| import OAI.Geometry.CylinderCovering.Angular.CartesianFailure | |
| namespace OAI | |
| namespace AngularCartesianValidation | |
| theorem sqrt_quarter_cubic_remainder {v : ℝ} (hv : |v| ≤ 1 / 8) : | |
| |Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)| ≤ 6 * |v| ^ 3 := by | |
| have hvl := (abs_le.mp hv).1 | |
| have hvr := (abs_le.mp hv).2 | |
| have hv2 : v ^ 2 ≤ 1 / 64 := by nlinarith only [hvl, hvr] | |
| have hq : 0 ≤ 1 / 4 + v := by linarith | |
| let r := Real.sqrt (1 / 4 + v) | |
| let p := 1 / 2 + v - v ^ 2 | |
| have hr0 : 0 ≤ r := Real.sqrt_nonneg _ | |
| have hrsq : r ^ 2 = 1 / 4 + v := Real.sq_sqrt hq | |
| have hr : 1 / 4 ≤ r := by nlinarith only [hr0, hrsq, hvl] | |
| have hp : 1 / 4 ≤ p := by dsimp [p]; linarith | |
| have hden : 1 / 2 ≤ r + p := by linarith | |
| have hden0 : 0 ≤ r + p := by linarith | |
| have hid : (r - p) * (r + p) = 2 * v ^ 3 - v ^ 4 := by | |
| dsimp [p] | |
| nlinarith only [hrsq] | |
| have hv4 : |v| ^ 4 ≤ |v| ^ 3 := by | |
| have hv1 : |v| ≤ 1 := hv.trans (by norm_num) | |
| nlinarith only [mul_le_mul_of_nonneg_left hv1 (pow_nonneg (abs_nonneg v) 3)] | |
| have hres : |2 * v ^ 3 - v ^ 4| ≤ 3 * |v| ^ 3 := by | |
| calc | |
| |2 * v ^ 3 - v ^ 4| ≤ |2 * v ^ 3| + |v ^ 4| := abs_sub _ _ | |
| _ = 2 * |v| ^ 3 + |v| ^ 4 := by rw [abs_mul, abs_pow, abs_pow]; norm_num | |
| _ ≤ 3 * |v| ^ 3 := by linarith | |
| have hprod : |r - p| * (r + p) ≤ 3 * |v| ^ 3 := by | |
| calc | |
| |r - p| * (r + p) = |(r - p) * (r + p)| := by | |
| rw [abs_mul, abs_of_nonneg hden0] | |
| _ = |2 * v ^ 3 - v ^ 4| := congrArg abs hid | |
| _ ≤ 3 * |v| ^ 3 := hres | |
| have hlo : |r - p| / 2 ≤ |r - p| * (r + p) := by | |
| nlinarith only [mul_nonneg (abs_nonneg (r - p)) (sub_nonneg.mpr hden)] | |
| change |r - p| ≤ 6 * |v| ^ 3 | |
| linarith only [hlo, hprod] | |
| theorem linear_quadratic_perturbation_bound {τ z a M : ℝ} | |
| (hτ : 0 ≤ τ) (hz : |z| ≤ 1) (ha : |a| ≤ M) | |
| (hsmall : τ * M ≤ 1) : | |
| |τ * z + τ ^ 2 * a| ≤ 2 * τ := by | |
| calc | |
| |τ * z + τ ^ 2 * a| ≤ |τ * z| + |τ ^ 2 * a| := abs_add_le _ _ | |
| _ = τ * |z| + τ ^ 2 * |a| := by | |
| rw [abs_mul, abs_of_nonneg hτ, abs_mul, abs_of_nonneg (sq_nonneg τ)] | |
| _ ≤ τ * 1 + τ ^ 2 * M := by gcongr | |
| _ ≤ 2 * τ := by | |
| nlinarith only [mul_le_mul_of_nonneg_left hsmall hτ] | |
| theorem linear_quadratic_square_error {τ z a M : ℝ} | |
| (hτ : 0 ≤ τ) (hz : |z| ≤ 1) (ha : |a| ≤ M) | |
| (hsmall : τ * M ≤ 1) : | |
| |(τ * z + τ ^ 2 * a) ^ 2 - (τ * z) ^ 2| ≤ 3 * M * τ ^ 3 := by | |
| have hv := linear_quadratic_perturbation_bound hτ hz ha hsmall | |
| have hzτ : |τ * z| ≤ τ := by | |
| rw [abs_mul, abs_of_nonneg hτ] | |
| nlinarith only [mul_le_mul_of_nonneg_left hz hτ] | |
| have hplus : |(τ * z + τ ^ 2 * a) + τ * z| ≤ 3 * τ := | |
| (abs_add_le _ _).trans (by linarith) | |
| have hminus : |(τ * z + τ ^ 2 * a) - τ * z| ≤ τ ^ 2 * M := by | |
| calc | |
| |(τ * z + τ ^ 2 * a) - τ * z| = τ ^ 2 * |a| := by | |
| rw [add_sub_cancel_left, abs_mul, abs_of_nonneg (sq_nonneg τ)] | |
| _ ≤ τ ^ 2 * M := mul_le_mul_of_nonneg_left ha (sq_nonneg τ) | |
| have hM : 0 ≤ M := (abs_nonneg a).trans ha | |
| calc | |
| |(τ * z + τ ^ 2 * a) ^ 2 - (τ * z) ^ 2| = | |
| |(τ * z + τ ^ 2 * a) - τ * z| * |(τ * z + τ ^ 2 * a) + τ * z| := by | |
| rw [← abs_mul] | |
| congr 1 | |
| ring | |
| _ ≤ (τ ^ 2 * M) * (3 * τ) := | |
| mul_le_mul hminus hplus (abs_nonneg _) (mul_nonneg (sq_nonneg τ) hM) | |
| _ = 3 * M * τ ^ 3 := by ring | |
| theorem paired_sqrt_cubic_remainder {τ z a b M : ℝ} | |
| (hτ : 0 ≤ τ) (hz : |z| ≤ 1) (ha : |a| ≤ M) (hb : |b| ≤ M) | |
| (hM : 1 ≤ M) (hsmall : τ * M ≤ 1 / 16) : | |
| |Real.sqrt (1 / 4 + τ * z + τ ^ 2 * a) + | |
| Real.sqrt (1 / 4 - τ * z + τ ^ 2 * b) - | |
| (1 + τ ^ 2 * (a + b - 2 * z ^ 2))| ≤ 128 * M * τ ^ 3 := by | |
| have hsmall1 : τ * M ≤ 1 := hsmall.trans (by norm_num) | |
| have hτ16 : τ ≤ 1 / 16 := by | |
| nlinarith only [mul_le_mul_of_nonneg_left hM hτ, hsmall] | |
| have hnz : |-z| ≤ 1 := by simpa only [abs_neg] using hz | |
| have hva := linear_quadratic_perturbation_bound hτ hz ha hsmall1 | |
| have hvb := linear_quadratic_perturbation_bound hτ hnz hb hsmall1 | |
| have hqa := sqrt_quarter_cubic_remainder (hva.trans (by linarith : 2 * τ ≤ 1 / 8)) | |
| have hqb := sqrt_quarter_cubic_remainder (hvb.trans (by linarith : 2 * τ ≤ 1 / 8)) | |
| have hpowA : |τ * z + τ ^ 2 * a| ^ 3 ≤ (2 * τ) ^ 3 := | |
| pow_le_pow_left₀ (abs_nonneg _) hva 3 | |
| have hpowB : |τ * -z + τ ^ 2 * b| ^ 3 ≤ (2 * τ) ^ 3 := | |
| pow_le_pow_left₀ (abs_nonneg _) hvb 3 | |
| have hqa' : |Real.sqrt (1 / 4 + (τ * z + τ ^ 2 * a)) - | |
| (1 / 2 + (τ * z + τ ^ 2 * a) - (τ * z + τ ^ 2 * a) ^ 2)| ≤ 48 * τ ^ 3 := by | |
| nlinarith only [hqa, hpowA] | |
| have hqb' : |Real.sqrt (1 / 4 + (τ * -z + τ ^ 2 * b)) - | |
| (1 / 2 + (τ * -z + τ ^ 2 * b) - (τ * -z + τ ^ 2 * b) ^ 2)| ≤ 48 * τ ^ 3 := by | |
| nlinarith only [hqb, hpowB] | |
| have hsa := linear_quadratic_square_error hτ hz ha hsmall1 | |
| have hsb := linear_quadratic_square_error hτ hnz hb hsmall1 | |
| have hMτ : τ ^ 3 ≤ M * τ ^ 3 := by | |
| nlinarith only [mul_le_mul_of_nonneg_right hM (pow_nonneg hτ 3)] | |
| rw [abs_le] at hqa' hqb' hsa hsb ⊢ | |
| have hrewA : 1 / 4 + (τ * z + τ ^ 2 * a) = 1 / 4 + τ * z + τ ^ 2 * a := by ring | |
| have hrewB : 1 / 4 + (τ * -z + τ ^ 2 * b) = 1 / 4 - τ * z + τ ^ 2 * b := by ring | |
| rw [hrewA] at hqa' | |
| rw [hrewB] at hqb' | |
| constructor <;> nlinarith only [hqa'.1, hqa'.2, hqb'.1, hqb'.2, | |
| hsa.1, hsa.2, hsb.1, hsb.2, hMτ, pow_nonneg hτ 3] | |
| theorem cartesian_root_sum_expansion {x y τ η : ℝ} | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hτ : 0 ≤ τ) (hη : 0 ≤ η) | |
| (hsmall : τ * (2 + η) ≤ 1 / 16) : | |
| |Real.sqrt (1 / 4 + τ * x * y + | |
| τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4)) + | |
| Real.sqrt (1 / 4 - τ * x * y + | |
| τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4)) - | |
| (1 + τ ^ 2 * ((x ^ 2 - y ^ 2) ^ 2 + η))| ≤ | |
| 128 * (2 + η) * τ ^ 3 := by | |
| have hx2 : x ^ 2 ≤ 1 := by | |
| nlinarith only [(abs_le.mp hx).1, (abs_le.mp hx).2] | |
| have hy2 : y ^ 2 ≤ 1 := by | |
| nlinarith only [(abs_le.mp hy).1, (abs_le.mp hy).2] | |
| have hx4 : x ^ 4 ≤ 1 := by | |
| nlinarith only [mul_nonneg (sq_nonneg x) (sub_nonneg.mpr hx2), hx2] | |
| have hy4 : y ^ 4 ≤ 1 := by | |
| nlinarith only [mul_nonneg (sq_nonneg y) (sub_nonneg.mpr hy2), hy2] | |
| have hz : |x * y| ≤ 1 := by | |
| rw [abs_mul] | |
| nlinarith only [mul_le_mul hx hy (abs_nonneg y) (by norm_num : (0 : ℝ) ≤ 1)] | |
| have ha : |(y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4| ≤ 2 + η := by | |
| rw [abs_le] | |
| constructor <;> nlinarith only [hx2, hy2, hx4, hy4, hη, | |
| sq_nonneg x, sq_nonneg y, sq_nonneg (x ^ 2), sq_nonneg (y ^ 2)] | |
| have hb : |(x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4| ≤ 2 + η := by | |
| rw [abs_le] | |
| constructor <;> nlinarith only [hx2, hy2, hx4, hy4, hη, | |
| sq_nonneg x, sq_nonneg y, sq_nonneg (x ^ 2), sq_nonneg (y ^ 2)] | |
| have h := paired_sqrt_cubic_remainder hτ hz ha hb (by linarith : 1 ≤ 2 + η) hsmall | |
| have hid : ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4) + | |
| ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4) - 2 * (x * y) ^ 2 = | |
| (x ^ 2 - y ^ 2) ^ 2 + η := by ring | |
| rw [hid] at h | |
| simpa only [mul_assoc] using h | |
| theorem cartesian_root_sum_gt_one {x y τ η : ℝ} | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hτ : 0 < τ) (hη : 0 < η) | |
| (hsmall : τ * (2 + η) ≤ 1 / 16) | |
| (hmargin : τ * (256 * (2 + η)) ≤ η) : | |
| 1 < Real.sqrt (1 / 4 + τ * x * y + | |
| τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4)) + | |
| Real.sqrt (1 / 4 - τ * x * y + | |
| τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4)) := by | |
| have hexp := cartesian_root_sum_expansion hx hy hτ.le hη.le hsmall | |
| have hbudget := mul_le_mul_of_nonneg_right hmargin (sq_nonneg τ) | |
| have hpositive := mul_pos hη (sq_pos_of_pos hτ) | |
| have hquartic := mul_nonneg (sq_nonneg τ) (sq_nonneg (x ^ 2 - y ^ 2)) | |
| nlinarith only [(abs_le.mp hexp).1, hbudget, hpositive, hquartic] | |
| end AngularCartesianValidation | |
| end OAI | |