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 := by
  /- To bound `μ (f '' s)`, we cover `s` by sets where `f` is well-approximated by linear maps
    `A n` (and where `f'` is almost everywhere close to `A n`), and then use that `f` expands the
    measure of such a set by at most `(A n).det + ε`. -/
  have :
    ∀ A : E →L[ℝ] E,
      ∃ δ : ℝ≥0,
        0 < δ ∧
          (∀ B : E →L[ℝ] E, ‖B - A‖ ≤ δ → |B.det - A.det| ≤ ε) ∧
            ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ →
              μ (g '' t) ≤ (ENNReal.ofReal |A.det| + ε) * μ t := by
    intro A
    let m : ℝ≥0 := Real.toNNReal |A.det| + ε
    have I : ENNReal.ofReal |A.det| < m := by
      simp only [ENNReal.ofReal, lt_add_iff_pos_right, εpos, ENNReal.coe_lt_coe]
    rcases ((addHaar_image_le_mul_of_det_lt μ A I).and self_mem_nhdsWithin).exists with ⟨δ, h, δpos⟩
    obtain ⟨δ', δ'pos, hδ'⟩ : ∃ (δ' : ℝ), 0 < δ' ∧ ∀ B, dist B A < δ' → dist B.det A.det < ↑ε :=
      continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt ε εpos
    let δ'' : ℝ≥0 := ⟨δ' / 2, (half_pos δ'pos).le⟩
    refine' ⟨min δ δ'', lt_min δpos (half_pos δ'pos), _, _⟩
    · intro B hB
      rw [← Real.dist_eq]
      apply (hδ' B _).le
      rw [dist_eq_norm]
      calc
        ‖B - A‖ ≤ (min δ δ'' : ℝ≥0) := hB
        _ ≤ δ'' := by simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]
        _ < δ' := half_lt_self δ'pos
    · intro t g htg
      exact h t g (htg.mono_num (min_le_left _ _))
  choose δ hδ using this
  obtain ⟨t, A, t_disj, t_meas, t_cover, ht, -⟩ :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] E),
      Pairwise (Disjoint on t) ∧
        (∀ n : ℕ, MeasurableSet (t n)) ∧
          (s ⊆ ⋃ n : ℕ, t n) ∧
            (∀ n : ℕ, ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))) ∧
              (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
    exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' δ fun A => (hδ A).1.ne'
  calc
    μ (f '' s) ≤ μ (⋃ n, f '' (s ∩ t n)) := by
      apply measure_mono
      rw [← image_iUnion, ← inter_iUnion]
      exact image_subset f (subset_inter Subset.rfl t_cover)
    _ ≤ ∑' n, μ (f '' (s ∩ t n)) := (measure_iUnion_le _)
    _ ≤ ∑' n, (ENNReal.ofReal |(A n).det| + ε) * μ (s ∩ t n) := by
      apply ENNReal.tsum_le_tsum fun n => ?_
      apply (hδ (A n)).2.2
      exact ht n
    _ = ∑' n, ∫⁻ _ in s ∩ t n, ENNReal.ofReal |(A n).det| + ε ∂μ := by
      simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
    _ ≤ ∑' n, ∫⁻ x in s ∩ t n, ENNReal.ofReal |(f' x).det| + 2 * ε ∂μ := by
      apply ENNReal.tsum_le_tsum fun n => ?_
      apply lintegral_mono_ae
      filter_upwards [(ht n).norm_fderiv_sub_le μ (hs.inter (t_meas n)) f' fun x hx =>
          (hf' x hx.1).mono (inter_subset_left _ _)]
      intro x hx
      have I : |(A n).det| ≤ |(f' x).det| + ε :=
        calc
          |(A n).det| = |(f' x).det - ((f' x).det - (A n).det)| := by congr 1; abel
          _ ≤ |(f' x).det| + |(f' x).det - (A n).det| := (abs_sub _ _)
          _ ≤ |(f' x).det| + ε := add_le_add le_rfl ((hδ (A n)).2.1 _ hx)
      

Proof state:
case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t
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, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
n : ℕ
x : E
hx : ‖f' x - A n‖₊ ≤ δ (A n)
I : |ContinuousLinearMap.det (A n)| ≤ |ContinuousLinearMap.det (f' x)| + ↑ε
⊢ ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ≤ ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε

Next tactic:
calc
        ENNReal.ofReal |(A n).det| + ε ≤ ENNReal.ofReal (|(f' x).det| + ε) + ε :=
          add_le_add (ENNReal.ofReal_le_ofReal I) le_rfl
        _ = ENNReal.ofReal |(f' x).det| + 2 * ε := by
          simp only [ENNReal.ofReal_add, abs_nonneg, two_mul, add_assoc, NNReal.zero_le_coe,
            ENNReal.ofReal_coe_nnreal]