File size: 3,412 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 | import OAI.Geometry.CylinderCovering.Beams.Finite
namespace OAI
open Filter
open scoped Topology
noncomputable section
namespace ShortenedBeam
open CylinderCovering
lemma radialDataPlus_frozenArea_eq (a : ℝ) (ha : a ≤ 2)
(N : ℕ) (tag : Fin N → ℝ) (i : Fin N) :
RadialSweep.frozenArea (radialDataPlus a ha) N tag i =
RadialSweep.frozenArea (radialData a ha) N tag i := by
unfold RadialSweep.frozenArea
rw [radialDataPlus_kappa, radialData_kappa]
simp only [RadialSweep.radialError, RadialSweep.collar, RadialSweep.scalarBound,
radialDataPlus, radialData, norm_neg]
lemma radialDataPlus_frozenArea {a : ℝ} (ha : 0 ≤ a) (ha' : a ≤ 2)
{N : ℕ} (hN : 0 < N) (tag : Fin N → ℝ) (i : Fin N) :
RadialSweep.frozenArea (radialDataPlus a ha') N tag i =
(1/(N:ℝ))*((2-a)+(8*a*(2-a)+1)/(N:ℝ))^2 /
(4*Real.sqrt (1+‖velocity a (tag i)‖^2)) := by
rw [radialDataPlus_frozenArea_eq]
exact radialData_frozenArea ha ha' hN tag i
theorem finiteBeams_exact (a : ℝ) (ha : 0 < a) (ha' : a < 1/2)
(tag : (N : ℕ) → Fin N → ℝ)
(htag : ∀ N, RadialRiemann.IsTagged 0 1 N (tag N)) :
∃ Cminus Cplus : (N : ℕ) → Fin N → TriangularCovering.TriCylinder,
(∀ N, 0 < N →
(∀ i : Fin N,
(Cminus N i).axis = axisMinus a (tag N i) ∧
(Cplus N i).axis = axisPlus a (tag N i) ∧
(Cminus N i).base = RadialSweep.frozenBase
(radialData a (by linarith)) N (tag N) i ∧
(Cplus N i).base = RadialSweep.frozenBase
(radialDataPlus a (by linarith)) N (tag N) i ∧
TriangularCovering.area (Cminus N i).base =
(1/(N:ℝ))*((2-a)+(8*a*(2-a)+1)/(N:ℝ))^2 /
(4*Real.sqrt (1+‖velocity a (tag N i)‖^2)) ∧
TriangularCovering.area (Cplus N i).base =
(1/(N:ℝ))*((2-a)+(8*a*(2-a)+1)/(N:ℝ))^2 /
(4*Real.sqrt (1+‖velocity a (tag N i)‖^2))) ∧
(∀ z ∈ beamMinus a, ∃ i : Fin N, z ∈ (Cminus N i).carrier) ∧
(∀ z ∈ beamPlus a, ∃ i : Fin N, z ∈ (Cplus N i).carrier)) ∧
Tendsto (fun N => TriangularCovering.totalArea (Cminus N) +
TriangularCovering.totalArea (Cplus N)) atTop
(𝓝 ((Real.sqrt 2/4)*BeamCost.beamCost a)) := by
have ha2 : a ≤ 2 := by linarith
obtain ⟨Cminus, hminus, hminuslim⟩ :=
RadialSweep.aligned_radial_sweep (radialData a ha2) tag htag
obtain ⟨Cplus, hplus, hpluslim⟩ :=
RadialSweep.aligned_radial_sweep (radialDataPlus a ha2) tag htag
refine ⟨Cminus, Cplus, ?_, ?_⟩
· intro N hN
obtain ⟨hminuslocal, hminuscover⟩ := hminus N hN
obtain ⟨hpluslocal, hpluscover⟩ := hplus N hN
refine ⟨?_, ?_, ?_⟩
· intro i
refine ⟨(hminuslocal i).1.trans (radialData_direction a (tag N i) ha2),
(hpluslocal i).1.trans (radialDataPlus_direction a (tag N i) ha2),
(hminuslocal i).2.1, (hpluslocal i).2.1, ?_, ?_⟩
· exact (hminuslocal i).2.2.trans (radialData_frozenArea ha.le ha2 hN (tag N) i)
· exact (hpluslocal i).2.2.trans (radialDataPlus_frozenArea ha.le ha2 hN (tag N) i)
· simpa only [radialData_sweep] using hminuscover
· simpa only [radialDataPlus_sweep] using hpluscover
· have hlim := hminuslim.add hpluslim
rw [radialData_total_weightedArea] at hlim
exact hlim
end ShortenedBeam
end
end OAI
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