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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 | import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions
import OAI.Geometry.CylinderCovering.Convexity.SimilarityDefinitions
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace CylinderCovering.HalfheightFinite
def NondegenerateTrapezoid (B : Set Space) : Prop :=
∃ v : Fin 4 → Space,
B = convexHull ℝ (range v) ∧ Function.Injective v ∧
AffineIndependent ℝ ![v 0, v 1, v 2] ∧
∃ a b : ℝ, 0 < a ∧ 0 < b ∧ ∃ w : Space, w ≠ 0 ∧
v 1 - v 0 = a • w ∧ v 3 - v 2 = b • w
structure GeometryStatement : Prop where
unitEdge : ∃ v : Fin 4 → Space, AffineIndependent ℝ v ∧
(∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = 1) ∧
tetrahedron = convexHull ℝ (range v)
minimumShadow : Similarity.minimumArea tetrahedron = height / 2
sourceTrapezoids : ∀ δ : ℝ, 0 < δ → ∀ N : ℕ, ∀ i : Fin N,
NondegenerateTrapezoid (firstBase δ (cubicCap δ) N i) ∧
NondegenerateTrapezoid (secondBase δ (cubicCap δ) N i) ∧
NondegenerateTrapezoid (firstBase δ (fractionalCap δ) N i) ∧
NondegenerateTrapezoid (secondBase δ (fractionalCap δ) N i)
eventualHalfMinimum : ∀ᶠ δ : ℝ in 𝓝[>] 0,
(∀ᶠ N in atTop, 0 < N ∧
totalArea δ (cubicCap δ) N < Similarity.minimumArea tetrahedron / 2) ∧
(∀ᶠ N in atTop, 0 < N ∧
totalArea δ (fractionalCap δ) N < Similarity.minimumArea tetrahedron / 2)
end CylinderCovering.HalfheightFinite
end
end OAI
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