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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import OAI.Geometry.TriangularCovering.Main
namespace OAI
noncomputable section
namespace TriangularCovering
open Set MeasureTheory
open scoped Interval
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 _)
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]
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]
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]
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
end TriangularCovering
end
end OAI
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