| import OAI.Geometry.CylinderCovering.Ruled.LocalCylinder | |
| import OAI.Geometry.CylinderCovering.Ruled.CellCost | |
| import OAI.Geometry.CylinderCovering.Convexity.CompactProductEigenframes | |
| namespace OAI | |
| noncomputable section | |
| open Set Metric MeasureTheory | |
| namespace CylinderCovering.ProductLocal | |
| open RuledApproximation | |
| lemma hyperbolicAt_trace_det {M : ℝ} (hM : 0 < M) | |
| {v : Plane → Plane} {p : Plane} (h : HyperbolicAt M v p) : | |
| LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 ∧ | |
| LinearMap.det (fderiv ℝ v p).toLinearMap < 0 := by | |
| obtain ⟨lam,hMl,_,hp,hm⟩ := h | |
| have hlam : 0 < lam := (mul_pos_iff_of_pos_left hM).mp hMl | |
| rw [Module.End.hasEigenvalue_iff_isRoot_charpoly,Polynomial.IsRoot,plane_charpoly] at hp hm | |
| simp only [Polynomial.eval_add,Polynomial.eval_sub,Polynomial.eval_pow, | |
| Polynomial.eval_X,Polynomial.eval_mul,Polynomial.eval_C] at hp hm | |
| have ht : LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 := by nlinarith | |
| refine ⟨ht,?_⟩ | |
| rw [ht] at hp | |
| nlinarith [sq_pos_of_pos hlam] | |
| lemma hyperbolicAt_exists_unit_eigenframe {M : ℝ} (hM : 0 < M) | |
| {v : Plane → Plane} {p : Plane} (h : HyperbolicAt M v p) : | |
| ∃ lam : ℝ, ∃ L : (ℝ × ℝ) ≃L[ℝ] Plane, | |
| 0 < M*lam ∧ M*lam < 1 ∧ IsUnitEigenframe (fderiv ℝ v p) lam L := by | |
| have htd := hyperbolicAt_trace_det hM h | |
| obtain ⟨lam,hMl,hMl1,hplus,_⟩ := h | |
| have hlam : 0 < lam := (mul_pos_iff_of_pos_left hM).mp hMl | |
| have heq := positive_eigenvalue_eq_sqrt (fderiv ℝ v p) htd.1 htd.2 hlam hplus | |
| obtain ⟨L,hL⟩ := exists_unit_eigenframe (fderiv ℝ v p) htd.1 htd.2 | |
| exact ⟨lam,L,hMl,hMl1,by simpa only [heq] using hL⟩ | |
| theorem local_proof : LocalStatement := by | |
| intro D v h M hD hDne hh hM hvnear hhyper eta tau gamma heta htau htau1 hgamma hpad hratio | |
| obtain ⟨U,hU,hDU,hv⟩ := hvnear | |
| have hder : ContinuousOn (fderiv ℝ v) D := by | |
| intro p hp | |
| exact ((hv.contDiffAt (hU.mem_nhds (hDU hp))).continuousAt_fderiv | |
| (by norm_num)).continuousWithinAt | |
| obtain ⟨N,hN,hNall⟩ := compact_uniform_eigenframe_inverse hD hder | |
| (fun p hp => (hyperbolicAt_trace_det hM (hhyper p hp)).1) | |
| (fun p hp => (hyperbolicAt_trace_det hM (hhyper p hp)).2) | |
| obtain ⟨rA,hrA,hA⟩ := uniform_intercept_containment hD hU hDU hv hM hN htau hgamma | |
| obtain ⟨rW,hrW,hW⟩ := uniform_weight_on_cells hD (hv.continuousOn.mono hDU) h heta | |
| refine ⟨min rA rW,lt_min hrA hrW,?_⟩ | |
| intro p hp lam L hL hML0 hML1 rho hrho hrho0 hGD | |
| have hlam : 0 < lam := (mul_pos_iff_of_pos_left hM).mp hML0 | |
| have hri : rho < rA := hrho0.trans_le (min_le_left _ _) | |
| have hrw : rho < rW := hrho0.trans_le (min_le_right _ _) | |
| let G := affineCell p rho L (ProductCells.parameterCell tau) | |
| let P := affineCell p rho L (ProductCells.interceptCell tau gamma) | |
| let C := projectedCylinder h (v p) P | |
| refine ⟨C,rfl,projectedCylinder_carrier h (v p) P,?_,?_,?_⟩ | |
| · exact ProductCells.pair_frame_productCylinder_openBounded L p (v p) hrho hh tau gamma | |
| · apply ruled_subset_projectedCylinder | |
| exact hA p hp lam L hL hlam.le hML1.le (hNall p hp lam L hlam hL).2 rho hrho hri | |
| · apply projectedCylinder_cost (parameter_affineCell_isCompact p rho tau L) | |
| (hv.continuousOn.mono (hGD.trans hDU)) hh heta.le p hratio | |
| · exact ProductCells.pair_frame_intercept_area L p hrho htau htau1 hgamma.le hpad | |
| · exact hW p hp lam rho tau L hL hrho hrw hGD | |
| end CylinderCovering.ProductLocal | |
| end | |
| end OAI | |