| import OAI.Geometry.TriangularCovering.Main |
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| namespace OAI |
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| noncomputable section |
| namespace TriangularCovering |
| open Set MeasureTheory |
| open scoped Interval |
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| lemma d_lipschitz_sharp {a q : ℝ} (ha : |a| ≤ 1) (hq : |q| ≤ 1) : |
| |d a - d q| ≤ (3 / 8 : ℝ) * |a - q| := by |
| have ha2 : a ^ 2 ≤ 1 := by |
| nlinarith only [(abs_le.mp ha).1, (abs_le.mp ha).2] |
| have hq2 : q ^ 2 ≤ 1 := by |
| nlinarith only [(abs_le.mp hq).1, (abs_le.mp hq).2] |
| have has : |a + q| ≤ 2 := (abs_add_le _ _).trans (by linarith only [ha, hq]) |
| have hs0 : 0 ≤ 1 + a ^ 2 + q ^ 2 := by positivity |
| have hs1 : 1 + a ^ 2 + q ^ 2 ≤ 3 := by linarith only [ha2, hq2] |
| have hid : d a - d q = (a - q) * ((a + q) * (1 + a ^ 2 + q ^ 2) / 16) := by |
| dsimp [d] |
| ring |
| have hfactor : |(a + q) * (1 + a ^ 2 + q ^ 2) / 16| ≤ (3 / 8 : ℝ) := by |
| rw [abs_div, abs_mul, abs_of_nonneg hs0] |
| norm_num |
| have hp := mul_le_mul has hs1 hs0 (by norm_num : (0 : ℝ) ≤ 2) |
| linarith only [hp] |
| rw [hid, abs_mul, mul_comm (3 / 8 : ℝ)] |
| exact mul_le_mul_of_nonneg_left hfactor (abs_nonneg _) |
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| lemma cutoffCoeff_error_sharp {a b q : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) |
| (hq : a ≤ q ∧ q ≤ b) : |
| |cutoffCoeff a b - (η + d q)| ≤ (3 / 8 : ℝ) * (b - a) := by |
| have hqab : |q| ≤ 1 := abs_le.mpr |
| ⟨(abs_le.mp ha).1.trans hq.1, hq.2.trans (abs_le.mp hb).2⟩ |
| have h1 := d_lipschitz_sharp ha hqab |
| have h2 := d_lipschitz_sharp hb hqab |
| rw [abs_of_nonpos (sub_nonpos.mpr hq.1)] at h1 |
| rw [abs_of_nonneg (sub_nonneg.mpr hq.2)] at h2 |
| have hlo := d_le_max_endpoints hq.1 hq.2 |
| have hhi : max (d a) (d b) ≤ d q + (3 / 8 : ℝ) * (b - a) := by |
| apply max_le <;> linarith only [(abs_le.mp h1).2, (abs_le.mp h2).2, hq.1, hq.2] |
| dsimp [cutoffCoeff] |
| rw [abs_le] |
| constructor <;> linarith only [hlo, hhi, hq.1, hq.2] |
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| lemma density_error_sharp {a b q : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) |
| (hq : a ≤ q ∧ q ≤ b) : |
| |cutoffCoeff a b - speedSq a b / 8 - areaDensity q| ≤ (b - a) / 2 := by |
| have hqab : |q| ≤ 1 := abs_le.mpr |
| ⟨(abs_le.mp ha).1.trans hq.1, hq.2.trans (abs_le.mp hb).2⟩ |
| have hc := coeff_bounds ha hb |
| have hal : |alpha a b| ≤ 1 / 2 := by |
| rw [abs_of_nonneg hc.1] |
| exact hc.2.1 |
| have hq2 : q ^ 2 ≤ 1 := by |
| nlinarith only [(abs_le.mp hqab).1, (abs_le.mp hqab).2] |
| have hql : |(1 + q ^ 2) / 4| ≤ 1 / 2 := by |
| rw [abs_of_nonneg (by positivity)] |
| linarith only [hq2] |
| have haq := sq_error hal hql (alpha_error ha hqab hq.1 hq.2) |
| have hqr : |q / 2| ≤ 1 / 2 := by |
| rw [abs_div] |
| norm_num |
| linarith only [hqab] |
| have hbq := sq_error hc.2.2.1 hqr (beta_error hq.1 hq.2) |
| have hspeed : |speedSq a b - (((1 + q ^ 2) / 4) ^ 2 + 2 * (q / 2) ^ 2)| ≤ b - a := by |
| have hid : speedSq a b - (((1 + q ^ 2) / 4) ^ 2 + 2 * (q / 2) ^ 2) = |
| ((alpha a b) ^ 2 - ((1 + q ^ 2) / 4) ^ 2) + |
| 2 * ((beta a b) ^ 2 - (q / 2) ^ 2) := by |
| dsimp [speedSq] |
| ring |
| rw [hid] |
| have h := abs_add_le ((alpha a b) ^ 2 - ((1 + q ^ 2) / 4) ^ 2) |
| (2 * ((beta a b) ^ 2 - (q / 2) ^ 2)) |
| rw [abs_mul] at h |
| norm_num at h |
| linarith only [h, haq, hbq] |
| have hM := cutoffCoeff_error_sharp ha hb hq |
| have hid : cutoffCoeff a b - speedSq a b / 8 - areaDensity q = |
| (cutoffCoeff a b - (η + d q)) - |
| (speedSq a b - (((1 + q ^ 2) / 4) ^ 2 + 2 * (q / 2) ^ 2)) / 8 := by |
| dsimp [areaDensity] |
| ring |
| rw [hid] |
| have hr := abs_add_le (cutoffCoeff a b - (η + d q)) |
| (-((speedSq a b - (((1 + q ^ 2) / 4) ^ 2 + 2 * (q / 2) ^ 2)) / 8)) |
| rw [← sub_eq_add_neg, abs_neg] at hr |
| rw [abs_div] at hr |
| norm_num at hr |
| linarith only [hr, hM, hspeed] |
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|
| lemma interval_density_error_sharp {a b : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) |
| (hab : a ≤ b) : |
| |(b - a) * (cutoffCoeff a b - speedSq a b / 8) - |
| (∫ q in a..b, areaDensity q)| ≤ (b - a) ^ 2 / 2 := by |
| have hu : ∀ q ∈ Ι a b, |
| ‖cutoffCoeff a b - speedSq a b / 8 - areaDensity q‖ ≤ (b - a) / 2 := by |
| intro q hq |
| rw [Set.uIoc_of_le hab] at hq |
| simpa only [Real.norm_eq_abs] using density_error_sharp ha hb ⟨hq.1.le, hq.2⟩ |
| have h := intervalIntegral.norm_integral_le_of_norm_le_const hu |
| rw [intervalIntegral.integral_sub (f := fun _ : ℝ => cutoffCoeff a b - speedSq a b / 8) |
| (g := areaDensity) (continuous_const.intervalIntegrable _ _) |
| (continuous_areaDensity.intervalIntegrable _ _), intervalIntegral.integral_const, |
| Real.norm_eq_abs, smul_eq_mul, abs_of_nonneg (sub_nonneg.mpr hab)] at h |
| nlinarith only [h] |
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| lemma coefficientSum_error_sharp {N : ℕ} (hN : 0 < N) : |
| |coefficientSum N + 13 / 6000| ≤ step N := by |
| have hi := integral_mesh_sum hN |
| have hid : coefficientSum N + 13 / 6000 = |
| ∑ j : Fin N, (step N * (cutoffCoeff (mesh N j) (mesh N (j + 1)) - |
| speedSq (mesh N j) (mesh N (j + 1)) / 8) - |
| ∫ q in mesh N j..mesh N (j + 1), areaDensity q) := by |
| rw [Finset.sum_sub_distrib, hi] |
| dsimp [coefficientSum] |
| ring |
| rw [hid] |
| calc |
| _ ≤ ∑ j : Fin N, |step N * (cutoffCoeff (mesh N j) (mesh N (j + 1)) - |
| speedSq (mesh N j) (mesh N (j + 1)) / 8) - |
| ∫ q in mesh N j..mesh N (j + 1), areaDensity q| := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ _j : Fin N, (step N) ^ 2 / 2 := by |
| apply Finset.sum_le_sum |
| intro j _ |
| have h := interval_density_error_sharp (mesh_abs_le hN j.isLt.le) |
| (mesh_abs_le hN j.isLt) |
| (le_of_lt (by have := mesh_succ N j; have := step_pos hN; linarith)) |
| simpa only [mesh_succ] using h |
| _ = step N := by |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
| calc |
| (N : ℝ) * ((step N) ^ 2 / 2) = ((N : ℝ) * step N) * step N / 2 := by ring |
| _ = step N := by rw [count_step hN]; ring |
|
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| end TriangularCovering |
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| end |
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| end OAI |
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