File size: 6,015 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Geometry.CylinderCovering.Ruled.LogarithmicLocalDefinitions
import OAI.Geometry.CylinderCovering.Ruled.LocalEndpoint
import OAI.Geometry.CylinderCovering.Ruled.Remainder
namespace OAI
noncomputable section
open Set MeasureTheory
namespace CylinderCovering.ProductLocal
open RuledApproximation
lemma affineCell_dilation (p : Plane) (t m : ℝ)
(L : (ℝ × ℝ) ≃L[ℝ] Plane) (S : Set (ℝ × ℝ)) :
affineCell p t L (ProductCells.dilation m '' S) =
affineCell p (m*t) L S := by
simp only [affineCell, image_image]
congr 1
funext s
have hd : ProductCells.dilation m s = m • s := by
ext <;> rfl
rw [hd, map_smul, smul_smul, mul_comm t m]
lemma affine_logarithmicCell (p : Plane) (t : ℝ)
(L : (ℝ × ℝ) ≃L[ℝ] Plane) {m : ℝ} (hm : 0 < m) :
affineCell p t L (ProductCells.logarithmicCell m) =
affineCell p (m*t) L (ProductCells.parameterCell (1/m^2)) := by
rw [ProductCells.logarithmicCell_eq_dilation m hm, affineCell_dilation]
lemma affine_logarithmicIntercept (p : Plane) (t gamma : ℝ)
(L : (ℝ × ℝ) ≃L[ℝ] Plane) {m : ℝ} (hm : 0 < m) :
affineCell p t L (ProductCells.logarithmicIntercept m gamma) =
affineCell p (m*t) L (ProductCells.interceptCell (1/m^2) gamma) := by
rw [ProductCells.logarithmicIntercept_eq_dilation m gamma hm, affineCell_dilation]
lemma logarithmic_projectedCylinder_eq (p g : Plane) (t gamma h : ℝ)
(L : (ℝ × ℝ) ≃L[ℝ] Plane) {m : ℝ} (hm : 0 < m) :
projectedCylinder h g (affineCell p t L (ProductCells.logarithmicIntercept m gamma)) =
projectedCylinder h g (affineCell p (m*t) L (ProductCells.interceptCell (1/m^2) gamma)) := by
rw [affine_logarithmicIntercept p t gamma L hm]
theorem logarithmic_local_proof : LogarithmicLocalStatement := by
intro D v h M hD hDne hh hM hv hhyper eta m gamma heta hm hgamma hpad hratio
have hm0 : 0 < m := lt_trans zero_lt_one hm
have htau : 0 < 1/m^2 := by positivity
have htau1 : 1/m^2 < 1 := by
apply (div_lt_one (sq_pos_of_pos hm0)).mpr
nlinarith
obtain ⟨rho0, hrho0, hlocal⟩ :=
local_proof D v h M hD hDne hh hM hv hhyper eta (1/m^2) gamma
heta htau htau1 hgamma hpad hratio
refine ⟨rho0/m, div_pos hrho0 hm0, ?_⟩
intro p hp lam L hL hlam0 hlam1 t ht ht0 hGD
have hmt : m*t < rho0 := by
have := (lt_div_iff₀ hm0).mp ht0
simpa only [mul_comm t m] using this
have hG : affineCell p (m*t) L (ProductCells.parameterCell (1/m^2)) ⊆ D := by
rwa [affine_logarithmicCell p t L hm0] at hGD
obtain ⟨C, haxis, hcarrier, hopen, hcover, hcost⟩ :=
hlocal p hp lam L hL hlam0 hlam1 (m*t) (mul_pos hm0 ht) hmt hG
refine ⟨C, haxis, ?_, hopen, ?_, ?_⟩
· simpa only [affine_logarithmicIntercept p t gamma L hm0] using hcarrier
· simpa only [affine_logarithmicCell p t L hm0] using hcover
· simpa only [affine_logarithmicCell p t L hm0] using hcost
theorem logarithmic_source_local : SourceLogarithmicLocalStatement := by
intro X k D v hX hk hD hDne hv hH eta heta
obtain ⟨U, hU, hDU, hv1⟩ := hv.smoothNear
have hw : SmoothNear 1 D (fun q => k • v q) :=
⟨U, hU, hDU, hv1.const_smul k⟩
have hf (p : Plane) (hp : p ∈ D) :
fderiv ℝ (fun q => k • v q) p = k • fderiv ℝ v p :=
(((hv1.contDiffAt (hU.mem_nhds (hDU hp))).differentiableAt
(by norm_num)).hasFDerivAt.const_smul k).fderiv
have hhyp : ∀ p ∈ D, HyperbolicAt X (fun q => k • v q) p := by
intro p hp
obtain ⟨htr, hdet, hsmall⟩ := hH p hp
obtain ⟨hplus, hminus⟩ :=
plane_hyperbolic_eigenvalues (fderiv ℝ v p).toLinearMap htr hdet
have hlam : 0 < Real.sqrt (-LinearMap.det (fderiv ℝ v p).toLinearMap) :=
Real.sqrt_pos.mpr (neg_pos.mpr hdet)
refine ⟨k*Real.sqrt (-LinearMap.det (fderiv ℝ v p).toLinearMap),
by positivity, by nlinarith, ?_, ?_⟩
· rw [hf p hp]
exact eigenvalue_smul _ hplus k
· rw [hf p hp]
change Module.End.HasEigenvalue (k • (fderiv ℝ v p).toLinearMap) (-(k*_))
rw [← mul_neg]
exact eigenvalue_smul _ hminus k
obtain ⟨m, hm, gamma, hgamma, hpad, hratio⟩ :=
ProductCells.exists_logarithmic_parameters heta
obtain ⟨t0, ht0, hlocal⟩ := logarithmic_local_proof D (fun q => k • v q) 1 X
hD hDne (by norm_num) hX hw hhyp eta m gamma heta hm hgamma hpad hratio.le
refine ⟨m, hm, gamma, hgamma, hpad, hratio, t0, ht0, ?_⟩
intro p hp lam L hlam hL t ht htt0 hGD
have heig : Module.End.HasEigenvalue (fderiv ℝ v p).toLinearMap lam := by
apply Module.End.hasEigenvalue_of_hasEigenvector (x := L (1,0))
refine ⟨Module.End.mem_eigenspace_iff.mpr hL.2.2.1, ?_⟩
intro hz
have hn := hL.1
rw [hz, norm_zero] at hn
norm_num at hn
have heq := positive_eigenvalue_eq_sqrt (fderiv ℝ v p)
(hH p hp).1 (hH p hp).2.1 hlam heig
have hsmall : X*(k*lam) < 1 := by
rw [heq]
nlinarith [(hH p hp).2.2]
have hscaled : IsUnitEigenframe (fderiv ℝ (fun q => k • v q) p) (k*lam) L := by
rw [hf p hp]
refine ⟨hL.1, hL.2.1, ?_, ?_⟩
· change k • (fderiv ℝ v p) (L (1,0)) = (k*lam) • L (1,0)
rw [hL.2.2.1, smul_smul]
· change k • (fderiv ℝ v p) (L (0,1)) = (-(k*lam)) • L (0,1)
rw [hL.2.2.2, smul_smul, mul_neg]
obtain ⟨C, haxis, hcarrier, hopen, hcover, hcost⟩ :=
hlocal p hp (k*lam) L hscaled (by positivity) hsmall t ht htt0 hGD
refine ⟨C, ?_, hcarrier, hopen, hcover, ?_⟩
· simpa only [PiLp.smul_apply, smul_eq_mul, one_mul] using haxis
· have hweight (q : Plane) : physicalWeight 1 (k • v q) =
1 / Real.sqrt (1+k^2*‖v q‖^2) := by
rw [physicalWeight_one, weight, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs]
simpa only [one_mul, hweight] using hcost
theorem complete_local_proof : CompleteLocalStatement :=
⟨ProductCells.geometry_proof, local_proof,
logarithmic_local_proof, logarithmic_source_local, logarithmic_remainder_proof⟩
end CylinderCovering.ProductLocal
end
end OAI
|