| import Mathlib |
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| namespace OAI |
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|
| noncomputable section |
| open Set Filter MeasureTheory Metric |
| open scoped Topology ENNReal NNReal |
| open Set Filter MeasureTheory BoxIntegral |
| open scoped Topology |
| namespace YauCounterexamples |
|
|
| def middleBox (I : Box (Fin 3)) : Box (Fin 3) where |
| lower i := (3*I.lower i+I.upper i)/4 |
| upper i := (I.lower i+3*I.upper i)/4 |
| lower_lt_upper i := by linarith [I.lower_lt_upper i] |
|
|
| lemma middleBox_Icc_subset (I : Box (Fin 3)) : Box.Icc (middleBox I) ⊆ Box.Ioo I := by |
| intro x hx i _ |
| have hlo := hx.1 i |
| have hup := hx.2 i |
| change (3*I.lower i+I.upper i)/4 ≤ x i at hlo |
| change x i ≤ (I.lower i+3*I.upper i)/4 at hup |
| constructor <;> linarith [I.lower_lt_upper i] |
|
|
| lemma middleBox_volume (I : Box (Fin 3)) : |
| (volume (Box.Icc (middleBox I))).toReal = (volume (Box.Ioo I)).toReal/8 := by |
| rw [←measure_congr (middleBox I).coe_ae_eq_Icc, |
| ←measure_congr (I.coe_ae_eq_Icc.trans I.Ioo_ae_eq_Icc.symm)] |
| simp only [Box.volume_apply'] |
| simp only [middleBox] |
| rw [Fin.prod_univ_three,Fin.prod_univ_three] |
| ring |
|
|
| lemma middleBox_shift_mem (I : Box (Fin 3)) {d : ℝ} (hd : 0 ≤ d) |
| (hsmall : ∀ i, d < (I.upper i-I.lower i)/4) |
| {x : Fin 3 → ℝ} (hx : x ∈ Box.Icc (middleBox I)) (j : Fin 3) : |
| x+Pi.single j d ∈ Box.Ioo I := by |
| intro i _ |
| have hxi := middleBox_Icc_subset I hx i trivial |
| have hu := hx.2 i |
| change x i ≤ (I.lower i+3*I.upper i)/4 at hu |
| by_cases hij : i=j |
| · subst i |
| simp only [Pi.add_apply,Pi.single_eq_same] |
| constructor <;> linarith [hsmall j,hxi.1] |
| · simpa only [Pi.add_apply,Pi.single_eq_of_ne hij,add_zero] using hxi |
|
|
| lemma middleBox_eventually_shift_mem (I : Box (Fin 3)) {ε ℓ : ℝ} |
| (hε : 0 < ε) (hℓ : 0 < ℓ) : |
| ∀ᶠ n : ℝ in atTop, ∀ x ∈ Box.Icc (middleBox I), ∀ j : Fin 3, |
| x+Pi.single j (ε/(n*ℓ)) ∈ Box.Ioo I := by |
| have hd : Tendsto (fun n : ℝ => ε/(n*ℓ)) atTop (𝓝 0) := by |
| have ht := (tendsto_const_nhds (x := ε)).mul (tendsto_inv_atTop_zero.mul_const (ℓ⁻¹)) |
| simpa only [mul_zero,zero_mul,div_eq_mul_inv,mul_inv_rev,mul_comm (ℓ⁻¹)] using ht |
| have he : ∀ᶠ n : ℝ in atTop, ∀ j : Fin 3, ε/(n*ℓ) < (I.upper j-I.lower j)/4 := by |
| rw [Filter.eventually_all] |
| intro j |
| exact (tendsto_order.mp hd).2 _ (by linarith [I.lower_lt_upper j]) |
| filter_upwards [he,eventually_ge_atTop (1:ℝ)] with n hn hn1 |
| intro x hx j |
| exact middleBox_shift_mem I (div_nonneg hε.le (mul_nonneg (by linarith) hℓ.le)) hn hx j |
|
|
| end YauCounterexamples |
| end |
|
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| end OAI |
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