| import Mathlib |
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| namespace OAI |
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| universe u_ι |
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| noncomputable section |
| namespace Problem310 |
| open MeasureTheory Set |
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| / |
| of the periodic grid with `q` cells. -/ |
| def gridBadCenters (q : ℕ) (δ : ℝ) : Set ℝ := |
| ⋃ k : ℤ, Ico ((k : ℝ) / q - δ) ((k : ℝ) / q) |
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| theorem mem_gridBadCenters_iff (q : ℕ) (δ x : ℝ) : |
| x ∈ gridBadCenters q δ ↔ ∃ k : ℤ, (k : ℝ) / q ∈ Ioc x (x + δ) := by |
| simp only [gridBadCenters, mem_iUnion, mem_Ico, mem_Ioc] |
| constructor |
| · rintro ⟨k, hk⟩ |
| exact ⟨k, by linarith [hk.1], by linarith [hk.2]⟩ |
| · rintro ⟨k, hk⟩ |
| exact ⟨k, by linarith [hk.2], by linarith [hk.1]⟩ |
|
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| theorem measurableSet_gridBadCenters (q : ℕ) (δ : ℝ) : |
| MeasurableSet (gridBadCenters q δ) := by |
| apply MeasurableSet.iUnion |
| intro k |
| exact measurableSet_Ico |
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| theorem gridBadCenters_add_one (q : ℕ) (hq : 0 < q) (δ x : ℝ) : |
| x + 1 ∈ gridBadCenters q δ ↔ x ∈ gridBadCenters q δ := by |
| rw [mem_gridBadCenters_iff, mem_gridBadCenters_iff] |
| have hq0 : (q : ℝ) ≠ 0 := by positivity |
| constructor |
| · rintro ⟨k, hk⟩ |
| refine ⟨k - q, ?_⟩ |
| have he : ((k - q : ℤ) : ℝ) / q = (k : ℝ) / q - 1 := by |
| push_cast |
| field_simp |
| rw [he] |
| exact ⟨by linarith [hk.1], by linarith [hk.2]⟩ |
| · rintro ⟨k, hk⟩ |
| refine ⟨k + q, ?_⟩ |
| have he : ((k + q : ℤ) : ℝ) / q = (k : ℝ) / q + 1 := by |
| push_cast |
| field_simp |
| rw [he] |
| exact ⟨by linarith [hk.1], by linarith [hk.2]⟩ |
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| / |
| intervals of length `δ`, provided `δ` is at most one cell width. -/ |
| theorem gridBadCenters_period_subset (q : ℕ) (hq : 0 < q) {δ : ℝ} |
| (hδ : δ ≤ (q : ℝ)⁻¹) : |
| gridBadCenters q δ ∩ Ico (0 : ℝ) 1 ⊆ |
| ⋃ i : Fin q, Ico (((i.val + 1 : ℕ) : ℝ) / q - δ) |
| (((i.val + 1 : ℕ) : ℝ) / q) := by |
| intro x hx |
| obtain ⟨k, hk⟩ := (mem_gridBadCenters_iff q δ x).mp hx.1 |
| have hqpos : (0 : ℝ) < q := by exact_mod_cast hq |
| have hkpos : 0 < k := by |
| have : (0 : ℝ) < (k : ℝ) / q := lt_of_le_of_lt hx.2.1 hk.1 |
| have := (div_pos_iff.mp this) |
| exact_mod_cast (by rcases this with h | h; exact h.1; linarith [h.2] : (0 : ℝ) < k) |
| have hklt : k < (q : ℤ) + 1 := by |
| have hstep : (k : ℝ) / q < 1 + (q : ℝ)⁻¹ := by linarith [hx.2.2, hk.2] |
| have hmul := (div_lt_iff₀ hqpos).mp hstep |
| have hi : (q : ℝ)⁻¹ * q = 1 := inv_mul_cancel₀ hqpos.ne' |
| have : (k : ℝ) < q + 1 := by nlinarith |
| exact_mod_cast this |
| have hkq : k.toNat ≤ q := by omega |
| have hkn : 0 < k.toNat := by omega |
| let i : Fin q := ⟨k.toNat - 1, by omega⟩ |
| apply mem_iUnion.mpr |
| refine ⟨i, ?_⟩ |
| have hki : ((i.val + 1 : ℕ) : ℝ) = (k : ℝ) := by |
| have : i.val + 1 = k.toNat := by dsimp [i]; omega |
| rw [this] |
| exact_mod_cast (Int.toNat_of_nonneg hkpos.le) |
| rw [hki] |
| exact ⟨by linarith [hk.2], hk.1⟩ |
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| theorem volume_gridBadCenters_period_le (q : ℕ) (hq : 0 < q) {δ : ℝ} |
| (_hδ0 : 0 ≤ δ) (hδ : δ ≤ (q : ℝ)⁻¹) : |
| volume (gridBadCenters q δ ∩ Icc (0 : ℝ) 1) ≤ ENNReal.ofReal ((q : ℝ) * δ) := by |
| have hae : (gridBadCenters q δ ∩ Icc (0 : ℝ) 1 : Set ℝ) =ᵐ[volume] |
| (gridBadCenters q δ ∩ Ico (0 : ℝ) 1 : Set ℝ) := by |
| exact Filter.EventuallyEqSet.inter Filter.EventuallyEq.rfl |
| (Ico_ae_eq_Icc (μ := volume) (a := (0 : ℝ)) (b := 1)).symm |
| rw [measure_congr hae] |
| calc |
| volume (gridBadCenters q δ ∩ Ico (0 : ℝ) 1) |
| ≤ volume (⋃ i : Fin q, Ico (((i.val + 1 : ℕ) : ℝ) / q - δ) |
| (((i.val + 1 : ℕ) : ℝ) / q)) := |
| measure_mono (gridBadCenters_period_subset q hq hδ) |
| _ ≤ ∑ i : Fin q, volume (Ico (((i.val + 1 : ℕ) : ℝ) / q - δ) |
| (((i.val + 1 : ℕ) : ℝ) / q)) := measure_iUnion_fintype_le _ _ |
| _ = ENNReal.ofReal ((q : ℝ) * δ) := by |
| simp only [Real.volume_Ico, sub_sub_cancel] |
| simp [ENNReal.ofReal_mul] |
|
|
| / |
| is bounded by the sum of its grid-width budgets. -/ |
| theorem volume_iUnion_gridBadCenters_period_le {ι : Type u_ι} [Fintype ι] |
| (q : ι → ℕ) (δ : ι → ℝ) (hq : ∀ i, 0 < q i) |
| (hδ0 : ∀ i, 0 ≤ δ i) (hδ : ∀ i, δ i ≤ (q i : ℝ)⁻¹) : |
| volume ((⋃ i, gridBadCenters (q i) (δ i)) ∩ Icc (0 : ℝ) 1) ≤ |
| ENNReal.ofReal (∑ i, (q i : ℝ) * δ i) := by |
| classical |
| rw [iUnion_inter] |
| calc |
| volume (⋃ i, gridBadCenters (q i) (δ i) ∩ Icc (0 : ℝ) 1) |
| ≤ ∑ i, volume (gridBadCenters (q i) (δ i) ∩ Icc (0 : ℝ) 1) := |
| measure_iUnion_fintype_le _ _ |
| _ ≤ ∑ i, ENNReal.ofReal ((q i : ℝ) * δ i) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| exact volume_gridBadCenters_period_le (q i) (hq i) (hδ0 i) (hδ i) |
| _ = ENNReal.ofReal (∑ i, (q i : ℝ) * δ i) := by |
| symm |
| apply ENNReal.ofReal_sum_of_nonneg |
| intro i hi |
| exact mul_nonneg (Nat.cast_nonneg _) (hδ0 i) |
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| end Problem310 |
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| end |
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| end OAI |
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