| import OAI.Geometry.CylinderCovering.Rational.CeilingIntegralDefinitions | |
| import OAI.Geometry.CylinderCovering.Rational.Integrals | |
| namespace OAI | |
| open Filter | |
| open scoped Topology | |
| noncomputable section | |
| namespace SlopeRational | |
| theorem ceilingIntegral_eq (d : ℝ) : | |
| ceilingIntegral d = alpha d * (ceilingConstant d / 96 + ceilingQuadratic d / 2560) := by | |
| exact CylinderCovering.RationalIntegrals.ceiling_integral _ _ _ | |
| theorem ceilingIntegral_formula (d : ℝ) : | |
| ceilingIntegral d = (2 + 5 * d) / (1920 * (1 + 2 * d) ^ 3) := by | |
| rw | |
| exact CylinderCovering.RationalIntegrals.rational_ceiling_coefficient_formula d | |
| theorem ceilingIntegral_tendsto_zero : | |
| Tendsto ceilingIntegral (𝓝 0) (𝓝 (1 / 960 : ℝ)) := by | |
| exact CylinderCovering.RationalIntegrals.rational_ceiling_integral_tendsto | |
| theorem ceilingIntegral_tendsto_zero_right : | |
| Tendsto ceilingIntegral (𝓝[Set.Ioi 0] 0) (𝓝 (16 / 15360 : ℝ)) := by | |
| have h : Tendsto ceilingIntegral (𝓝[Set.Ioi 0] 0) (𝓝 (1 / 960 : ℝ)) := | |
| ceilingIntegral_tendsto_zero.mono_left nhdsWithin_le_nhds | |
| norm_num at h ⊢ | |
| exact h | |
| theorem ceilingIntegral_pos {d : ℝ} (hd : 0 ≤ d) : 0 < ceilingIntegral d := by | |
| rw | |
| positivity | |
| end SlopeRational | |
| end | |
| end OAI | |