝ → E} {x₀ : 𝕜}
    (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, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, ‖F' x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_diff : ∀ᵐ t ∂μ, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, HasDerivAt (fun x => F x t) (F' x t) x) :
    IntervalIntegrable (F' x₀) μ a b ∧
      HasDerivAt (fun x => ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' x₀ t ∂μ) x₀ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {μ : MeasureTheory.Measure ℝ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {a b ε : ℝ} {bound : ℝ → ℝ} {F F' : 𝕜 → ℝ → E} {x₀ : 𝕜} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) 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, t ∈ 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, t ∈ Ι a b → ∀ x ∈ Metric.ball x₀ ε, HasDerivAt (fun x => F x t) (F' x t) x)
Goal: IntervalIntegrable (F' x₀) μ a b ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀
-- LinearIsometry.norm_map'
structure LinearIsometry (σ₁₂ : R →+* R₂) (E E₂ : Type*) [SeminormedAddCommGroup E]
  [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] extends E →ₛₗ[σ₁₂] E₂ where
  norm_map' : ∀ x, ‖toLinearMap x‖ = ‖x‖
Arguments: {R : Type u_1} {R₂ : Type u_2} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {E : Type u_11} {E₂ : Type u_12} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (self : E →ₛₗᵢ[σ₁₂] E₂) (x : E)
Goal: ‖self.toLinearMap x‖ = ‖x‖
-- SemilinearIsometryClass.norm_map
class SemilinearIsometryClass (𝓕 : Type*) {R R₂ : outParam Type*} [Semiring R] [Semiring R₂]
  (σ₁₂ : outParam <| R →+* R₂) (E E₂ : outParam Type*) [SeminormedAddCommGroup E]
  [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] extends
  SemilinearMapClass 𝓕 σ₁₂ E E₂ : Prop where
  norm_map : ∀ (f : 𝓕) (x : E), ‖f x‖ = ‖x‖
Arguments: {𝓕 : Type u_11} {R : outParam (Type u_12)} {R₂ : outParam (Type u_13)} [Semiring R] [Semiring R₂] {σ₁₂ : outParam (R →+* R₂)} {E : outParam (Type u_14)} {E₂ : outParam (Type u_15)} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [self : SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (x : E)
Goal: ‖f x‖ = ‖x‖
-- SemilinearIsometryClass.isometry
protected theorem isometry [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) : Isometry f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕)
Goal: Isometry ⇑f
-- SemilinearIsometryClass.continuous
@[continuity]
protected theorem continuous [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) : Continuous f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕)
Goal: Continuous ⇑f
-- SemilinearIsometryClass.nnnorm_map
theorem nnnorm_map [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (x : E) : ‖f x‖₊ = ‖x‖₊ 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (x : E)
Goal: ‖f x‖₊ = ‖x‖₊
-- SemilinearIsometryClass.lipschitz
protected theorem lipschitz [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) : LipschitzWith 1 f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕)
Goal: LipschitzWith 1 ⇑f
-- SemilinearIsometryClass.antilipschitz
protected theorem antilipschitz [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :
    AntilipschitzWith 1 f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕)
Goal: AntilipschitzWith 1 ⇑f
-- SemilinearIsometryClass.ediam_image
theorem ediam_image [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : Set E) :
    EMetric.diam (f '' s) = EMetric.diam s 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : Set E)
Goal: EMetric.diam (⇑f '' s) = EMetric.diam s
-- SemilinearIsometryClass.ediam_range
theorem ediam_range [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :
    EMetric.diam (range f) = EMetric.diam (univ : Set E) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕)
Goal: EMetric.diam (Set.range ⇑f) = EMetric.diam Set.univ