| import OAI.Geometry.CylinderCovering.Footprints.QuadraticPhysicalFootprintsDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions |
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|
| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology |
| namespace CylinderCovering.CanonicalFootprintFinite |
| open RuledApproximation RadialRiemann Footprints |
|
|
| def rotation (z : Space) : Space := !₂[z 1,-z 0,-z 2] |
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| def midpoint (N : ℕ) (i : Fin N) : ℝ := |
| (RadialRiemann.point (-1) 1 N i + RadialRiemann.point (-1) 1 N (i+1))/2 |
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| def leftTag (N : ℕ) (i : Fin N) : ℝ := RadialRiemann.point (-1) 1 N i |
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| def absoluteFeet (R : ℝ → ℝ) (A B : ℝ) (N : ℕ) (i : Fin N) : Set (ℝ × ℝ) := |
| absoluteFootprint (midpoint N i) (capMax R (-1) 1 N i) (step (-1) 1 N) A B |
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| def affineFeet (R : ℝ → ℝ) (A B : ℝ) (N : ℕ) (i : Fin N) : Set (ℝ × ℝ) := |
| affineFootprint (midpoint N i) (capMax R (-1) 1 N i) (step (-1) 1 N) A B |
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| def endpointFeet (R : ℝ → ℝ) (ε : ℝ) (N : ℕ) (i : Fin N) : Set (ℝ × ℝ) := |
| endpointFootprint (leftTag N i) (capMax R (-1) 1 N i) (step (-1) 1 N) ε |
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| def localCost (h ε c : ℝ) (F : Set (ℝ × ℝ)) : ℝ := |
| h * volume.real F / Real.sqrt (1+ε^2*(((1+c^2)/2)^2+h^2*c^2)) |
|
|
| def FamilyConclusion (h ε : ℝ) (R : ℝ → ℝ) |
| (tag : (N : ℕ) → Fin N → ℝ) |
| (F : (N : ℕ) → Fin N → Set (ℝ × ℝ)) : Prop := |
| ∃ U L : (N : ℕ) → Fin N → Cylinder, |
| (∀ N : ℕ, 0<N → |
| (∀ i : Fin N, |
| (U N i).axis = RadialPairedSweep.upperAxis h ε (tag N i) ∧ |
| (L N i).axis = rotation (RadialPairedSweep.upperAxis h ε (tag N i)) ∧ |
| Subtype.val '' (U N i).base = physicalFootprintProjection h ε (tag N i) '' F N i ∧ |
| Subtype.val '' (L N i).base = rotation '' |
| (physicalFootprintProjection h ε (tag N i) '' F N i) ∧ |
| IsCompact (U N i).base ∧ IsCompact (L N i).base ∧ |
| MeasurableSet (U N i).base ∧ MeasurableSet (L N i).base ∧ |
| volume (U N i).base < ⊤ ∧ volume (L N i).base < ⊤ ∧ |
| (Convex ℝ (U N i).base ↔ Convex ℝ (F N i)) ∧ |
| (Convex ℝ (L N i).base ↔ Convex ℝ (F N i)) ∧ |
| (U N i).area = localCost h ε (tag N i) (F N i) ∧ |
| (L N i).area = localCost h ε (tag N i) (F N i)) ∧ |
| RadialPairedSweep.upperSweep h ε R ⊆ familySet (U N) ∧ |
| RadialPairedSweep.lowerSweep h ε R ⊆ familySet (L N)) ∧ |
| Tendsto (fun N => familyArea (U N)+familyArea (L N)) atTop |
| (𝓝 (2*h*RadialPairedSweep.physicalCost h ε R)) |
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|
| def Statement : Prop := |
| ∀ h ε : ℝ, 0<h → 0≤ε → ∀ R : ℝ → ℝ, |
| ContinuousOn R (Icc (-1:ℝ) 1) → (∀ q ∈ Icc (-1:ℝ) 1, 0≤R q) → |
| (∀ A B : ℝ, 2*ε≤A → ε≤B → |
| ∀ tag : (N : ℕ) → Fin N → ℝ, |
| (∀ N, IsTagged (-1) 1 N (tag N)) → |
| FamilyConclusion h ε R tag (absoluteFeet R A B)) ∧ |
| (∀ A B : ℝ, ε≤A → ε/4≤B → |
| FamilyConclusion h ε R midpoint (absoluteFeet R A B)) ∧ |
| (∀ A B : ℝ, ε≤A → 3*ε/4≤B → A/2≤B → |
| FamilyConclusion h ε R midpoint (affineFeet R A B)) ∧ |
| FamilyConclusion h ε R leftTag (endpointFeet R ε) |
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| end CylinderCovering.CanonicalFootprintFinite |
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| end |
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| end OAI |
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