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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 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 | import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions
namespace OAI
noncomputable section
open Set Filter
open scoped Topology
namespace CylinderCovering.RadialCapVariants
abbrev Space := TriangularCovering.Space
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))
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
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
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))
end CylinderCovering.RadialCapVariants
end
end OAI
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