| import OAI.Geometry.CylinderCovering.Physical.CoordinatesDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.SweepDefinitions |
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|
| namespace OAI |
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| open Set Filter MeasureTheory |
| open scoped Topology InnerProductSpace |
| noncomputable section |
| namespace RadialPhysical |
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| def axialBound (centered : Bool) (b : ℝ) : ℝ := if centered then 2*b else b |
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| def modelBody (centered : Bool) (b : ℝ) : Set Space := |
| if centered then |
| {z | ∃ x y t : ℝ, |x| ≤ b*(1+t) ∧ |y| ≤ b*(1-t) ∧ z = point b x y t} |
| else tetrahedron b |
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| def upperPoint (b eta rho s x : ℝ) : Space := |
| point b x (rho*s-eta*x*(s^2+b^2)/2) (1-rho+eta*x*s) |
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| def lowerPoint (centered : Bool) (b eta rho s y : ℝ) : Space := |
| point b (rho*s+eta*y*(s^2+b^2)/2) y |
| ((if centered then (-1:ℝ) else 0)+rho+eta*y*s) |
|
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| def upperSweepOn (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) (I : Set ℝ) : Set Space := |
| {z | ∃ s ∈ I, ∃ rho ∈ Icc (0:ℝ) (T s), ∃ x : ℝ, |
| |x| ≤ axialBound centered b ∧ z = upperPoint b eta rho s x} |
|
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| def lowerSweepOn (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) (I : Set ℝ) : Set Space := |
| {z | ∃ s ∈ I, ∃ rho ∈ Icc (0:ℝ) (T s), ∃ y : ℝ, |
| |y| ≤ axialBound centered b ∧ z = lowerPoint centered b eta rho s y} |
|
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| def upperPhysicalAxis (b eta s : ℝ) : Space := |
| let v := point b 1 (-eta*(s^2+b^2)/2) (eta*s) |
| ‖v‖⁻¹ • v |
|
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| def lowerPhysicalAxis (b eta s : ℝ) : Space := |
| let v := point b (eta*(s^2+b^2)/2) 1 (eta*s) |
| ‖v‖⁻¹ • v |
|
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| def upperPhysicalBase (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : Set Space := |
| TriangularCovering.projection (upperPhysicalAxis b eta (tag i)) '' |
| upperSweepOn centered b eta T (CylinderCovering.RadialRiemann.cell (-b) b N i) |
|
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| def lowerPhysicalBase (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : Set Space := |
| TriangularCovering.projection (lowerPhysicalAxis b eta (tag i)) '' |
| lowerSweepOn centered b eta T (CylinderCovering.RadialRiemann.cell (-b) b N i) |
|
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| def physicalActualCost (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tagU tagL : Fin N → ℝ) : ℝ := |
| (∑ i : Fin N, TriangularCovering.area (upperPhysicalBase centered b eta T N tagU i)) + |
| (∑ i : Fin N, TriangularCovering.area (lowerPhysicalBase centered b eta T N tagL i)) |
|
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| def physicalBaseProperties (u : Space) (B : Set Space) : Prop := |
| ‖u‖ = 1 ∧ IsCompact B ∧ MeasurableSet B ∧ |
| Measure.euclideanHausdorffMeasure 2 B ≠ ⊤ ∧ (∀ z ∈ B, inner ℝ u z = 0) |
|
|
| def PhysicalFiniteStatement : Prop := |
| ∀ (centered : Bool) (b eta : ℝ), 0 < b → ∀ T : ℝ → ℝ, |
| ContinuousOn T (Icc (-b) b) → (∀ s ∈ Icc (-b) b, 0 ≤ T s) → |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, |
| (∀ N, CylinderCovering.RadialRiemann.IsTagged (-b) b N (tagU N)) → |
| (∀ N, CylinderCovering.RadialRiemann.IsTagged (-b) b N (tagL N)) → |
| ∃ U L : (N : ℕ) → Fin N → TriangularCovering.TriCylinder, |
| (∀ N, 0 < N → |
| (∀ i, (U N i).axis = upperPhysicalAxis b eta (tagU N i) ∧ |
| (L N i).axis = lowerPhysicalAxis b eta (tagL N i) ∧ |
| upperPhysicalBase centered b eta T N (tagU N) i ⊆ (U N i).base ∧ |
| lowerPhysicalBase centered b eta T N (tagL N) i ⊆ (L N i).base ∧ |
| physicalBaseProperties (upperPhysicalAxis b eta (tagU N i)) |
| (upperPhysicalBase centered b eta T N (tagU N) i) ∧ |
| physicalBaseProperties (lowerPhysicalAxis b eta (tagL N i)) |
| (lowerPhysicalBase centered b eta T N (tagL N) i)) ∧ |
| (∀ z ∈ upperSweepOn centered b eta T (Icc (-b) b), |
| (∃ i : Fin N, z ∈ (U N i).carrier) ∧ |
| (∃ i : Fin N, z ∈ CylinderCovering.RadialSweep.cylinder |
| (upperPhysicalAxis b eta (tagU N i)) |
| (upperPhysicalBase centered b eta T N (tagU N) i))) ∧ |
| (∀ z ∈ lowerSweepOn centered b eta T (Icc (-b) b), |
| (∃ i : Fin N, z ∈ (L N i).carrier) ∧ |
| (∃ i : Fin N, z ∈ CylinderCovering.RadialSweep.cylinder |
| (lowerPhysicalAxis b eta (tagL N i)) |
| (lowerPhysicalBase centered b eta T N (tagL N) i)))) ∧ |
| Tendsto (fun N => TriangularCovering.totalArea (U N) + |
| TriangularCovering.totalArea (L N)) atTop (𝓝 (scaledCost b eta T)) ∧ |
| (∀ δ : ℝ, 0 < δ → ∀ᶠ N in atTop, |
| physicalActualCost centered b eta T N (tagU N) (tagL N) ≤ scaledCost b eta T + δ) ∧ |
| Filter.limsup (fun N => physicalActualCost centered b eta T N (tagU N) (tagL N)) |
| atTop ≤ scaledCost b eta T |
|
|
| end RadialPhysical |
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| end |
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| end OAI |
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