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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 | import OAI.Geometry.CylinderCovering.Angular.GeneralDefinitions
import OAI.Geometry.CylinderCovering.Convexity.GeneralCapArea
namespace OAI
open Set MeasureTheory
namespace CylinderCovering.AngularGeneral
theorem integralCost_expansion_from_uniform_errors {τ η E : ℝ} {D : Data}
(hD : Admissible τ η E D) (hη0 : 0 ≤ η) (hη1 : η ≤ 1 / 8)
(hE : 0 ≤ E) (hτ : τ ^ 2 ≤ 1) (hsmall : E * τ ^ 2 ≤ 1 / 16) :
|integralCost τ D - (1 / 2 + (2 * η - 1 / 240) * τ ^ 2)| ≤
2 * (3 * E + 1) * τ ^ 4 := by
rcases hD with ⟨hN, h0, hN', hnode, hbound, hboundary, hcont, hpos, herr⟩
have h := GeneralCapArea.general_cap_area_expansion h0 hN'
(fun j hj => (hnode j hj).le) hcont hη0 hη1 hE hτ hsmall
(fun j hj q hq => (herr j hj q hq).1)
(fun j hj q hq => (herr j hj q hq).2.1)
(fun j hj q hq => (herr j hj q hq).2.2)
unfold integralCost
rw [Fin.sum_univ_eq_sum_range (fun j : ℕ =>
∫ q in D.node j..D.node (j+1),
(D.radius j q)^2 / Real.sqrt (1+τ^2*((D.alpha j)^2+2*(D.beta j)^2))) D.count]
exact h
end CylinderCovering.AngularGeneral
end OAI
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