File size: 3,432 Bytes
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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.Geometry.CylinderCovering.Beams.MainDefinitions
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped InnerProductSpace
namespace ShortenedBeam
def vertices : Fin 4 → Space :=
![point 2 0 (-1), point (-2) 0 (-1), point 0 2 1, point 0 (-2) 1]
def GeometryStatement : Prop :=
tetrahedron = convexHull ℝ (range vertices) ∧ AffineIndependent ℝ vertices ∧
(∀ i j : Fin 4, i ≠ j → dist (vertices i) (vertices j) = 2*Real.sqrt 2) ∧
volume.real tetrahedron = 8/3 ∧
CylinderCovering.minimumShadowArea tetrahedron = 2*Real.sqrt 2
abbrev MixedShape := HasMixedBase
def FiniteMixedStatement : Prop :=
∀ a : ℝ, 0 < a → a < 1/2 → ∀ ε : ℝ, 0 < ε →
∃ n : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder,
(∀ i, MixedShape (C i) ∧ IsCompact (C i).base) ∧
tetrahedron ⊆ RuledApproximation.familySet C ∧
RuledApproximation.familyArea C ≤
(Real.sqrt 2/4)*(CylinderCovering.BeamCost.beamCost a+
CylinderCovering.BeamCost.stripCost a)+ε
abbrev RegularCounterexampleStatement := RegularMixedCounterexampleStatement
abbrev ParameterPlane := EuclideanSpace ℝ (Fin 2)
def doubleSector (a : ℝ) (N : ℕ) (i : Fin N) : Set ParameterPlane :=
{z | ∃ p ∈ CylinderCovering.RadialRiemann.cell 0 1 N i,
∃ r ∈ Icc (-(1/(N:ℝ))) ((2-a)+1/(N:ℝ)),
z = !₂[r*p,r*(1-p)]}
def midpointTag (N : ℕ) (i : Fin N) : ℝ :=
(CylinderCovering.RadialRiemann.point 0 1 N i+
CylinderCovering.RadialRiemann.point 0 1 N (i+1))/2
def neighborhoodSector (a : ℝ) (N : ℕ) (i : Fin N) : Set ParameterPlane :=
Metric.cthickening ((1/(N:ℝ))^2) (doubleSector a N i)
def neighborhoodBaseMinus (a : ℝ) (N : ℕ) (i : Fin N) : Set Space :=
TriangularCovering.projection (axisMinus a (midpointTag N i)) ''
((fun z : ParameterPlane => point (-1+z 0) (-1+z 1) 0) '' neighborhoodSector a N i)
def neighborhoodBasePlus (a : ℝ) (N : ℕ) (i : Fin N) : Set Space :=
TriangularCovering.projection (axisPlus a (midpointTag N i)) ''
((fun z : ParameterPlane => point (1-z 0) (1-z 1) 0) '' neighborhoodSector a N i)
def NeighborhoodAlternativeStatement : Prop :=
∀ a : ℝ, 0 < a → a < 1/2 →
∃ errorConstant : ℝ, 0 ≤ errorConstant ∧
∃ Cminus Cplus : (N : ℕ) → Fin N → RuledApproximation.Cylinder,
∀ N : ℕ, 0 < N →
(∀ i : Fin N,
(Cminus N i).axis = axisMinus a (midpointTag N i) ∧
(Cplus N i).axis = axisPlus a (midpointTag N i) ∧
Subtype.val '' (Cminus N i).base = neighborhoodBaseMinus a N i ∧
Subtype.val '' (Cplus N i).base = neighborhoodBasePlus a N i ∧
IsCompact (Cminus N i).base ∧ IsCompact (Cplus N i).base) ∧
beamMinus a ⊆ RuledApproximation.familySet (Cminus N) ∧
beamPlus a ⊆ RuledApproximation.familySet (Cplus N) ∧
(RuledApproximation.familyArea (Cminus N)+RuledApproximation.familyArea (Cplus N)) /
(Real.sqrt 2/4) ≤
CylinderCovering.BeamCost.beamCost a+errorConstant/(N:ℝ)
def FullStatement : Prop :=
GeometryStatement ∧ CoverageStatement ∧ FiniteBeamsStatement ∧ FiniteStripStatement ∧
CylinderCovering.BeamCost.CostStatement ∧ FiniteMixedStatement ∧
CanonicalCounterexampleStatement ∧ RegularCounterexampleStatement ∧
NeighborhoodAlternativeStatement
end ShortenedBeam
end
end OAI
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