| import OAI.Geometry.CylinderCovering.Physical.CapNormalizedDefinitions |
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| namespace OAI |
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| open Set Filter |
| open scoped Topology |
| noncomputable section |
| namespace RadialPhysical |
| open CylinderCovering |
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| def paddedNumerator (fractional : Bool) (b eta c s : ℝ) : ℝ := |
| 1 + eta^2 * (s^4 + b^2*s^2)/2 + |
| c * (if fractional then eta^(5/2 : ℝ) else eta^3) |
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| def physicalPaddedCap (centered fractional : Bool) (b eta c : ℝ) : ℝ → ℝ := |
| if centered then paddedNumerator fractional b eta c |
| else fun s => paddedNumerator fractional b eta c s / 2 |
|
|
| def PaddedFiniteAt (centered fractional : Bool) (b eta c : ℝ) : Prop := |
| ∀ 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 |
| (physicalPaddedCap centered fractional b eta c) tagU tagL C |
|
|
| def PaddedStrictFiniteAt (centered fractional : Bool) (b eta c : ℝ) : Prop := |
| ∀ 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 |
| (physicalPaddedCap centered fractional b eta c) tagU tagL C ∧ |
| ∀ᶠ N in atTop, 0 < N ∧ |
| TriangularCovering.totalArea (C N) < physicalHalfReference centered b ∧ |
| physicalActualCost centered b eta |
| (physicalPaddedCap centered fractional b eta c) N (tagU N) (tagL N) < |
| physicalHalfReference centered b |
|
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| def PaddedNormalizedFiniteAt (centered fractional : Bool) (b eta c : ℝ) : Prop := |
| ∀ 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 |
| (physicalPaddedCap centered fractional b eta c) tagU tagL C ∧ |
| ∀ᶠ N in atTop, 0 < N ∧ |
| PhysicalBothHalfBounds centered b eta (physicalPaddedCap centered fractional b eta c) |
| N (tagU N) (tagL N) (C N) |
|
|
| structure PhysicalPaddedFiniteStatement : Prop where |
| cubic_all : ∀ (centered : Bool) (b c : ℝ), 0 < b → (9/2 : ℝ)*b^6 ≤ c → |
| ∀ eta : ℝ, 0 < eta → PaddedFiniteAt centered false b eta c |
| fractional_threshold : ∀ (centered : Bool) (b c : ℝ), 0 < b → 0 < c → |
| ∀ eta : ℝ, 0 < eta → eta*b^2 ≤ (2*(c/b^5)/9)^2 → |
| PaddedFiniteAt centered true b eta c |
| eventual_strict : ∀ (centered fractional : Bool) (b c : ℝ), 0 < b → |
| (if fractional then 0 < c else (9/2 : ℝ)*b^6 ≤ c) → |
| ∀ᶠ eta : ℝ in 𝓝[>] 0, PaddedStrictFiniteAt centered fractional b eta c |
| eventual_normalized : ∀ (centered fractional : Bool) (b c : ℝ), 0 < b → |
| (if fractional then 0 < c else (9/2 : ℝ)*b^6 ≤ c) → |
| ∀ᶠ eta : ℝ in 𝓝[>] 0, PaddedNormalizedFiniteAt centered fractional b eta c |
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| end RadialPhysical |
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| end |
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| end OAI |
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