| import OAI.Geometry.CylinderCovering.Angular.ApertureDefinitions |
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| namespace OAI |
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| open Set |
| noncomputable section |
| namespace AngularAperture |
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| lemma transverse_nonneg (y : ℝ) : 0 ≤ transverse y := by |
| unfold transverse |
| positivity |
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| lemma transverse_half (q : ℝ) : transverse (q / 2) = profile q := by |
| unfold transverse profile |
| ring |
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| 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 |
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| 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 |
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| 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] |
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| 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 |
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| 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 |
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| 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 |
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| end AngularAperture |
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| end |
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| end OAI |
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