File size: 4,676 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.Geometry.CylinderCovering.Physical.CoordinatesDefinitions
import OAI.Geometry.CylinderCovering.Radial.SweepDefinitions
namespace OAI
open Set Filter MeasureTheory
open scoped Topology InnerProductSpace
noncomputable section
namespace RadialPhysical
def axialBound (centered : Bool) (b : ℝ) : ℝ := if centered then 2*b else b
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
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)
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)
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}
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}
def upperPhysicalAxis (b eta s : ℝ) : Space :=
let v := point b 1 (-eta*(s^2+b^2)/2) (eta*s)
‖v‖⁻¹ • v
def lowerPhysicalAxis (b eta s : ℝ) : Space :=
let v := point b (eta*(s^2+b^2)/2) 1 (eta*s)
‖v‖⁻¹ • v
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)
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)
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))
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
end
end OAI
|