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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 | import OAI.Geometry.CylinderCovering.Physical.CapFiniteDefinitions
import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace RadialPhysical
open CylinderCovering
def physicalActualRelativeCost (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ)
(N : ℕ) (tagU tagL : Fin N → ℝ) : ℝ :=
(∑ i : Fin N, TriangularCovering.area (upperPhysicalBase centered b eta T N tagU i) /
shadowArea (modelBody centered b) (upperPhysicalAxis b eta (tagU i))) +
(∑ i : Fin N, TriangularCovering.area (lowerPhysicalBase centered b eta T N tagL i) /
shadowArea (modelBody centered b) (lowerPhysicalAxis b eta (tagL i)))
def PhysicalBothHalfBounds (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ)
(N : ℕ) (tagU tagL : Fin N → ℝ)
(C : Fin (2*N) → TriangularCovering.TriCylinder) : Prop :=
TriangularCovering.totalArea C < minimumShadowArea (modelBody centered b)/2 ∧
triangularRelativeCost (modelBody centered b) C < 1/2 ∧
physicalActualCost centered b eta T N tagU tagL < minimumShadowArea (modelBody centered b)/2 ∧
physicalActualRelativeCost centered b eta T N tagU tagL < 1/2
def PhysicalCapNormalizedStatement : Prop :=
∀ (centered square : Bool) (b beta : ℝ), 0 < b → 0 < beta → beta < 1/60 →
∀ᶠ eta : ℝ in 𝓝[>] (0 : ℝ),
∀ tagU tagL : (N : ℕ) → Fin N → ℝ,
(∀ N, RadialRiemann.IsTagged (-b) b N (tagU N)) →
(∀ N, RadialRiemann.IsTagged (-b) b N (tagL N)) →
∃ C : (N : ℕ) → Fin (2*N) → TriangularCovering.TriCylinder,
PhysicalFamilyConclusion centered b eta (physicalModelCap centered square b eta beta)
tagU tagL C ∧
∀ᶠ N in atTop, 0 < N ∧
PhysicalBothHalfBounds centered b eta (physicalModelCap centered square b eta beta)
N (tagU N) (tagL N) (C N)
end RadialPhysical
end
end OAI
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