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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | 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 [ceilingIntegral_eq]
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 [ceilingIntegral_formula]
positivity
end SlopeRational
end
end OAI
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