| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions | |
| import OAI.Geometry.CylinderCovering.Caps.CapsDefinitions | |
| namespace OAI | |
| open Set | |
| noncomputable section | |
| namespace CylinderCovering.HalfheightCertificates | |
| abbrev Space := TriangularCovering.Space | |
| def IsTrapezoid (B : Set Space) : Prop := | |
| ∃ v : Fin 4 → Space, | |
| B = convexHull ℝ (range v) ∧ Function.Injective v ∧ | |
| AffineIndependent ℝ ![v 0,v 1,v 2] ∧ | |
| ∃ d : Space, ∃ a b : ℝ, d ≠ 0 ∧ 0 < a ∧ 0 < b ∧ | |
| v 1-v 0 = a • d ∧ v 3-v 2 = b • d | |
| def ShapeStatement : Prop := | |
| ∀ δ : ℝ, 0 < δ → ∀ R : ℝ → ℝ, | |
| ContinuousOn R (Icc (-1) 1) → (∀ r ∈ Icc (-1:ℝ) 1, 0 ≤ R r) → | |
| ∀ N : ℕ, ∀ i : Fin N, | |
| IsTrapezoid (HalfheightFinite.firstBase δ R N i) ∧ | |
| IsTrapezoid (HalfheightFinite.secondBase δ R N i) | |
| open HalfHeightCaps | |
| def QuantitativeStatement : Prop := | |
| (∀ δ X Y Z r s u v : ℝ, | |
| 0 ≤ δ → δ ≤ 1/2 → |X| ≤ 1 → |Y| ≤ 1 → |r| ≤ 1 → |s| ≤ 1 → | |
| 1/2 ≤ u → 1/2 ≤ v → | |
| Y = u*r+δ*X*alpha r → Z = u+δ*X*beta r → | |
| X = v*s-δ*Y*alpha s → 1-Z = v-δ*Y*beta s → | |
| (1/2 ≤ 1-δ*X*r ∧ 1/2 ≤ 1+δ*Y*s) ∧ | |
| (0 ≤ u-1/2 ∧ u-1/2 ≤ 2*δ^2 ∧ 0 ≤ v-1/2 ∧ v-1/2 ≤ 2*δ^2) ∧ | |
| (|Y-r/2| ≤ 3*δ/2 ∧ |X-s/2| ≤ 3*δ/2) ∧ | |
| (X^2*alpha r+Y^2*alpha s ≤ (s^2*alpha r+r^2*alpha s)/4+9*δ/4 ∧ | |
| X^2*alpha r+Y^2*alpha s ≤ F r+F s+9*δ/4)) ∧ | |
| (∀ δ r s : ℝ, 0 ≤ δ → δ ≤ 1/2 → |r| ≤ 1 → |s| ≤ 1 → | |
| F r+F s+15*δ/4 ≤ (1-δ)*(F r+F s+8*δ)) ∧ | |
| (∀ δ X Y Z r s u v : ℝ, | |
| 0 < δ → δ ≤ 1/2 → |X| ≤ 1 → |Y| ≤ 1 → |r| ≤ 1 → |s| ≤ 1 → | |
| Y = u*r+δ*X*alpha r → Z = u+δ*X*beta r → | |
| X = v*s-δ*Y*alpha s → 1-Z = v-δ*Y*beta s → | |
| ¬ (cubicCap δ r < u ∧ cubicCap δ s < v)) | |
| def Statement : Prop := ShapeStatement ∧ QuantitativeStatement | |
| end CylinderCovering.HalfheightCertificates | |
| end | |
| end OAI | |