| import OAI.Geometry.CylinderCovering.Physical.FiniteDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.FinitePairDefinitions |
|
|
| namespace OAI |
|
|
| open Set Filter |
| open scoped Topology InnerProductSpace |
| noncomputable section |
| namespace RadialPhysical |
| open CylinderCovering |
|
|
| def capChoice (square : Bool) (b eta beta : ℝ) : ℝ → ℝ := |
| if square then squareCap b eta beta else linearCap b eta beta |
|
|
| def physicalModelCap (centered square : Bool) (b eta beta : ℝ) : ℝ → ℝ := |
| if centered then fun s => 2 * capChoice square b eta beta s |
| else capChoice square b eta beta |
|
|
| def physicalHalfReference (centered : Bool) (b : ℝ) : ℝ := |
| if centered then 2*b*height b else b*height b/2 |
|
|
| structure PhysicalFamilyConclusion (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (tagU tagL : (N : ℕ) → Fin N → ℝ) |
| (C : (N : ℕ) → Fin (2*N) → TriangularCovering.TriCylinder) : Prop where |
| local_properties : ∀ N, 0 < N → ∀ i : Fin N, |
| (C N (RadialPairedSweep.upperIndex i)).axis = upperPhysicalAxis b eta (tagU N i) ∧ |
| (C N (RadialPairedSweep.lowerIndex i)).axis = lowerPhysicalAxis b eta (tagL N i) ∧ |
| upperPhysicalBase centered b eta T N (tagU N) i ⊆ |
| (C N (RadialPairedSweep.upperIndex i)).base ∧ |
| lowerPhysicalBase centered b eta T N (tagL N) i ⊆ |
| (C N (RadialPairedSweep.lowerIndex 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) |
| covers_body : ∀ N, 0 < N → ∀ z ∈ modelBody centered b, |
| ∃ i : Fin (2*N), z ∈ (C N i).carrier |
| covers_sweeps : ∀ N, 0 < N → |
| ∀ z ∈ upperSweepOn centered b eta T (Icc (-b) b) ∪ |
| lowerSweepOn centered b eta T (Icc (-b) b), |
| ∃ i : Fin (2*N), z ∈ (C N i).carrier |
| actual_covers_body : ∀ N, 0 < N → ∀ z ∈ modelBody centered b, |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder (upperPhysicalAxis b eta (tagU N i)) |
| (upperPhysicalBase centered b eta T N (tagU N) i)) ∨ |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder (lowerPhysicalAxis b eta (tagL N i)) |
| (lowerPhysicalBase centered b eta T N (tagL N) i)) |
| triangle_limit : Tendsto (fun N => TriangularCovering.totalArea (C N)) atTop |
| (𝓝 (scaledCost b eta T)) |
| actual_bound : ∀ δ : ℝ, 0 < δ → ∀ᶠ N in atTop, |
| physicalActualCost centered b eta T N (tagU N) (tagL N) ≤ scaledCost b eta T + δ |
| actual_limsup : Filter.limsup |
| (fun N => physicalActualCost centered b eta T N (tagU N) (tagL N)) atTop ≤ |
| scaledCost b eta T |
|
|
| def PhysicalCapFiniteStatement : Prop := |
| ∀ (centered square : Bool) (b beta : ℝ), 0 < b → 0 < beta → beta < 1/60 → |
| ∀ᶠ eta : ℝ in 𝓝[>] (0 : ℝ), |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagU N)) → |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagL N)) → |
| ∃ C : (N : ℕ) → Fin (2*N) → TriangularCovering.TriCylinder, |
| PhysicalFamilyConclusion centered b eta (physicalModelCap centered square b eta beta) tagU tagL C ∧ |
| ∀ᶠ N in atTop, 0 < N ∧ |
| TriangularCovering.totalArea (C N) < physicalHalfReference centered b ∧ |
| physicalActualCost centered b eta (physicalModelCap centered square b eta beta) N |
| (tagU N) (tagL N) < physicalHalfReference centered b |
|
|
| def PhysicalCapCoverageStatement : Prop := |
| ∀ (centered square : Bool) (b beta : ℝ), 0 < b → 0 < beta → |
| ∀ᶠ eta : ℝ in 𝓝[>] (0 : ℝ), |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagU N)) → |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagL N)) → |
| ∃ C : (N : ℕ) → Fin (2*N) → TriangularCovering.TriCylinder, |
| PhysicalFamilyConclusion centered b eta (physicalModelCap centered square b eta beta) |
| tagU tagL C |
|
|
| end RadialPhysical |
|
|
| end |
|
|
| end OAI |
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