| import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions |
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| namespace OAI |
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| noncomputable section |
| open Set Filter |
| open scoped Topology |
| namespace CylinderCovering.RadialCapVariants |
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| abbrev Space := TriangularCovering.Space |
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| def CoverLimit (ε : ℝ) (R : ℝ → ℝ) : Prop := |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, |
| (∀ N, RadialRiemann.IsTagged (-1) 1 N (tagU N)) → |
| (∀ N, RadialRiemann.IsTagged (-1) 1 N (tagL N)) → |
| ∃ U L : (N : ℕ) → Fin N → TriangularCovering.TriCylinder, |
| (∀ N, 0 < N → |
| (∀ i, (U N i).axis = RadialPairedSweep.upperAxis (Real.sqrt 2) ε (tagU N i) ∧ |
| (L N i).axis = RadialPairedSweep.lowerAxis (Real.sqrt 2) ε (tagL N i)) ∧ |
| (∀ z ∈ RadialPairedSweep.upperSweep (Real.sqrt 2) ε R, |
| ∃ i : Fin N, z ∈ (U N i).carrier) ∧ |
| (∀ z ∈ RadialPairedSweep.lowerSweep (Real.sqrt 2) ε R, |
| ∃ i : Fin N, z ∈ (L N i).carrier) ∧ |
| (∀ z ∈ RadialPairedSweep.tetrahedron (Real.sqrt 2), |
| (∃ i : Fin N, z ∈ (U N i).carrier) ∨ (∃ i : Fin N, z ∈ (L N i).carrier))) ∧ |
| Tendsto (fun N => TriangularCovering.totalArea (U N) + |
| TriangularCovering.totalArea (L N)) atTop (𝓝 (2 * Real.sqrt 2 * RadialCaps.Q R ε)) |
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| def FiniteSaving (ε : ℝ) : Prop := |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, |
| (∀ N, RadialRiemann.IsTagged (-1) 1 N (tagU N)) → |
| (∀ N, RadialRiemann.IsTagged (-1) 1 N (tagL N)) → |
| ∃ N : ℕ, 0 < N ∧ ∃ U L : Fin N → TriangularCovering.TriCylinder, |
| (∀ i, (U i).axis = RadialPairedSweep.upperAxis (Real.sqrt 2) ε (tagU N i) ∧ |
| (L i).axis = RadialPairedSweep.lowerAxis (Real.sqrt 2) ε (tagL N i)) ∧ |
| (∀ z ∈ RadialPairedSweep.tetrahedron (Real.sqrt 2), |
| (∃ i : Fin N, z ∈ (U i).carrier) ∨ (∃ i : Fin N, z ∈ (L i).carrier)) ∧ |
| TriangularCovering.totalArea U + TriangularCovering.totalArea L < 2 * Real.sqrt 2 |
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| def SweepsCover (ε : ℝ) (R : ℝ → ℝ) : Prop := |
| RadialPairedSweep.tetrahedron (Real.sqrt 2) ⊆ |
| RadialPairedSweep.upperSweep (Real.sqrt 2) ε R ∪ |
| RadialPairedSweep.lowerSweep (Real.sqrt 2) ε R |
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| def LCFStatement : Prop := |
| (∀ β : ℝ, 0 < β → |
| (∀ᶠ ε : ℝ in 𝓝[>] 0, |
| SweepsCover ε (RadialCaps.linearCap β ε) ∧ CoverLimit ε (RadialCaps.linearCap β ε)) ∧ |
| ((fun ε => RadialCaps.Q (RadialCaps.linearCap β ε) ε - |
| (1 + (2 * β - 1 / 30) * ε ^ 2)) =O[𝓝[>] (0 : ℝ)] (fun ε => ε ^ 4)) ∧ |
| (β < 1 / 60 → ∀ᶠ ε : ℝ in 𝓝[>] 0, |
| RadialCaps.Q (RadialCaps.linearCap β ε) ε < 1 ∧ FiniteSaving ε)) ∧ |
| (∀ c : ℝ, 9 / 2 ≤ c → |
| (∀ ε : ℝ, 0 < ε → |
| SweepsCover ε (RadialCaps.cubicCap c ε) ∧ CoverLimit ε (RadialCaps.cubicCap c ε)) ∧ |
| ((fun ε => RadialCaps.Q (RadialCaps.cubicCap c ε) ε - |
| (1 - ε ^ 2 / 30 + 2 * c * ε ^ 3)) =O[𝓝[>] (0 : ℝ)] (fun ε => ε ^ 4)) ∧ |
| (∀ᶠ ε : ℝ in 𝓝[>] 0, |
| RadialCaps.Q (RadialCaps.cubicCap c ε) ε < 1 ∧ FiniteSaving ε)) ∧ |
| (∀ c : ℝ, 0 < c → |
| (∀ᶠ ε : ℝ in 𝓝[>] 0, |
| SweepsCover ε (RadialCaps.fractionalCap c ε) ∧ CoverLimit ε (RadialCaps.fractionalCap c ε)) ∧ |
| ((fun ε => RadialCaps.Q (RadialCaps.fractionalCap c ε) ε - |
| (1 - ε ^ 2 / 30 + 2 * c * ε ^ (5 / 2 : ℝ))) =O[𝓝[>] (0 : ℝ)] (fun ε => ε ^ 4)) ∧ |
| (∀ᶠ ε : ℝ in 𝓝[>] 0, |
| RadialCaps.Q (RadialCaps.fractionalCap c ε) ε < 1 ∧ FiniteSaving ε)) |
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| end CylinderCovering.RadialCapVariants |
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| end |
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| end OAI |
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