| import OAI.Geometry.CylinderCovering.Angular.GeneralAssembly |
| import OAI.Geometry.TriangularCovering.Main |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
|
|
| namespace CylinderCovering.AngularGeneral |
|
|
| theorem relativeCost_le_totalArea {τ : ℝ} {D : Data} |
| (hunit : ∀ j < D.count, |
| ‖lowerAxis τ D j‖ = 1 ∧ ‖upperAxis τ D j‖ = 1) : |
| relativeCost τ D ≤ totalArea τ D / TriangularCovering.H := by |
| unfold relativeCost totalArea |
| rw [Finset.sum_div] |
| apply Finset.sum_le_sum |
| intro j _ |
| rw [add_div] |
| exact add_le_add |
| (div_le_div_of_nonneg_left ENNReal.toReal_nonneg TriangularCovering.H_pos |
| (TriangularCovering.shadowArea_ge_H (hunit j j.isLt).1)) |
| (div_le_div_of_nonneg_left ENNReal.toReal_nonneg TriangularCovering.H_pos |
| (TriangularCovering.shadowArea_ge_H (hunit j j.isLt).2)) |
|
|
| theorem strict_costs_of_ratio {τ : ℝ} {D : Data} |
| (hunit : ∀ j < D.count, |
| ‖lowerAxis τ D j‖ = 1 ∧ ‖upperAxis τ D j‖ = 1) |
| (hratio : totalArea τ D / TriangularCovering.H < 1/2) : |
| totalArea τ D < TriangularCovering.A_min/2 ∧ relativeCost τ D < 1/2 := by |
| constructor |
| · rw [TriangularCovering.projection_minimum] |
| have hh := (div_lt_iff₀ TriangularCovering.H_pos).mp hratio |
| linarith only [hh] |
| · exact (relativeCost_le_totalArea hunit).trans_lt hratio |
|
|
| theorem cost_conclusions {τ η E : ℝ} {D : Data} |
| (hD : Admissible τ η E D) (hτ : 0 < τ) |
| (hη : 0 ≤ η) (hη1 : η ≤ 1/8) (hE : 0 ≤ E) |
| (hτ1 : τ^2 ≤ 1) (hsmall : E*τ^2 ≤ 1/16) |
| (hsave : 2*(3*E+1)*τ^2 < 1/240-2*η) |
| (hunit : ∀ j < D.count, |
| ‖lowerAxis τ D j‖ = 1 ∧ ‖upperAxis τ D j‖ = 1) |
| (harea : totalArea τ D / TriangularCovering.H = integralCost τ D) : |
| |totalArea τ D / TriangularCovering.H-(1/2+(2*η-1/240)*τ^2)| ≤ |
| 2*(3*E+1)*τ^4 ∧ |
| totalArea τ D < TriangularCovering.A_min/2 ∧ relativeCost τ D < 1/2 := by |
| have herror := integralCost_expansion hD hη hη1 hE hτ1 hsmall |
| rw [← harea] at herror |
| have hratio := GeneralCapArea.strict_half_of_area_error hτ herror hsave |
| exact ⟨herror,strict_costs_of_ratio hunit hratio⟩ |
|
|
| end CylinderCovering.AngularGeneral |
|
|
| end |
|
|
| end OAI |
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|
