ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Idempotents.DoldKan.η_hom_app_f
@[simps!]
def η : Γ ⋙ N ≅ 𝟭 (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : ChainComplex C ℕ) (i : ℕ)
Goal: (CategoryTheory.Idempotents.DoldKan.η.hom.app X).f i =
  ((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex C ℕ)).inverse.map
          (AlgebraicTopology.DoldKan.N₁Γ₀.hom.app X)).f
      i ≫
    ((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex C ℕ)).unitIso.inv.app X).f i
-- CategoryTheory.Idempotents.DoldKan.equivalence_counitIso
theorem equivalence_counitIso :
    DoldKan.equivalence.counitIso = (η : Γ ⋙ N ≅ 𝟭 (ChainComplex C ℕ)) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Idempotents.DoldKan.equivalence.counitIso = CategoryTheory.Idempotents.DoldKan.η
-- CategoryTheory.Idempotents.DoldKan.hε
theorem hε :
    Compatibility.υ (isoN₁) =
      (Γ₂N₁ : (toKaroubiEquivalence _).functor ≅
          (N₁ : SimplicialObject C ⥤ _) ⋙ Preadditive.DoldKan.equivalence.inverse) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: AlgebraicTopology.DoldKan.Compatibility.υ CategoryTheory.Idempotents.DoldKan.isoN₁ = AlgebraicTopology.DoldKan.Γ₂N₁
-- CategoryTheory.Idempotents.DoldKan.equivalence_unitIso
theorem equivalence_unitIso :
    DoldKan.equivalence.unitIso = (ε : 𝟭 (SimplicialObject C) ≅ N ⋙ Γ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Idempotents.DoldKan.equivalence.unitIso = CategoryTheory.Idempotents.DoldKan.ε
-- CategoryTheory.nerve_map
@[simps]
def nerve (C : Type u) [Category.{v} C] : SSet.{max u v} 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : SimplexCategoryᵒᵖ} (f : X ⟶ Y) (x : CategoryTheory.ComposableArrows C (Opposite.unop X).len),
  (CategoryTheory.nerve C).map f x = x.whiskerLeft (SimplexCategory.toCat.map f.unop)
-- CategoryTheory.nerve_obj
@[simps]
def nerve (C : Type u) [Category.{v} C] : SSet.{max u v} 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (Δ : SimplexCategoryᵒᵖ)
Goal: (CategoryTheory.nerve C).obj Δ = CategoryTheory.ComposableArrows C (Opposite.unop Δ).len
-- CategoryTheory.nerveFunctor_map_app
@[simps]
def nerveFunctor : Cat ⥤ SSet 
Goal: ∀ {X Y : CategoryTheory.Cat} (F : X ⟶ Y) (Δ : SimplexCategoryᵒᵖ)
  (a : CategoryTheory.ComposableArrows (↑X) (Opposite.unop Δ).len),
  (CategoryTheory.nerveFunctor.map F).app Δ a =
    (CategoryTheory.Functor.mapComposableArrows F (Opposite.unop Δ).len).obj a
-- CategoryTheory.nerveFunctor_obj
@[simps]
def nerveFunctor : Cat ⥤ SSet 
Arguments: (C : CategoryTheory.Cat)
Goal: CategoryTheory.nerveFunctor.obj C = CategoryTheory.nerve ↑C
-- CategoryTheory.Nerve.δ₀_eq
lemma δ₀_eq {x : nerve C _[n + 1]} : (nerve C).δ (0 : Fin (n + 2)) x = x.δ₀ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {n : ℕ} {x : CategoryTheory.nerve C _[n + 1]}
Goal: CategoryTheory.SimplicialObject.δ (CategoryTheory.nerve C) 0 x = CategoryTheory.ComposableArrows.δ₀ x
-- ContinuousLinearMap.measurable_apply₂
theorem measurable_apply₂ [MeasurableSpace E] [OpensMeasurableSpace E]
    [SecondCountableTopologyEither (E →L[𝕜] F) E]
    [MeasurableSpace F] [BorelSpace F] : Measurable fun p : (E →L[𝕜] F) × E => p.1 p.2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopologyEither (E →L[𝕜] F) E] [MeasurableSpace F] [BorelSpace F]
Goal: Measurable fun p => p.1 p.2
-- FDerivMeasurableAux.isOpen_A
theorem isOpen_A (L : E →L[𝕜] F) (r ε : ℝ) : IsOpen (A f L r ε) 
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} (L : E →L[𝕜] F) (r ε : ℝ)
Goal: IsOpen (FDerivMeasurableAux.A f L r ε)
-- FDerivMeasurableAux.isOpen_B
theorem isOpen_B {K : Set (E →L[𝕜] F)} {r s ε : ℝ} : IsOpen (B f K r s ε) 
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} {K : Set (E →L[𝕜] F)} {r s ε : ℝ}
Goal: IsOpen (FDerivMeasurableAux.B f K r s ε)
-- FDerivMeasurableAux.A_mono
theorem A_mono (L : E →L[𝕜] F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) : A f L r ε ⊆ A f L r δ 
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} (L : E →L[𝕜] F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ)
Goal: FDerivMeasurableAux.A f L r ε ⊆ FDerivMeasurableAux.A f L r δ
-- FDerivMeasurableAux.le_of_mem_A
theorem le_of_mem_A {r ε : ℝ} {L : E →L[𝕜] F} {x : E} (hx : x ∈ A f L r ε) {y z : E}
    (hy : y ∈ closedBall x (r / 2)) (hz : z ∈ closedBall x (r / 2)) :
    ‖f z - f y - L (z - y)‖ ≤ ε * r 
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} {r ε : ℝ} {L : E →L[𝕜] F} {x : E} (hx : x ∈ FDerivMeasurableAux.A f L r ε) {y z : E} (hy : y ∈ Metric.closedBall x (r / 2)) (hz : z ∈ Metric.closedBall x (r / 2))
Goal: ‖f z - f y - L (z - y)‖ ≤ ε * r
-- FDerivMeasurableAux.mem_A_of_differentiable
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : E} (hx : DifferentiableAt 𝕜 f x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (fderiv 𝕜 f x) r ε 
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} {ε : ℝ} (hε : 0 < ε) {x : E} (hx : DifferentiableAt 𝕜 f x)
Goal: ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, x ∈ FDerivMeasurableAux.A f (fderiv 𝕜 f x) r ε
-- FDerivMeasurableAux.norm_sub_le_of_mem_A
theorem norm_sub_le_of_mem_A {c : 𝕜} (hc : 1 < ‖c‖) {r ε : ℝ} (hε : 0 < ε) (hr : 0 < r) {x : E}
    {L₁ L₂ : E →L[𝕜] F} (h₁ : x ∈ A f L₁ r ε) (h₂ : x ∈ A f L₂ r ε) : ‖L₁ - L₂‖ ≤ 4 * ‖c‖ * ε 
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} {c : 𝕜} (hc : 1 < ‖c‖) {r ε : ℝ} (hε : 0 < ε) (hr : 0 < r) {x : E} {L₁ L₂ : E →L[𝕜] F} (h₁ : x ∈ FDerivMeasurableAux.A f L₁ r ε) (h₂ : x ∈ FDerivMeasurableAux.A f L₂ r ε)
Goal: ‖L₁ - L₂‖ ≤ 4 * ‖c‖ * ε
-- FDerivMeasurableAux.differentiable_set_subset_D
theorem differentiable_set_subset_D :
    { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } ⊆ D f K 
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} (K : Set (E →L[𝕜] F))
Goal: {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} ⊆ FDerivMeasurableAux.D f K
-- FDerivMeasurableAux.D_subset_differentiable_set
theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
    D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } 
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} {K : Set (E →L[𝕜] F)} (hK : IsComplete K)
Goal: FDerivMeasurableAux.D f K ⊆ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
-- FDerivMeasurableAux.differentiable_set_eq_D
theorem differentiable_set_eq_D (hK : IsComplete K) :
    { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } = D f K 
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} (K : Set (E →L[𝕜] F)) (hK : IsComplete K)
Goal: {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} = FDerivMeasurableAux.D f K
-- measurableSet_of_differentiableAt_of_isComplete
theorem measurableSet_of_differentiableAt_of_isComplete {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
    MeasurableSet { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } 
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) [MeasurableSpace E] [OpensMeasurableSpace E] {K : Set (E →L[𝕜] F)} (hK : IsComplete K)
Goal: MeasurableSet {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
-- measurableSet_of_differentiableAt
theorem measurableSet_of_differentiableAt : MeasurableSet { 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] (f : E → F) [MeasurableSpace E] [OpensMeasurableSpace E] [CompleteSpace F]
Goal: MeasurableSet {x | DifferentiableAt 𝕜 f x}
-- measurable_fderiv
@[measurability, fun_prop]
theorem measurable_fderiv : Measurable (fderiv 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] (f : E → F) [MeasurableSpace E] [OpensMeasurableSpace E] [CompleteSpace F]
Goal: Measurable (fderiv 𝕜 f)
-- measurable_fderiv_apply_const
@[measurability, fun_prop]
theorem measurable_fderiv_apply_const [MeasurableSpace F] [BorelSpace F] (y : E) :
    Measurable fun x => fderiv 𝕜 f x 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] (f : E → F) [MeasurableSpace E] [OpensMeasurableSpace E] [CompleteSpace F] [MeasurableSpace F] [BorelSpace F] (y : E)
Goal: Measurable fun x => (fderiv 𝕜 f x) y
-- measurable_deriv
@[measurability, fun_prop]
theorem measurable_deriv [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [MeasurableSpace F]
    [BorelSpace F] (f : 𝕜 → F) : Measurable (deriv f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [MeasurableSpace F] [BorelSpace F] (f : 𝕜 → F)
Goal: Measurable (deriv f)
-- stronglyMeasurable_deriv
theorem stronglyMeasurable_deriv [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜]
    [h : SecondCountableTopologyEither 𝕜 F] (f : 𝕜 → F) : StronglyMeasurable (deriv f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [h : SecondCountableTopologyEither 𝕜 F] (f : 𝕜 → F)
Goal: MeasureTheory.StronglyMeasurable (deriv f)
-- aemeasurable_deriv
theorem aemeasurable_deriv [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [MeasurableSpace F]
    [BorelSpace F] (f : 𝕜 → F) (μ : Measure 𝕜) : AEMeasurable (deriv f) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [MeasurableSpace F] [BorelSpace F] (f : 𝕜 → F) (μ : MeasureTheory.Measure 𝕜)
Goal: AEMeasurable (deriv f) μ
-- aestronglyMeasurable_deriv
theorem aestronglyMeasurable_deriv [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜]
    [SecondCountableTopologyEither 𝕜 F] (f : 𝕜 → F) (μ : Measure 𝕜) :
    AEStronglyMeasurable (deriv f) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [SecondCountableTopologyEither 𝕜 F] (f : 𝕜 → F) (μ : MeasureTheory.Measure 𝕜)
Goal: MeasureTheory.AEStronglyMeasurable (deriv f) μ
-- RightDerivMeasurableAux.A_mem_nhdsWithin_Ioi
theorem A_mem_nhdsWithin_Ioi {L : F} {r ε x : ℝ} (hx : x ∈ A f L r ε) : A f L r ε ∈ 𝓝[>] x 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} {L : F} {r ε x : ℝ} (hx : x ∈ RightDerivMeasurableAux.A f L r ε)
Goal: RightDerivMeasurableAux.A f L r ε ∈ 𝓝[>] x
-- RightDerivMeasurableAux.B_mem_nhdsWithin_Ioi
theorem B_mem_nhdsWithin_Ioi {K : Set F} {r s ε x : ℝ} (hx : x ∈ B f K r s ε) :
    B f K r s ε ∈ 𝓝[>] x 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} {K : Set F} {r s ε x : ℝ} (hx : x ∈ RightDerivMeasurableAux.B f K r s ε)
Goal: RightDerivMeasurableAux.B f K r s ε ∈ 𝓝[>] x
-- RightDerivMeasurableAux.measurableSet_B
theorem measurableSet_B {K : Set F} {r s ε : ℝ} : MeasurableSet (B f K r s ε) 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} {K : Set F} {r s ε : ℝ}
Goal: MeasurableSet (RightDerivMeasurableAux.B f K r s ε)
-- RightDerivMeasurableAux.A_mono
theorem A_mono (L : F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) : A f L r ε ⊆ A f L r δ 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} (L : F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ)
Goal: RightDerivMeasurableAux.A f L r ε ⊆ RightDerivMeasurableAux.A f L r δ
-- RightDerivMeasurableAux.le_of_mem_A
theorem le_of_mem_A {r ε : ℝ} {L : F} {x : ℝ} (hx : x ∈ A f L r ε) {y z : ℝ}
    (hy : y ∈ Icc x (x + r / 2)) (hz : z ∈ Icc x (x + r / 2)) :
  ‖f z - f y - (z - y) • L‖ ≤ ε * r 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} {r ε : ℝ} {L : F} {x : ℝ} (hx : x ∈ RightDerivMeasurableAux.A f L r ε) {y z : ℝ} (hy : y ∈ Set.Icc x (x + r / 2)) (hz : z ∈ Set.Icc x (x + r / 2))
Goal: ‖f z - f y - (z - y) • L‖ ≤ ε * r
-- RightDerivMeasurableAux.mem_A_of_differentiable
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} {ε : ℝ} (hε : 0 < ε) {x : ℝ} (hx : DifferentiableWithinAt ℝ f (Set.Ici x) x)
Goal: ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, x ∈ RightDerivMeasurableAux.A f (derivWithin f (Set.Ici x) x) r ε
-- RightDerivMeasurableAux.norm_sub_le_of_mem_A
theorem norm_sub_le_of_mem_A {r x : ℝ} (hr : 0 < r) (ε : ℝ) {L₁ L₂ : F} (h₁ : x ∈ A f L₁ r ε)
    (h₂ : x ∈ A f L₂ r ε) : ‖L₁ - L₂‖ ≤ 4 * ε 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} {r x : ℝ} (hr : 0 < r) (ε : ℝ) {L₁ L₂ : F} (h₁ : x ∈ RightDerivMeasurableAux.A f L₁ r ε) (h₂ : x ∈ RightDerivMeasurableAux.A f L₂ r ε)
Goal: ‖L₁ - L₂‖ ≤ 4 * ε
-- RightDerivMeasurableAux.differentiable_set_subset_D
theorem differentiable_set_subset_D :
    { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } ⊆ D f K 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} (K : Set F)
Goal: {x | DifferentiableWithinAt ℝ f (Set.Ici x) x ∧ derivWithin f (Set.Ici x) x ∈ K} ⊆ RightDerivMeasurableAux.D f K
-- RightDerivMeasurableAux.D_subset_differentiable_set
theorem D_subset_differentiable_set {K : Set F} (hK : IsComplete K) :
    D f K ⊆ { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} {K : Set F} (hK : IsComplete K)
Goal: RightDerivMeasurableAux.D f K ⊆ {x | DifferentiableWithinAt ℝ f (Set.Ici x) x ∧ derivWithin f (Set.Ici x) x ∈ K}
-- RightDerivMeasurableAux.differentiable_set_eq_D
theorem differentiable_set_eq_D (hK : IsComplete K) :
    { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } = D f K 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} (K : Set F) (hK : IsComplete K)
Goal: {x | DifferentiableWithinAt ℝ f (Set.Ici x) x ∧ derivWithin f (Set.Ici x) x ∈ K} = RightDerivMeasurableAux.D f K
-- measurableSet_of_differentiableWithinAt_Ici_of_isComplete
theorem measurableSet_of_differentiableWithinAt_Ici_of_isComplete {K : Set F} (hK : IsComplete K) :
    MeasurableSet { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) {K : Set F} (hK : IsComplete K)
Goal: MeasurableSet {x | DifferentiableWithinAt ℝ f (Set.Ici x) x ∧ derivWithin f (Set.Ici x) x ∈ K}
-- measurableSet_of_differentiableWithinAt_Ici
theorem measurableSet_of_differentiableWithinAt_Ici :
    MeasurableSet { x | DifferentiableWithinAt ℝ f (Ici x) x } 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F]
Goal: MeasurableSet {x | DifferentiableWithinAt ℝ f (Set.Ici x) x}
-- measurable_derivWithin_Ici
@[measurability, fun_prop]
theorem measurable_derivWithin_Ici [MeasurableSpace F] [BorelSpace F] :
    Measurable fun x => derivWithin f (Ici x) x 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F] [MeasurableSpace F] [BorelSpace F]
Goal: Measurable fun x => derivWithin f (Set.Ici x) x
-- stronglyMeasurable_derivWithin_Ici
theorem stronglyMeasurable_derivWithin_Ici :
    StronglyMeasurable (fun x ↦ derivWithin f (Ici x) x) 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F]
Goal: MeasureTheory.StronglyMeasurable fun x => derivWithin f (Set.Ici x) x
-- aemeasurable_derivWithin_Ici
theorem aemeasurable_derivWithin_Ici [MeasurableSpace F] [BorelSpace F] (μ : Measure ℝ) :
    AEMeasurable (fun x => derivWithin f (Ici x) x) μ 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F] [MeasurableSpace F] [BorelSpace F] (μ : MeasureTheory.Measure ℝ)
Goal: AEMeasurable (fun x => derivWithin f (Set.Ici x) x) μ
-- aestronglyMeasurable_derivWithin_Ici
theorem aestronglyMeasurable_derivWithin_Ici (μ : Measure ℝ) :
    AEStronglyMeasurable (fun x => derivWithin f (Ici x) x) μ 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F] (μ : MeasureTheory.Measure ℝ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => derivWithin f (Set.Ici x) x) μ
-- measurableSet_of_differentiableWithinAt_Ioi
theorem measurableSet_of_differentiableWithinAt_Ioi :
    MeasurableSet { x | DifferentiableWithinAt ℝ f (Ioi x) x } 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F]
Goal: MeasurableSet {x | DifferentiableWithinAt ℝ f (Set.Ioi x) x}
-- measurable_derivWithin_Ioi
@[measurability, fun_prop]
theorem measurable_derivWithin_Ioi [MeasurableSpace F] [BorelSpace F] :
    Measurable fun x => derivWithin f (Ioi x) x 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F] [MeasurableSpace F] [BorelSpace F]
Goal: Measurable fun x => derivWithin f (Set.Ioi x) x
-- stronglyMeasurable_derivWithin_Ioi
theorem stronglyMeasurable_derivWithin_Ioi :
    StronglyMeasurable (fun x ↦ derivWithin f (Ioi x) x) 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F]
Goal: MeasureTheory.StronglyMeasurable fun x => derivWithin f (Set.Ioi x) x
-- aemeasurable_derivWithin_Ioi
theorem aemeasurable_derivWithin_Ioi [MeasurableSpace F] [BorelSpace F] (μ : Measure ℝ) :
    AEMeasurable (fun x => derivWithin f (Ioi x) x) μ 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F] [MeasurableSpace F] [BorelSpace F] (μ : MeasureTheory.Measure ℝ)
Goal: AEMeasurable (fun x => derivWithin f (Set.Ioi x) x) μ
-- aestronglyMeasurable_derivWithin_Ioi
theorem aestronglyMeasurable_derivWithin_Ioi (μ : Measure ℝ) :
    AEStronglyMeasurable (fun x => derivWithin f (Ioi x) x) μ 
Arguments: {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : ℝ → F) [CompleteSpace F] (μ : MeasureTheory.Measure ℝ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => derivWithin f (Set.Ioi x) x) μ
-- FDerivMeasurableAux.isOpen_A_with_param
lemma isOpen_A_with_param {r s : ℝ} (hf : Continuous f.uncurry) (L : E →L[𝕜] F) :
    IsOpen {p : α × E | p.2 ∈ A (f p.1) L r s} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] {f : α → E → F} {r s : ℝ} (hf : Continuous (Function.uncurry f)) (L : E →L[𝕜] F)
Goal: IsOpen {p | p.2 ∈ FDerivMeasurableAux.A (f p.1) L r s}
-- FDerivMeasurableAux.isOpen_B_with_param
lemma isOpen_B_with_param {r s t : ℝ} (hf : Continuous f.uncurry) (K : Set (E →L[𝕜] F)) :
    IsOpen {p : α × E | p.2 ∈ B (f p.1) K r s t} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] {f : α → E → F} {r s t : ℝ} (hf : Continuous (Function.uncurry f)) (K : Set (E →L[𝕜] F))
Goal: IsOpen {p | p.2 ∈ FDerivMeasurableAux.B (f p.1) K r s t}
-- measurableSet_of_differentiableAt_of_isComplete_with_param
theorem measurableSet_of_differentiableAt_of_isComplete_with_param
    (hf : Continuous f.uncurry) {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
    MeasurableSet {p : α × E | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [MeasurableSpace E] [OpensMeasurableSpace α] [OpensMeasurableSpace E] {f : α → E → F} (hf : Continuous (Function.uncurry f)) {K : Set (E →L[𝕜] F)} (hK : IsComplete K)
Goal: MeasurableSet {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K}
-- measurableSet_of_differentiableAt_with_param
theorem measurableSet_of_differentiableAt_with_param (hf : Continuous f.uncurry) :
    MeasurableSet {p : α × E | DifferentiableAt 𝕜 (f p.1) p.2} 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [MeasurableSpace E] [OpensMeasurableSpace α] [OpensMeasurableSpace E] {f : α → E → F} [CompleteSpace F] (hf : Continuous (Function.uncurry f))
Goal: MeasurableSet {p | DifferentiableAt 𝕜 (f p.1) p.2}
-- measurable_fderiv_with_param
theorem measurable_fderiv_with_param (hf : Continuous f.uncurry) :
    Measurable (fun (p : α × E) ↦ fderiv 𝕜 (f p.1) p.2) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [MeasurableSpace E] [OpensMeasurableSpace α] [OpensMeasurableSpace E] {f : α → E → F} [CompleteSpace F] (hf : Continuous (Function.uncurry f))
Goal: Measurable fun p => fderiv 𝕜 (f p.1) p.2
-- measurable_fderiv_apply_const_with_param
theorem measurable_fderiv_apply_const_with_param [MeasurableSpace F] [BorelSpace F]
    (hf : Continuous f.uncurry) (y : E) :
    Measurable (fun (p : α × E) ↦ fderiv 𝕜 (f p.1) p.2 y) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [MeasurableSpace E] [OpensMeasurableSpace α] [OpensMeasurableSpace E] {f : α → E → F} [CompleteSpace F] [MeasurableSpace F] [BorelSpace F] (hf : Continuous (Function.uncurry f)) (y : E)
Goal: Measurable fun p => (fderiv 𝕜 (f p.1) p.2) y