first_exception'

lemma first_exception' : volume (G₁ : Set X) ≤ 2 ^ (- 5 : ℤ) * volume G := by  -- Handle trivial cases  classical  by_cases hF : volume F = 0  · simp [G₁_empty hF]  by_cases hG : volume G = 0  · exact (G₁_empty' hG ▸ OuterMeasureClass.measure_empty volume) ▸ zero_le  -- Define constant `K` and prove 0 < K < ⊤  let K := 2 ^ (2 * a + 5) * volume F / volume G  have K0 : K > 0 := by    refine ENNReal.div_pos (ne_of_gt ?_) volume_G_ne_top    exact mul_pos_iff.2 ⟨ENNReal.pow_pos two_pos _, measure_pos_of_superset subset_rfl hF⟩  have K_ne_top : K ≠ ⊤ := by    simp only [K]    finiteness [volume_F_ne_top (X := X)]  -- Define function `r : 𝔓 X → ℝ`, with garbage value `0` for `p ∉ highDensityTiles`  have : ∀ p ∈ highDensityTiles, ∃ r ≥ 4 * (D : ℝ) ^ 𝔰 p,      volume (F ∩ (ball (𝔠 p) r)) ≥ K * volume (ball (𝔠 p) r) := by    intro p hp    simp_rw [highDensityTiles, mem_setOf_eq, dens₂, lt_iSup_iff, mem_singleton_iff] at hp    rcases hp with ⟨p, rfl, r, hr, h⟩    use r, hr    refine ENNReal.lt_div_iff_mul_lt ?_ (Or.inl measure_ball_ne_top) |>.mp h |>.le    have r0 : r > 0 := lt_of_lt_of_le (by have := realD_pos a; positivity) hr    exact Or.inl <| (measure_ball_pos volume (𝔠 p) r0).ne.symm  let r (p : 𝔓 X) := dite (p ∈ highDensityTiles) (fun hp ↦ Classical.choose (this p hp)) (fun _ ↦ 0)  have hr {p : 𝔓 X} (hp : p ∈ highDensityTiles) := Classical.choose_spec (this p hp)  -- Show that balls with centers in `highDensityTiles` covers `G₁`.  let 𝓑 : Finset (𝔓 X) := highDensityTiles.toFinset  have : (G₁ : Set X) ⊆ ⋃ p ∈ 𝓑, (ball (𝔠 p) (r p)) := by    refine fun x hx ↦ mem_iUnion.2 ?_    simp only [G₁, mem_iUnion, exists_prop] at hx    rcases hx with ⟨p, hp, xp⟩    use p    simp only [mem_iUnion, exists_prop, 𝓑, mem_toFinset]    refine ⟨hp, ?_⟩    suffices (𝓘 p : Set X) ⊆ ball (𝔠 p) (r p) from this xp    apply Grid_subset_ball.trans ∘ ball_subset_ball    convert! (hr hp).1.le    simp only [r, dif_pos hp]  apply (OuterMeasureClass.measure_mono volume this).trans  -- Apply `measure_biUnion_le_lintegral` to `u := F.indicator 1` to bound the volume of ⋃ 𝓑.  let u := F.indicator (1 : X → ℝ≥0∞)  have h2u : ∀ p ∈ 𝓑, K * volume (Metric.ball (𝔠 p) (r p)) ≤      ∫⁻ (x : X) in ball (𝔠 p) (r p), u x := by    intro p h    simp_rw [𝓑, mem_toFinset] at h    simpa [u, lintegral_indicator, Measure.restrict_apply, measurableSet_F, r, h] using (hr h).2.le  have ineq := measure_biUnion_le_lintegral (A := defaultA a) _ K u h2u  simp only [u, lintegral_indicator, measurableSet_F, Pi.one_apply, lintegral_const,    MeasurableSet.univ, Measure.restrict_apply, univ_inter, one_mul] at ineq  rw [← ENNReal.mul_le_mul_iff_right K0.ne.symm K_ne_top]  apply ineq.trans_eq  -- Prove that the desired bound for the volume of ⋃ 𝓑 is equal to the bound proven above.  simp_rw [defaultA, Nat.cast_pow, Nat.cast_ofNat, ENNReal.coe_pow, coe_ofNat, K]  have : (volume G)⁻¹ * (2 ^ (2 * a + 5) * volume F) * (2 ^ (-5 : ℤ) * volume G) =      (2 ^ (2 * a + 5) * 2 ^ (-5 : ℤ)) * volume F * ((volume G)⁻¹ * volume G) := by ring  rw [ENNReal.div_eq_inv_mul, ← mul_one (_ * _), this]  congr  · have h : (2 : ℝ≥0∞) ^ (2 * a + 5) = (2 : ℝ≥0∞) ^ (2 * a + 5 : ℤ) := by norm_cast    rw [h, ← ENNReal.zpow_add (NeZero.ne 2) ofNat_ne_top, add_neg_cancel_right, ← pow_mul, mul_comm 2]    norm_cast  · exact ENNReal.inv_mul_cancel hG volume_G_ne_top |>.symm