ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.integral_sub_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [μ.IsAddRightInvariant] (f : G → E) (g : G)
Goal: ∫ (x : G), f (x - g) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_div_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [μ.IsMulRightInvariant] (f : G → E) (g : G)
Goal: ∫ (x : G), f (x / g) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_eq_zero_of_mul_left_eq_neg
@[to_additive
      "If some left-translate of a function negates it, then the integral of the function with
      respect to a left-invariant measure is 0."]
theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
    ∫ x, f x ∂μ = 0 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} {f : G → E} {g : G} [Group G] [MeasurableMul G] [μ.IsMulLeftInvariant] (hf' : ∀ (x : G), f (g * x) = -f x)
Goal: ∫ (x : G), f x ∂μ = 0
-- MeasureTheory.integral_eq_zero_of_add_left_eq_neg
@[to_additive
      "If some left-translate of a function negates it, then the integral of the function with
      respect to a left-invariant measure is 0."]
theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
    ∫ x, f x ∂μ = 0 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} {f : G → E} {g : G} [AddGroup G] [MeasurableAdd G] [μ.IsAddLeftInvariant] (hf' : ∀ (x : G), f (g + x) = -f x)
Goal: ∫ (x : G), f x ∂μ = 0
-- MeasureTheory.integral_eq_zero_of_mul_right_eq_neg
@[to_additive
      "If some right-translate of a function negates it, then the integral of the function with
      respect to a right-invariant measure is 0."]
theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
    ∫ x, f x ∂μ = 0 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} {f : G → E} {g : G} [Group G] [MeasurableMul G] [μ.IsMulRightInvariant] (hf' : ∀ (x : G), f (x * g) = -f x)
Goal: ∫ (x : G), f x ∂μ = 0
-- MeasureTheory.integral_eq_zero_of_add_right_eq_neg
@[to_additive
      "If some right-translate of a function negates it, then the integral of the function with
      respect to a right-invariant measure is 0."]
theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
    ∫ x, f x ∂μ = 0 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure G} {f : G → E} {g : G} [AddGroup G] [MeasurableAdd G] [μ.IsAddRightInvariant] (hf' : ∀ (x : G), f (x + g) = -f x)
Goal: ∫ (x : G), f x ∂μ = 0
-- MeasureTheory.Integrable.comp_mul_left
@[to_additive]
theorem Integrable.comp_mul_left {f : G → F} [IsMulLeftInvariant μ] (hf : Integrable f μ) (g : G) :
    Integrable (fun t => f (g * t)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] {f : G → F} [μ.IsMulLeftInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (g * t)) μ
-- MeasureTheory.Integrable.comp_add_left
@[to_additive]
theorem Integrable.comp_mul_left {f : G → F} [IsMulLeftInvariant μ] (hf : Integrable f μ) (g : G) :
    Integrable (fun t => f (g * t)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] {f : G → F} [μ.IsAddLeftInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (g + t)) μ
-- MeasureTheory.Integrable.comp_mul_right
@[to_additive]
theorem Integrable.comp_mul_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t * g)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] {f : G → F} [μ.IsMulRightInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (t * g)) μ
-- MeasureTheory.Integrable.comp_add_right
@[to_additive]
theorem Integrable.comp_mul_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t * g)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] {f : G → F} [μ.IsAddRightInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (t + g)) μ
-- MeasureTheory.Integrable.comp_sub_right
@[to_additive]
theorem Integrable.comp_div_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t / g)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] {f : G → F} [μ.IsAddRightInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (t - g)) μ
-- MeasureTheory.Integrable.comp_div_right
@[to_additive]
theorem Integrable.comp_div_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t / g)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] {f : G → F} [μ.IsMulRightInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (t / g)) μ
-- MeasureTheory.Integrable.comp_div_left
@[to_additive]
theorem Integrable.comp_div_left {f : G → F} [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (hf : Integrable f μ) (g : G) : Integrable (fun t => f (g / t)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [MeasurableInv G] {f : G → F} [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (g / t)) μ
-- MeasureTheory.Integrable.comp_sub_left
@[to_additive]
theorem Integrable.comp_div_left {f : G → F} [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (hf : Integrable f μ) (g : G) : Integrable (fun t => f (g / t)) μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] {f : G → F} [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (hf : MeasureTheory.Integrable f μ) (g : G)
Goal: MeasureTheory.Integrable (fun t => f (g - t)) μ
-- MeasureTheory.integrable_comp_sub_left
@[to_additive] -- Porting note: was `@[simp]`
theorem integrable_comp_div_left (f : G → F) [IsInvInvariant μ] [IsMulLeftInvariant μ] (g : G) :
    Integrable (fun t => f (g / t)) μ ↔ Integrable f μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] (f : G → F) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.Integrable (fun t => f (g - t)) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_comp_div_left
@[to_additive] -- Porting note: was `@[simp]`
theorem integrable_comp_div_left (f : G → F) [IsInvInvariant μ] [IsMulLeftInvariant μ] (g : G) :
    Integrable (fun t => f (g / t)) μ ↔ Integrable f μ 
Arguments: {G : Type u_4} {F : Type u_6} [MeasurableSpace G] [NormedAddCommGroup F] {μ : MeasureTheory.Measure G} [Group G] [MeasurableMul G] [MeasurableInv G] (f : G → F) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.Integrable (fun t => f (g / t)) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integral_sub_left_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_left_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ]
    [IsMulLeftInvariant μ] (x' : G) : (∫ x, f (x' / x) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] (f : G → E) (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (x' : G)
Goal: ∫ (x : G), f (x' - x) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_div_left_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_left_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ]
    [IsMulLeftInvariant μ] (x' : G) : (∫ x, f (x' / x) ∂μ) = ∫ x, f x ∂μ 
Arguments: {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] [Group G] [MeasurableMul G] [MeasurableInv G] (f : G → E) (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (x' : G)
Goal: ∫ (x : G), f (x' / x) ∂μ = ∫ (x : G), f x ∂μ
-- MeasureTheory.integral_vadd_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_smul_eq_self {μ : Measure α} [SMulInvariantMeasure G α μ] (f : α → E) {g : G} :
    (∫ x, f (g • x) ∂μ) = ∫ x, f x ∂μ 
Arguments: {α : Type u_3} {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] [AddGroup G] [MeasurableSpace α] [AddAction G α] [MeasurableVAdd G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (f : α → E) {g : G}
Goal: ∫ (x : α), f (g +ᵥ x) ∂μ = ∫ (x : α), f x ∂μ
-- MeasureTheory.integral_smul_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_smul_eq_self {μ : Measure α} [SMulInvariantMeasure G α μ] (f : α → E) {g : G} :
    (∫ x, f (g • x) ∂μ) = ∫ x, f x ∂μ 
Arguments: {α : Type u_3} {G : Type u_4} {E : Type u_5} [MeasurableSpace G] [NormedAddCommGroup E] [NormedSpace ℝ E] [Group G] [MeasurableSpace α] [MulAction G α] [MeasurableSMul G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (f : α → E) {g : G}
Goal: ∫ (x : α), f (g • x) ∂μ = ∫ (x : α), f x ∂μ
-- exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [SecondCountableTopology F] (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0)
Goal: ∃ t A,
  (∀ (n : ℕ), IsClosed (t n)) ∧
    s ⊆ ⋃ n, t n ∧
      (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
-- exists_partition_approximatesLinearOn_of_hasFDerivWithinAt
theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      Pairwise (Disjoint on t) ∧
        (∀ n, MeasurableSet (t n)) ∧
          (s ⊆ ⋃ n, t n) ∧
            (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
              (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology F] (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0)
Goal: ∃ t A,
  Pairwise (Disjoint on t) ∧
    (∀ (n : ℕ), MeasurableSet (t n)) ∧
      s ⊆ ⋃ n, t n ∧
        (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
-- MeasureTheory.addHaar_image_le_mul_of_det_lt
theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : ENNReal.ofReal |A.det| < m) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (A : E →L[ℝ] E) {m : ℝ≥0} (hm : ENNReal.ofReal |A.det| < ↑m)
Goal: ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ ↑m * μ s
-- MeasureTheory.mul_le_addHaar_image_of_lt_det
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (A : E →L[ℝ] E) {m : ℝ≥0} (hm : ↑m < ENNReal.ofReal |A.det|)
Goal: ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * μ s ≤ μ (f '' s)
-- ApproximatesLinearOn.norm_fderiv_sub_le
theorem _root_.ApproximatesLinearOn.norm_fderiv_sub_le {A : E →L[ℝ] E} {δ : ℝ≥0}
    (hf : ApproximatesLinearOn f A s δ) (hs : MeasurableSet s) (f' : E → E →L[ℝ] E)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : ∀ᵐ x ∂μ.restrict s, ‖f' x - A‖₊ ≤ δ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {A : E →L[ℝ] E} {δ : ℝ≥0} (hf : ApproximatesLinearOn f A s δ) (hs : MeasurableSet s) (f' : E → E →L[ℝ] E) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
Goal: ∀ᵐ (x : E) ∂μ.restrict s, ‖f' x - A‖₊ ≤ δ
-- MeasureTheory.addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero
theorem addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero (hf : DifferentiableOn ℝ f s)
    (hs : μ s = 0) : μ (f '' s) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hf : DifferentiableOn ℝ f s) (hs : μ s = 0)
Goal: μ (f '' s) = 0
-- MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux
theorem addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (R : ℝ) (hs : s ⊆ closedBall 0 R) (ε : ℝ≥0)
    (εpos : 0 < ε) (h'f' : ∀ x ∈ s, (f' x).det = 0) : μ (f '' s) ≤ ε * μ (closedBall 0 R) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (R : ℝ) (hs : s ⊆ Metric.closedBall 0 R) (ε : ℝ≥0) (εpos : 0 < ε) (h'f' : ∀ x ∈ s, (f' x).det = 0)
Goal: μ (f '' s) ≤ ↑ε * μ (Metric.closedBall 0 R)
-- MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero
theorem addHaar_image_eq_zero_of_det_fderivWithin_eq_zero
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (h'f' : ∀ x ∈ s, (f' x).det = 0) :
    μ (f '' s) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (h'f' : ∀ x ∈ s, (f' x).det = 0)
Goal: μ (f '' s) = 0
-- MeasureTheory.aemeasurable_fderivWithin
theorem aemeasurable_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : AEMeasurable f' (μ.restrict s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
Goal: AEMeasurable f' (μ.restrict s)
-- MeasureTheory.aemeasurable_ofReal_abs_det_fderivWithin
theorem aemeasurable_ofReal_abs_det_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    AEMeasurable (fun x => ENNReal.ofReal |(f' x).det|) (μ.restrict s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
Goal: AEMeasurable (fun x => ENNReal.ofReal |(f' x).det|) (μ.restrict s)
-- MeasureTheory.aemeasurable_toNNReal_abs_det_fderivWithin
theorem aemeasurable_toNNReal_abs_det_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    AEMeasurable (fun x => |(f' x).det|.toNNReal) (μ.restrict s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
Goal: AEMeasurable (fun x => |(f' x).det|.toNNReal) (μ.restrict s)
-- MeasureTheory.measurable_image_of_fderivWithin
theorem measurable_image_of_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) : MeasurableSet (f '' s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s)
Goal: MeasurableSet (f '' s)
-- MeasureTheory.measurableEmbedding_of_fderivWithin
theorem measurableEmbedding_of_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    MeasurableEmbedding (s.restrict f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s)
Goal: MeasurableEmbedding (s.restrict f)
-- MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux1
theorem addHaar_image_le_lintegral_abs_det_fderiv_aux1 (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) {ε : ℝ≥0} (εpos : 0 < ε) :
    μ (f '' s) ≤ (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) + 2 * ε * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) {ε : ℝ≥0} (εpos : 0 < ε)
Goal: μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ + 2 * ↑ε * μ s
-- MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2
theorem addHaar_image_le_lintegral_abs_det_fderiv_aux2 (hs : MeasurableSet s) (h's : μ s ≠ ∞)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    μ (f '' s) ≤ ∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (h's : μ s ≠ ⊤) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
Goal: μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ
-- MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv
theorem addHaar_image_le_lintegral_abs_det_fderiv (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    μ (f '' s) ≤ ∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
Goal: μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ
-- MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1
theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) + 2 * ε * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε)
Goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s) + 2 * ↑ε * μ s
-- MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux2
theorem lintegral_abs_det_fderiv_le_addHaar_image_aux2 (hs : MeasurableSet s) (h's : μ s ≠ ∞)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (h's : μ s ≠ ⊤) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s)
Goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)
-- MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image
theorem lintegral_abs_det_fderiv_le_addHaar_image (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s)
Goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)
-- MeasureTheory.lintegral_abs_det_fderiv_eq_addHaar_image
theorem lintegral_abs_det_fderiv_eq_addHaar_image (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) = μ (f '' s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s)
Goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = μ (f '' s)