| import OAI.Geometry.CylinderCovering.Cartesian.CapsDefinitions | |
| namespace OAI | |
| namespace CylinderCovering.CartesianCaps | |
| theorem sqrt_quarter_cubic_remainder {v : ℝ} (hv : -(1 / 4 : ℝ) ≤ v) : | |
| |Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)| ≤ 12 * |v| ^ 3 := by | |
| let z : ℝ := Real.sqrt (1 / 4 + v) - 1 / 2 | |
| have ht0 := Real.sqrt_nonneg (1 / 4 + v) | |
| have ht2 := Real.sq_sqrt (show 0 ≤ 1 / 4 + v by linarith) | |
| have hzid : v = z + z ^ 2 := by dsimp [z]; nlinarith only [ht2] | |
| have hzlower : -(1 / 2 : ℝ) ≤ z := by dsimp [z]; linarith only [ht0] | |
| have hzbound : |z| ≤ 2 * |v| := by | |
| by_cases hz : 0 ≤ z | |
| · rw [abs_of_nonneg hz] | |
| have hv0 : 0 ≤ v := by nlinarith only [hzid, hz, sq_nonneg z] | |
| rw [abs_of_nonneg hv0] | |
| nlinarith only [hzid, hz, sq_nonneg z] | |
| · have hz0 := le_of_not_ge hz | |
| rw [abs_of_nonpos hz0] | |
| have hv0 : v ≤ 0 := by nlinarith only [hzid, hz0, hzlower] | |
| rw [abs_of_nonpos hv0] | |
| nlinarith only [hzid, hz0, hzlower] | |
| have hvz : |v + z| ≤ 3 * |v| := (abs_add_le _ _).trans (by linarith) | |
| have hzsq : |z| ^ 2 ≤ (2 * |v|) ^ 2 := | |
| (sq_le_sq₀ (abs_nonneg _) (by positivity)).2 hzbound | |
| calc | |
| _ = |z ^ 2 * (v + z)| := by | |
| congr 1 | |
| dsimp [z] at * | |
| nlinarith only [ht2] | |
| _ = |z| ^ 2 * |v + z| := by rw [abs_mul, abs_pow] | |
| _ ≤ (2 * |v|) ^ 2 * (3 * |v|) := | |
| mul_le_mul hzsq hvz (abs_nonneg _) (by positivity) | |
| _ = 12 * |v| ^ 3 := by ring | |
| theorem sqrt_quarter_perturbed_remainder {τ M a B : ℝ} | |
| (hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hM : 0 ≤ M) | |
| (ha : |a| ≤ 1) (hB : |B| ≤ M) (hsmall : (M + 1) * τ ≤ 1 / 4) : | |
| |Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) - | |
| (1 / 2 + τ * a + τ ^ 2 * (B - a ^ 2))| ≤ | |
| (12 * (M + 1) ^ 3 + M * (M + 2)) * τ ^ 3 := by | |
| let v : ℝ := τ * a + τ ^ 2 * B | |
| have hτ2 : τ ^ 2 ≤ τ := by nlinarith | |
| have haτ : |τ * a| ≤ τ := by | |
| rw [abs_mul, abs_of_nonneg hτ0] | |
| simpa using mul_le_mul_of_nonneg_left ha hτ0 | |
| have hBτ : |τ ^ 2 * B| ≤ τ ^ 2 * M := by | |
| rw [abs_mul, abs_of_nonneg (sq_nonneg τ)] | |
| exact mul_le_mul_of_nonneg_left hB (sq_nonneg τ) | |
| have hv : |v| ≤ (M + 1) * τ := by | |
| have hMτ := mul_le_mul_of_nonneg_left hτ2 hM | |
| exact (abs_add_le _ _).trans (by nlinarith only [haτ, hBτ, hMτ]) | |
| have hvdom : -(1 / 4 : ℝ) ≤ v := by | |
| linarith only [(abs_le.mp hv).1, hsmall] | |
| have hvpow : |v| ^ 3 ≤ ((M + 1) * τ) ^ 3 := by gcongr | |
| have htaylor : |Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)| ≤ | |
| 12 * (M + 1) ^ 3 * τ ^ 3 := by | |
| calc | |
| _ ≤ 12 * |v| ^ 3 := sqrt_quarter_cubic_remainder hvdom | |
| _ ≤ 12 * ((M + 1) * τ) ^ 3 := by gcongr | |
| _ = 12 * (M + 1) ^ 3 * τ ^ 3 := by ring | |
| have hplus : |v + τ * a| ≤ (M + 2) * τ := | |
| (abs_add_le _ _).trans (by nlinarith only [hv, haτ]) | |
| have hminus : |v - τ * a| ≤ τ ^ 2 * M := by | |
| simpa [v] using hBτ | |
| have hsq : |v ^ 2 - (τ * a) ^ 2| ≤ M * (M + 2) * τ ^ 3 := by | |
| calc | |
| _ = |v - τ * a| * |v + τ * a| := by | |
| rw [← abs_mul] | |
| congr 1 | |
| ring | |
| _ ≤ (τ ^ 2 * M) * ((M + 2) * τ) := | |
| mul_le_mul hminus hplus (abs_nonneg _) (by positivity) | |
| _ = M * (M + 2) * τ ^ 3 := by ring | |
| have hsplit : Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) - | |
| (1 / 2 + τ * a + τ ^ 2 * (B - a ^ 2)) = | |
| (Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)) - | |
| (v ^ 2 - (τ * a) ^ 2) := by | |
| dsimp [v] | |
| rw [show 1 / 4 + (τ * a + τ ^ 2 * B) = 1 / 4 + τ * a + τ ^ 2 * B by ring] | |
| ring | |
| rw [hsplit] | |
| exact (abs_sub _ _).trans (by nlinarith only [htaylor, hsq]) | |
| theorem sqrt_quarter_pair_remainder {τ M a B C : ℝ} | |
| (hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hM : 0 ≤ M) | |
| (ha : |a| ≤ 1) (hB : |B| ≤ M) (hC : |C| ≤ M) | |
| (hsmall : (M + 1) * τ ≤ 1 / 4) : | |
| |Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) + | |
| Real.sqrt (1 / 4 - τ * a + τ ^ 2 * C) - | |
| (1 + τ ^ 2 * (B + C - 2 * a ^ 2))| ≤ | |
| (24 * (M + 1) ^ 3 + 2 * M * (M + 2)) * τ ^ 3 := by | |
| have hfirst := sqrt_quarter_perturbed_remainder hτ0 hτ1 hM ha hB hsmall | |
| have hsecond := sqrt_quarter_perturbed_remainder hτ0 hτ1 hM | |
| (a := -a) (by simpa only [abs_neg] using ha) hC hsmall | |
| have hneg : 1 / 4 + τ * -a + τ ^ 2 * C = 1 / 4 - τ * a + τ ^ 2 * C := by ring | |
| rw [hneg] at hsecond | |
| have hsplit : Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) + | |
| Real.sqrt (1 / 4 - τ * a + τ ^ 2 * C) - | |
| (1 + τ ^ 2 * (B + C - 2 * a ^ 2)) = | |
| (Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) - | |
| (1 / 2 + τ * a + τ ^ 2 * (B - a ^ 2))) + | |
| (Real.sqrt (1 / 4 - τ * a + τ ^ 2 * C) - | |
| (1 / 2 + τ * -a + τ ^ 2 * (C - (-a) ^ 2))) := by ring | |
| rw [hsplit] | |
| exact (abs_add_le _ _).trans (by nlinarith only [hfirst, hsecond]) | |
| theorem cartesian_second_coefficient_bound {η x y : ℝ} (hη : 0 ≤ η) | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) : | |
| |D y + η / 2 - x ^ 2 / 4| ≤ η + 2 := 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 hy4 : y ^ 4 ≤ 1 := by | |
| nlinarith only [mul_nonneg (sq_nonneg y) (sub_nonneg.mpr hy2), hy2] | |
| unfold D | |
| rw [abs_le] | |
| constructor <;> nlinarith only [hx2, hy2, hy4, sq_nonneg x, sq_nonneg y, sq_nonneg (y ^ 2), hη] | |
| theorem cartesian_pair_expansion {τ η x y : ℝ} | |
| (hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hη : 0 ≤ η) | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hsmall : (η + 3) * τ ≤ 1 / 4) : | |
| |Real.sqrt (1 / 4 + τ * (x * y) + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4)) + | |
| Real.sqrt (1 / 4 - τ * (x * y) + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4)) - | |
| (1 + τ ^ 2 * ((x ^ 2 - y ^ 2) ^ 2 + η))| ≤ | |
| (24 * (η + 3) ^ 3 + 2 * (η + 2) * (η + 4)) * τ ^ 3 := by | |
| have hxy : |x * y| ≤ 1 := by | |
| rw [abs_mul] | |
| calc | |
| |x| * |y| ≤ 1 * 1 := mul_le_mul hx hy (abs_nonneg _) (by norm_num) | |
| _ = 1 := by norm_num | |
| have h := sqrt_quarter_pair_remainder hτ0 hτ1 (M := η + 2) (by positivity) | |
| hxy (cartesian_second_coefficient_bound hη hx hy) | |
| (cartesian_second_coefficient_bound hη hy hx) (by linarith) | |
| have hcoef : (D y + η / 2 - x ^ 2 / 4) + (D x + η / 2 - y ^ 2 / 4) - | |
| 2 * (x * y) ^ 2 = (x ^ 2 - y ^ 2) ^ 2 + η := by unfold D; ring | |
| rw [hcoef] at h | |
| convert h using 1 | |
| ring | |
| theorem cartesian_pair_gt_one {τ η x y : ℝ} | |
| (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hη : 0 < η) | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hsmall : (η + 3) * τ ≤ 1 / 4) | |
| (hmargin : (24 * (η + 3) ^ 3 + 2 * (η + 2) * (η + 4)) * τ ≤ η / 2) : | |
| 1 < Real.sqrt (1 / 4 + τ * (x * y) + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4)) + | |
| Real.sqrt (1 / 4 - τ * (x * y) + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4)) := by | |
| have h := (abs_le.mp (cartesian_pair_expansion hτ.le hτ1 hη.le hx hy hsmall)).1 | |
| have hm := mul_le_mul_of_nonneg_right hmargin (sq_nonneg τ) | |
| have hp := mul_pos (sq_pos_of_pos hτ) hη | |
| have hq := mul_nonneg (sq_nonneg τ) (sq_nonneg (x ^ 2 - y ^ 2)) | |
| nlinarith only [h, hm, hp, hq] | |
| theorem cartesian_edge_one_obstruction_identity (ε x y : ℝ) : | |
| (ε / 2) ^ 2 * (((2 * x) ^ 2 - (2 * y) ^ 2) ^ 2 + 1 / 1000) = | |
| ε ^ 2 * (4 * (x ^ 2 - y ^ 2) ^ 2 + 1 / 4000) := by ring | |
| end CylinderCovering.CartesianCaps | |
| end OAI | |