openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Beams /NeighborhoodRealizationDefinitions.lean
| import OAI.Geometry.CylinderCovering.Beams.NeighborhoodDefinitions | |
| import OAI.Geometry.CylinderCovering.Radial.SweepDefinitions | |
| import OAI.Geometry.CylinderCovering.Beams.sDefinitions | |
| import OAI.Geometry.CylinderCovering.Beams.CostDefinitions | |
| namespace OAI | |
| noncomputable section | |
| namespace CylinderCovering.BeamNeighborhoodRealization | |
| open Set MeasureTheory RadialSweep | |
| def physicalMap (D : Data) (z : BeamNeighborhood.Plane) : Space := | |
| D.p₀ + (z 0 + z 1) • D.a + z 0 • D.b | |
| def rawBase (D : Data) (a l r δ : ℝ) : Set Space := | |
| physicalMap D '' BeamNeighborhood.paddedSector a l r δ | |
| def base (D : Data) (a l r δ : ℝ) : Set Space := | |
| TriangularCovering.projection (direction D ((l+r)/2)) '' rawBase D a l r δ | |
| def cellBase (D : Data) (a : ℝ) (N : ℕ) (i : Fin N) : Set Space := | |
| base D a (RadialRiemann.point 0 1 N i) (RadialRiemann.point 0 1 N (i+1)) | |
| (RadialRiemann.step 0 1 N) | |
| def cost (D : Data) (a : ℝ) (N : ℕ) : ℝ := | |
| ∑ i : Fin N, TriangularCovering.area (cellBase D a N i) | |
| def cellMidpoint (N : ℕ) (i : Fin N) : ℝ := | |
| (RadialRiemann.point 0 1 N i+RadialRiemann.point 0 1 N (i+1))/2 | |
| def beamBaseMinus (a : ℝ) (N : ℕ) (i : Fin N) : Set Space := | |
| TriangularCovering.projection (ShortenedBeam.axisMinus a (cellMidpoint N i)) '' | |
| ((fun z : BeamNeighborhood.Plane => ShortenedBeam.point (-1+z 0) (-1+z 1) 0) '' | |
| BeamNeighborhood.paddedSector a (RadialRiemann.point 0 1 N i) | |
| (RadialRiemann.point 0 1 N (i+1)) (RadialRiemann.step 0 1 N)) | |
| def beamBasePlus (a : ℝ) (N : ℕ) (i : Fin N) : Set Space := | |
| TriangularCovering.projection (ShortenedBeam.axisPlus a (cellMidpoint N i)) '' | |
| ((fun z : BeamNeighborhood.Plane => ShortenedBeam.point (1-z 0) (1-z 1) 0) '' | |
| BeamNeighborhood.paddedSector a (RadialRiemann.point 0 1 N i) | |
| (RadialRiemann.point 0 1 N (i+1)) (RadialRiemann.step 0 1 N)) | |
| def alternativeCost (a : ℝ) (N : ℕ) : ℝ := | |
| (∑ i : Fin N, TriangularCovering.area (beamBaseMinus a N i))+ | |
| ∑ i : Fin N, TriangularCovering.area (beamBasePlus a N i) | |
| def AlternativeStatement : Prop := | |
| ∀ a : ℝ, 0 < a → a < 1/2 → ∃ C : ℝ, 0 < C ∧ ∀ N : ℕ, 0 < N → | |
| (∀ i : Fin N, | |
| ‖ShortenedBeam.axisMinus a (cellMidpoint N i)‖ = 1 ∧ | |
| ‖ShortenedBeam.axisPlus a (cellMidpoint N i)‖ = 1 ∧ | |
| IsCompact (beamBaseMinus a N i) ∧ IsCompact (beamBasePlus a N i) ∧ | |
| MeasurableSet (beamBaseMinus a N i) ∧ MeasurableSet (beamBasePlus a N i) ∧ | |
| Measure.euclideanHausdorffMeasure 2 (beamBaseMinus a N i) ≠ ⊤ ∧ | |
| Measure.euclideanHausdorffMeasure 2 (beamBasePlus a N i) ≠ ⊤ ∧ | |
| (∀ z ∈ beamBaseMinus a N i, inner ℝ (ShortenedBeam.axisMinus a (cellMidpoint N i)) z=0) ∧ | |
| (∀ z ∈ beamBasePlus a N i, inner ℝ (ShortenedBeam.axisPlus a (cellMidpoint N i)) z=0)) ∧ | |
| (∀ z ∈ ShortenedBeam.beamMinus a, ∃ i : Fin N, | |
| z ∈ cylinder (ShortenedBeam.axisMinus a (cellMidpoint N i)) (beamBaseMinus a N i)) ∧ | |
| (∀ z ∈ ShortenedBeam.beamPlus a, ∃ i : Fin N, | |
| z ∈ cylinder (ShortenedBeam.axisPlus a (cellMidpoint N i)) (beamBasePlus a N i)) ∧ | |
| alternativeCost a N ≤ (Real.sqrt 2/4)*BeamCost.beamCost a+C/(N:ℝ) | |
| end CylinderCovering.BeamNeighborhoodRealization | |
| end | |
| end OAI | |