| import Mathlib | |
| namespace OAI | |
| namespace CylinderCovering.AngularCubic | |
| theorem coarse_excess_bound {ε P : ℝ} (hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) | |
| (hP : P ≤ ε * P + 4 * ε ^ 2 + 4 * ε ^ 3) : | |
| P ≤ 12 * ε ^ 2 := by | |
| have hcube : ε ^ 3 ≤ ε ^ 2 / 2 := by | |
| nlinarith [mul_nonneg (sq_nonneg ε) (sub_nonneg.mpr hεhalf)] | |
| by_contra h | |
| have hlarge : 12 * ε ^ 2 < P := lt_of_not_ge h | |
| have hPnonneg : 0 ≤ P := | |
| le_trans (mul_nonneg (by norm_num) (pow_nonneg hε 2)) hlarge.le | |
| have hprod : ε * P ≤ P / 2 := by | |
| nlinarith [mul_nonneg hPnonneg (sub_nonneg.mpr hεhalf)] | |
| nlinarith | |
| theorem fine_excess_bound {ε Δ P A : ℝ} (hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) | |
| (hΔ : Δ ≤ ε ^ 2) (hP : P ≤ 12 * ε ^ 2) | |
| (hfine : P ≤ ε ^ 2 * A + ε * P + 2 * ε ^ 2 * P + | |
| 4 * ε ^ 3 + 4 * ε * Δ) : | |
| P ≤ ε ^ 2 * A + 32 * ε ^ 3 := by | |
| have h₁ : ε * P ≤ 12 * ε ^ 3 := by | |
| nlinarith [mul_le_mul_of_nonneg_left hP hε] | |
| have h₂ : 2 * ε ^ 2 * P ≤ 24 * ε ^ 4 := by | |
| nlinarith [mul_le_mul_of_nonneg_left hP (show 0 ≤ 2 * ε ^ 2 by positivity)] | |
| have h₃ : 24 * ε ^ 4 ≤ 12 * ε ^ 3 := by | |
| nlinarith [mul_nonneg (show 0 ≤ ε ^ 3 by positivity) (sub_nonneg.mpr hεhalf)] | |
| have h₄ : 4 * ε * Δ ≤ 4 * ε ^ 3 := by | |
| nlinarith [mul_le_mul_of_nonneg_left hΔ (show 0 ≤ 4 * ε by positivity)] | |
| linarith | |
| theorem coefficient_bound {a b q r Δ : ℝ} | |
| (hq : |q| ≤ 1) (hr : |r| ≤ 1) (hΔ : 0 ≤ Δ) | |
| (ha : |a - (1 + r ^ 2) / 2| ≤ Δ) | |
| (hb : |b - (1 + q ^ 2) / 2| ≤ Δ) : | |
| a * q ^ 2 + b * r ^ 2 ≤ | |
| (q ^ 2 + q ^ 4) / 2 + (r ^ 2 + r ^ 4) / 2 + 2 * Δ := by | |
| have hq2 : q ^ 2 ≤ 1 := by | |
| have h := abs_le.mp hq | |
| nlinarith | |
| have hr2 : r ^ 2 ≤ 1 := by | |
| have h := abs_le.mp hr | |
| nlinarith | |
| have ha' : a ≤ (1 + r ^ 2) / 2 + Δ := by | |
| have := (abs_le.mp ha).2 | |
| linarith | |
| have hb' : b ≤ (1 + q ^ 2) / 2 + Δ := by | |
| have := (abs_le.mp hb).2 | |
| linarith | |
| have h₁ := mul_le_mul_of_nonneg_right ha' (sq_nonneg q) | |
| have h₂ := mul_le_mul_of_nonneg_right hb' (sq_nonneg r) | |
| have h₃ := mul_le_mul_of_nonneg_left hq2 hΔ | |
| have h₄ := mul_le_mul_of_nonneg_left hr2 hΔ | |
| nlinarith [sq_nonneg (q ^ 2 - r ^ 2)] | |
| theorem profile_excess_bound {ε Δ P A D E : ℝ} | |
| (hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) (hΔ : Δ ≤ ε ^ 2) | |
| (hP : P ≤ ε ^ 2 * A + 32 * ε ^ 3) (hA : A ≤ D + E + 2 * Δ) : | |
| P ≤ ε ^ 2 * (D + E) + 33 * ε ^ 3 := by | |
| have h₁ := mul_le_mul_of_nonneg_left hA (sq_nonneg ε) | |
| have h₂ := mul_le_mul_of_nonneg_left hΔ (show 0 ≤ 2 * ε ^ 2 by positivity) | |
| have h₃ : 2 * ε ^ 4 ≤ ε ^ 3 := by | |
| nlinarith [mul_nonneg (show 0 ≤ ε ^ 3 by positivity) (sub_nonneg.mpr hεhalf)] | |
| nlinarith | |
| theorem no_joint_failure {ε P D E : ℝ} (hε : 0 < ε) | |
| (hP : P ≤ ε ^ 2 * (D + E) + 33 * ε ^ 3) : | |
| ¬ (ε ^ 2 * D + 100 * ε ^ 3) + | |
| (ε ^ 2 * E + 100 * ε ^ 3) < P := by | |
| have hcube : 0 < ε ^ 3 := pow_pos hε 3 | |
| nlinarith | |
| end CylinderCovering.AngularCubic | |
| end OAI | |