File size: 4,045 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 | 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
|