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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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | 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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