File size: 3,172 Bytes
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 | import OAI.Geometry.CylinderCovering.Physical.CapNormalizedDefinitions
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace RadialPhysical
open CylinderCovering
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)
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
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
end RadialPhysical
end
end OAI
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