| 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 |
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|
