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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 | import OAI.Geometry.CylinderCovering.Physical.FiniteDefinitions
import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions
namespace OAI
open Set
noncomputable section
namespace RadialPhysical.PhysicalMinimum
open CylinderCovering
def actualRelativeCost (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ)
(N : ℕ) (tagU tagL : Fin N → ℝ) : ℝ :=
(∑ i, TriangularCovering.area (upperPhysicalBase centered b eta T N tagU i) /
CylinderCovering.shadowArea (modelBody centered b) (upperPhysicalAxis b eta (tagU i))) +
(∑ i, TriangularCovering.area (lowerPhysicalBase centered b eta T N tagL i) /
CylinderCovering.shadowArea (modelBody centered b) (lowerPhysicalAxis b eta (tagL i)))
def Statement : Prop :=
(∀ (centered : Bool) (b : ℝ), 0 < b →
minimumShadowArea (modelBody centered b) = (if centered then 4 else 1)*b*height b ∧
(∃ u : Space, ‖u‖ = 1 ∧
CylinderCovering.shadowArea (modelBody centered b) u =
minimumShadowArea (modelBody centered b)) ∧
(∀ u : Space, ‖u‖ = 1 →
(if centered then 4 else 1)*b*height b ≤
CylinderCovering.shadowArea (modelBody centered b) u) ∧
(∀ (n : ℕ) (C : Fin n → TriangularCovering.TriCylinder),
TriangularCovering.totalArea C < ((if centered then 4 else 1)*b*height b)/2 →
TriangularCovering.totalArea C < minimumShadowArea (modelBody centered b)/2 ∧
triangularRelativeCost (modelBody centered b) C < 1/2)) ∧
minimumShadowArea {z : Space | ∃ p q t : ℝ,
|p| ≤ 1+t ∧ |q| ≤ 1-t ∧ z = nonorthPoint p q t} = 2*Real.sqrt 2 ∧
(∀ (centered : Bool) (b : ℝ), 0 < b → ∀ (eta : ℝ) (T : ℝ → ℝ)
(N : ℕ) (tagU tagL : Fin N → ℝ),
physicalActualCost centered b eta T N tagU tagL <
((if centered then 4 else 1)*b*height b)/2 →
physicalActualCost centered b eta T N tagU tagL <
minimumShadowArea (modelBody centered b)/2 ∧
actualRelativeCost centered b eta T N tagU tagL < 1/2)
end RadialPhysical.PhysicalMinimum
end
end OAI
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