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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | import OAI.Geometry.CylinderCovering.Angular.ApertureDefinitions
namespace OAI
open Set
noncomputable section
namespace AngularAperture
lemma transverse_nonneg (y : ℝ) : 0 ≤ transverse y := by
unfold transverse
positivity
lemma transverse_half (q : ℝ) : transverse (q / 2) = profile q := by
unfold transverse profile
ring
lemma transverse_le {y : ℝ} (hy : |y| ≤ 1) : transverse y ≤ 5 / 4 := by
have hy2 : y ^ 2 ≤ 1 := by nlinarith [abs_le.mp hy, sq_abs y]
have hy4 : y ^ 4 ≤ 1 := by nlinarith [sq_nonneg (y ^ 2 - 1), sq_nonneg y]
unfold transverse
linarith
lemma transverse_lipschitz {x y : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1) :
|transverse x - transverse y| ≤ (9 / 2 : ℝ) * |x - y| := by
have hx2 : |x ^ 2| ≤ 1 := by simpa only [abs_pow] using pow_le_one₀ (abs_nonneg x) hx
have hy2 : |y ^ 2| ≤ 1 := by simpa only [abs_pow] using pow_le_one₀ (abs_nonneg y) hy
have hxy : |x + y| ≤ 2 := (abs_add_le x y).trans (by linarith)
have hsum : |1 / 4 + x ^ 2 + y ^ 2| ≤ 9 / 4 := by
calc
|1 / 4 + x ^ 2 + y ^ 2| ≤ |1 / 4 + x ^ 2| + |y ^ 2| := abs_add_le _ _
_ ≤ |(1 / 4 : ℝ)| + |x ^ 2| + |y ^ 2| := by gcongr; exact abs_add_le _ _
_ ≤ 9 / 4 := by
have hquarter : |(1 / 4 : ℝ)| = 1 / 4 := by norm_num
rw [hquarter]
linarith
have hid : transverse x - transverse y =
(x - y) * (x + y) * (1 / 4 + x ^ 2 + y ^ 2) := by
unfold transverse
ring
rw [hid, abs_mul, abs_mul]
calc
|x - y| * |x + y| * |1 / 4 + x ^ 2 + y ^ 2| ≤ |x - y| * 2 * (9 / 4) := by gcongr
_ = (9 / 2 : ℝ) * |x - y| := by ring
lemma tau_sq_le {τ : ℝ} (hτ : |τ| ≤ 1 / 4) : τ ^ 2 ≤ 1 / 16 := by
nlinarith [(abs_le.mp hτ).1, (abs_le.mp hτ).2, sq_abs τ]
lemma level_strictMonoOn {τ η q : ℝ} (hτ : |τ| ≤ 1 / 4) (hq : |q| ≤ 1) :
StrictMonoOn (level τ η q) (Icc 0 1) := by
intro x hx y hy hxy
have hxabs : |q * x| ≤ 1 := by
rw [abs_mul, abs_of_nonneg hx.1]
exact (mul_le_of_le_one_left hx.1 hq).trans hx.2
have hyabs : |q * y| ≤ 1 := by
rw [abs_mul, abs_of_nonneg hy.1]
exact (mul_le_of_le_one_left hy.1 hq).trans hy.2
have hD := transverse_lipschitz hxabs hyabs
have hdist : |q * x - q * y| ≤ y - x := by
rw [← mul_sub, abs_mul, abs_of_neg (sub_neg.mpr hxy)]
nlinarith [abs_nonneg q]
have hD' : transverse (q * y) - transverse (q * x) ≤ (9 / 2 : ℝ) * (y - x) := by
have := (abs_le.mp hD).1
nlinarith
have hmul := mul_le_mul_of_nonneg_left hD' (sq_nonneg τ)
have ht := tau_sq_le hτ
have hbound := mul_le_mul_of_nonneg_right ht (sub_nonneg.mpr hxy.le)
unfold level
nlinarith
lemma root_remainder {τ η q r : ℝ} (hq : |q| ≤ 1)
(hη1 : η ≤ 1)
(hr : r ∈ Icc (1 / 2 : ℝ) (3 / 4)) (hroot : level τ η q r = 0) :
|r - (1 / 2 + τ ^ 2 * (profile q + η))| ≤ (81 / 8 : ℝ) * τ ^ 4 := by
have hr0 : 0 ≤ r := by linarith [hr.1]
have habs : |q * r| ≤ 1 := by
rw [abs_mul, abs_of_nonneg hr0]
nlinarith [hr.2]
have hhalf : |q / 2| ≤ 1 := by
rw [abs_div]
norm_num
linarith
have hD := transverse_le habs
have hDelta : r - 1 / 2 ≤ (9 / 4 : ℝ) * τ ^ 2 := by
have hh : transverse (q * r) + η ≤ 9 / 4 := by linarith
have hm := mul_le_mul_of_nonneg_left hh (sq_nonneg τ)
unfold level at hroot
nlinarith
have hlip := transverse_lipschitz habs hhalf
have hdist : |q * r - q / 2| ≤ r - 1 / 2 := by
have hid : q * r - q / 2 = q * (r - 1 / 2) := by ring
rw [hid, abs_mul, abs_of_nonneg (sub_nonneg.mpr hr.1)]
exact mul_le_of_le_one_left (sub_nonneg.mpr hr.1) hq
have hb : |transverse (q * r) - transverse (q / 2)| ≤
(9 / 2 : ℝ) * (r - 1 / 2) := hlip.trans (by gcongr)
have heq : r - (1 / 2 + τ ^ 2 * (profile q + η)) =
τ ^ 2 * (transverse (q * r) - transverse (q / 2)) := by
rw [transverse_half]
unfold level at hroot
nlinarith
rw [heq, abs_mul, abs_of_nonneg (sq_nonneg τ)]
have hm1 := mul_le_mul_of_nonneg_left hb (sq_nonneg τ)
have hm2 := mul_le_mul_of_nonneg_left hDelta (sq_nonneg τ)
nlinarith
lemma roots_lipschitz {τ η q p r s : ℝ} (hτ : |τ| ≤ 1 / 4)
(hq : |q| ≤ 1) (hp : |p| ≤ 1)
(hr : r ∈ Icc (1 / 2 : ℝ) (3 / 4))
(hs : s ∈ Icc (1 / 2 : ℝ) (3 / 4))
(hrootr : level τ η q r = 0) (hroots : level τ η p s = 0) :
|r - s| ≤ 7 * τ ^ 2 * |q - p| := by
have hr0 : 0 ≤ r := by linarith [hr.1]
have hs0 : 0 ≤ s := by linarith [hs.1]
have hqr : |q * r| ≤ 1 := by
rw [abs_mul, abs_of_nonneg hr0]
nlinarith [hr.2]
have hps : |p * s| ≤ 1 := by
rw [abs_mul, abs_of_nonneg hs0]
nlinarith [hs.2]
have hcoord : |q * r - p * s| ≤ |r - s| + |q - p| := by
have hid : q * r - p * s = q * (r - s) + (q - p) * s := by ring
rw [hid]
calc
|q * (r - s) + (q - p) * s| ≤ |q * (r - s)| + |(q - p) * s| := abs_add_le _ _
_ = |q| * |r - s| + |q - p| * s := by rw [abs_mul, abs_mul, abs_of_nonneg hs0]
_ ≤ |r - s| + |q - p| := by
have h1 := mul_le_of_le_one_left (abs_nonneg (r - s)) hq
have h2 := mul_le_of_le_one_right (abs_nonneg (q - p)) (show s ≤ 1 by linarith [hs.2])
linarith
have hD : |transverse (q * r) - transverse (p * s)| ≤
(9 / 2 : ℝ) * (|r - s| + |q - p|) :=
(transverse_lipschitz hqr hps).trans (by gcongr)
have heq : r - s = τ ^ 2 * (transverse (q * r) - transverse (p * s)) := by
unfold level at hrootr hroots
nlinarith
have hself : |r - s| ≤ τ ^ 2 * ((9 / 2 : ℝ) * (|r - s| + |q - p|)) := by
calc
|r - s| = τ ^ 2 * |transverse (q * r) - transverse (p * s)| := by
rw [heq, abs_mul, abs_of_nonneg (sq_nonneg τ)]
_ ≤ _ := mul_le_mul_of_nonneg_left hD (sq_nonneg τ)
have habs := mul_le_mul_of_nonneg_right (tau_sq_le hτ) (abs_nonneg (r - s))
have hn := mul_nonneg (sq_nonneg τ) (abs_nonneg (q - p))
nlinarith
end AngularAperture
end
end OAI
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