| 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 |
|
|
