awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
i j : ℕ
hij : i ≠ j
⊢ Disjoint (s ∩ t i) (s ∩ t j)

Tactic:
exact Disjoint.mono inter_subset_right inter_subset_right (t_disj hij)

Premises used: Disjoint.mono, Set.inter_subset_right
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
i : ℕ
⊢ MeasurableSet (f '' (s ∩ t i))

Tactic:
measurable_image_of_fderivWithin (hs.inter (t_meas i))
            (fun x hx => (hf' x hx.1).mono inter_subset_left)
            (hf.mono inter_subset_left)

Premises used: MeasureTheory.measurable_image_of_fderivWithin, Set.inter_subset_left, And.left, Set.InjOn.mono, HasFDerivWithinAt.mono, MeasurableSet.inter
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
i : ℕ
⊢ MeasurableSet (f '' (s ∩ t i))

Tactic:
exact
          measurable_image_of_fderivWithin (hs.inter (t_meas i))
            (fun x hx => (hf' x hx.1).mono inter_subset_left)
            (hf.mono inter_subset_left)

Premises used: MeasureTheory.measurable_image_of_fderivWithin, Set.inter_subset_left, And.left, Set.InjOn.mono, HasFDerivWithinAt.mono, MeasurableSet.inter
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
rw :
  ∑' (n : ℕ),
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t n)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) =
    ∑' (i : ℕ), (μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) +
      2 * (↑ε : ℝ≥0∞) * ∑' (i : ℕ), (μ : (a : Set E) → ℝ≥0∞) (s ∩ t i)
hn : Pairwise (α := ℕ) (Function.onFun.{1, u_1 + 1, 1} (β := Set E) Disjoint fun (n : ℕ) ↦ s ∩ t n)
h : ∀ (i : ℕ), MeasurableSet (s ∩ t i)
⊢ ∑' (n : ℕ),
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t n)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) =
    ∑' (i : ℕ), (μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) +
      HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) (⋃ (n : ℕ), s ∩ t n) : ℝ≥0∞)

Tactic:
rw [measure_iUnion]

Premises used: MeasureTheory.measure_iUnion
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
⊢ Pairwise (α := ℕ) (Function.onFun.{1, u_1 + 1, 1} (β := Set E) Disjoint fun (n : ℕ) ↦ s ∩ t n)

Tactic:
pairwise_disjoint_mono t_disj fun i => inter_subset_right

Premises used: pairwise_disjoint_mono, Set.inter_subset_right
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
⊢ Pairwise (α := ℕ) (Function.onFun.{1, u_1 + 1, 1} (β := Set E) Disjoint fun (n : ℕ) ↦ s ∩ t n)

Tactic:
exact pairwise_disjoint_mono t_disj fun i => inter_subset_right

Premises used: pairwise_disjoint_mono, Set.inter_subset_right
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
⊢ ∀ (i : ℕ), MeasurableSet (s ∩ t i)

Tactic:
fun i => hs.inter (t_meas i)

Premises used: MeasurableSet.inter
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
⊢ ∀ (i : ℕ), MeasurableSet (s ∩ t i)

Tactic:
exact fun i => hs.inter (t_meas i)

Premises used: MeasurableSet.inter
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
rw :
  Eq (α := ℝ≥0∞)
    (∑' (n : ℕ),
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t n)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)))
    (∑' (a : ℕ),
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t a)) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t a) : ℝ≥0∞)))
⊢ ∑' (n : ℕ),
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t n)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) =
    ∑' (i : ℕ), (μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) +
      2 * (↑ε : ℝ≥0∞) * ∑' (i : ℕ), (μ : (a : Set E) → ℝ≥0∞) (s ∩ t i)

Tactic:
rw [← ENNReal.tsum_mul_left, ← ENNReal.tsum_add]

Premises used: ENNReal.tsum_mul_left, ENNReal.tsum_add
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
e_f :
  Eq (α := ℕ → ℝ≥0∞)
    (fun (n : ℕ) ↦
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t n)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞))
    fun (a : ℕ) ↦
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t a)) : ℝ≥0∞)
      (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t a) : ℝ≥0∞))
⊢ Eq (α := ℝ≥0∞)
    (∑' (n : ℕ),
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t n)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)))
    (∑' (a : ℕ),
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t a)) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t a) : ℝ≥0∞)))

Tactic:
congr 1
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
e_f :
  ∀ (i : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞) =
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞))
⊢ Eq (α := ℕ → ℝ≥0∞)
    (fun (n : ℕ) ↦
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t n)) : ℝ≥0∞)
          (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞)) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t n) : ℝ≥0∞))
    fun (a : ℕ) ↦
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t a)) : ℝ≥0∞)
      (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t a) : ℝ≥0∞))

Tactic:
ext1 i
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (A : E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ (↑(δ A) : ℝ) → |B.det - A.det| ≤ (↑ε : ℝ)) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) →
            HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal |A.det|) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (g '' t) : ℝ≥0∞)
                (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) t : ℝ≥0∞))
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ (n : ℕ), t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ (n : ℕ), s ∩ t n
i : ℕ
e_f :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞)) +
      HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞) =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
      (2 * HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞))
e_f₁ :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞)) +
      HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞) =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
      (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞))
e_f₂ e_f₃ :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
      (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞)) =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
      (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞) +
        HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞))
⊢ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞)) +
      HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞) =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' (s ∩ t i)) : ℝ≥0∞)
      (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ t i) : ℝ≥0∞))

Tactic:
rw [mul_assoc, two_mul, add_assoc]

Premises used: add_assoc, mul_assoc, two_mul
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
h's : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
apply :
  Tendsto
    (fun (ε : ℝ≥0) ↦
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞)))
    (𝓝 0)
    (𝓝
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑0 : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞))))
⊢ Tendsto
    (fun (ε : ℝ≥0) ↦
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞)))
    (𝓝[>] 0)
    (𝓝
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑0 : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞))))

Tactic:
apply Tendsto.mono_left _ nhdsWithin_le_nhds

Premises used: nhdsWithin_le_nhds, Filter.Tendsto.mono_left
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
h's : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
⊢ Tendsto
    (fun (ε : ℝ≥0) ↦
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞)))
    (𝓝 0)
    (𝓝
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑0 : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞))))

Tactic:
tendsto_const_nhds.add ?_

Premises used: tendsto_const_nhds, Filter.Tendsto.add
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
hs : MeasurableSet s
h's : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
refine :
  Tendsto (fun (ε : ℝ≥0) ↦ HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞)) (𝓝 0)
    (𝓝 (HMul.hMul (β := ℝ≥0∞) (2 * (↑0 : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞)))
⊢ Tendsto
    (fun (ε : ℝ≥0) ↦
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑ε : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞)))
    (𝓝 0)
    (𝓝
      (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (f '' s) : ℝ≥0∞)
        (HMul.hMul (β := ℝ≥0∞) (2 * (↑0 : ℝ≥0∞)) ((μ : (a : Set E) → ℝ≥0∞) s : ℝ≥0∞))))

Tactic:
refine tendsto_const_nhds.add ?_

Premises used: tendsto_const_nhds, Filter.Tendsto.add