| import OAI.Geometry.CylinderCovering.Beams.Finite |
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| namespace OAI |
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| open Filter |
| open scoped Topology |
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| noncomputable section |
| namespace ShortenedBeam |
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| open CylinderCovering |
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| 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] |
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| 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 |
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| 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 |
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| end ShortenedBeam |
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| end |
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| end OAI |
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