| import OAI.Geometry.IsometricImmersion.Immersions.HeightNumericParameters |
| import OAI.Geometry.IsometricImmersion.Energy.DirectedParameterChoice |
| import OAI.Geometry.IsometricImmersion.Energy.PatchDirectedCoercivity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Function |
| open scoped ContDiff |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model |
|
|
| theorem exists_patch_uniform_numeric_directed_flow |
| {g0 eta : MetricField} {z : Coord → ℝ} {U : Set Coord} {G Z d c e0 kappa : ℝ} |
| (hg : SmoothPositiveOn (g0 + eta) U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hz : ContDiffOn ℝ ∞ z U) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0) |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => (g0 + eta) p i j) modelSquare 4 G) |
| (hzB : CoordinateBound z modelSquare 5 Z) |
| (hdet : ∀ p ∈ modelSquare, d ≤ |((g0 + eta) p).det|) |
| (hyy : ∀ p ∈ modelSquare, c ≤ |covHessian (g0 + eta) z p 1 1|) |
| (hEfloor : ∀ p ∈ modelSquare, e0 ≤ heightEnergy (g0 + eta) z p) |
| (hsmall : ∀ p ∈ modelSquare, |hessianQuotient (g0 + eta) z p| ≤ (1 : ℝ) / 100) |
| (hD : ∀ p ∈ modelSquare, |
| (covHessian (g0 + eta) z p).det = gaussianCurvature (g0 + eta) p * heightEnergy (g0 + eta) z p) |
| (hk : 0 < kappa) |
| (hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p) |
| (hsupport : tsupport eta ⊆ patchBox) |
| (hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2) |
| (ell : ℕ) : |
| let epsilon := heightDirectedEpsilon G Z d c e0 kappa ell |
| let lambda := heightDirectedLambda G Z d c e0 kappa ell |
| let c1 := heightDirectedSlope G Z d c e0 kappa ell |
| let c2 := heightDirectedCoercivity G Z d c e0 kappa ell |
| ∃ Y : ℝ → ℝ → ℝ, |
| 0 < epsilon ∧ epsilon ≤ 1 ∧ 0 < lambda ∧ 0 < c1 ∧ 0 < c2 ∧ |
| ContDiffOn ℝ ∞ (capChart Y) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2) ∧ |
| (∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s) ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| HasDerivWithinAt (Y s) (-hessianQuotient (g0 + eta) z (coordinatePoint t (Y s t))) |
| (Icc (-2 : ℝ) 2) t) ∧ |
| (∀ p ∈ capChartDomain, |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50) ∧ |
| ContDiffOn ℝ ∞ (heightChartA (g0 + eta) z Y) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (heightChartB (g0 + eta) z Y ell) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (heightChartC (g0 + eta) z Y ell) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (coordinatePrimitive (heightChartB (g0 + eta) z Y ell)) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (capPullback Y (gaussianCurvature (g0 + eta))) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (heightChartG1 (g0 + eta) z Y) capChartDomain ∧ |
| (∀ p ∈ capChartDomain, |
| heightG1Lower G Z d c e0 ≤ heightChartG1 (g0 + eta) z Y p ∧ |
| |coordinatePrimitive (heightChartB (g0 + eta) z Y ell) p| ≤ 2*heightBJetBound G Z d c ell ∧ |
| (∀ i : Fin 2, |coordPartial i (coordinatePrimitive (heightChartB (g0 + eta) z Y ell)) p| ≤ |
| heightCapCoefficientBound G Z d c e0 ell) ∧ |
| (∀ i : Fin 2, |coordPartial i (capPullback Y (gaussianCurvature (g0 + eta))) p| ≤ |
| heightCapCurvatureBound G Z d c) ∧ |
| |heightChartA (g0 + eta) z Y p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧ |
| (∀ i : Fin 2, |coordPartial i (heightChartA (g0 + eta) z Y) p| ≤ |
| heightCapCoefficientBound G Z d c e0 ell) ∧ |
| |heightChartB (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧ |
| |heightChartC (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell) ∧ |
| ∀ edge : ℝ, edge ≤ 0 → ∀ p ∈ capChartDomain, p 1 < edge → |
| capPullback Y (gaussianCurvature (g0 + eta)) p ≤ c1 * (edge - p 1) → |
| ∀ x y : ℝ, |
| let A := heightChartA (g0 + eta) z Y |
| let B := heightChartB (g0 + eta) z Y ell |
| let C := heightChartC (g0 + eta) z Y ell |
| let I := coordinatePrimitive B |
| c2 * directedWeight edge lambda I p * (x^2 + y^2) ≤ |
| multiplierT B (directedM edge lambda I) (directedN edge lambda epsilon I) p * x^2 + |
| multiplierS A C (directedM edge lambda I) (directedN edge lambda epsilon I) p * y^2 + |
| multiplierJ A B C (directedM edge lambda I) (directedN edge lambda epsilon I) p * x * y := by |
| let M := heightQuotientJetBound G Z d c |
| let cG := heightG1Lower G Z d c e0 |
| let rhoMax := 1 / Real.exp (-2 * M) |
| let MG := heightG1JetBound G Z d c |
| let MB := heightBJetBound G Z d c ell |
| let Mr := heightRemainderBound G Z d c e0 ell |
| let MA := heightAJetBound G Z d c |
| have hcG : 0 < cG := heightG1Lower_pos hG hZ hd he0 |
| have hrhoMax : 0 < rhoMax := by dsimp [rhoMax]; positivity |
| have hMG : 0 ≤ MG := heightG1JetBound_nonneg hG hZ hd hc |
| have hMB : 0 ≤ MB := heightBJetBound_nonneg hG hZ hd hc ell |
| have hMr : 0 ≤ Mr := heightRemainderBound_nonneg hG hZ hd hc he0 ell |
| have hMA : 0 ≤ MA := heightAJetBound_nonneg hG hZ hd hc |
| let epsilon := heightDirectedEpsilon G Z d c e0 kappa ell |
| let lambda := heightDirectedLambda G Z d c e0 kappa ell |
| let c1 := heightDirectedSlope G Z d c e0 kappa ell |
| obtain ⟨heps, heps1, hlambda, hc1, hqbudget, hTbudget, hEbudget, |
| hcross, hloss, hfarbudget, hnearstrip, hstripbudget⟩ := |
| heightDirectedNumbers_spec ⟨hG,hZ,hd,hc,he0,hk⟩ ell |
| let margin := modelCoercivityMargin cG kappa rhoMax epsilon ell |
| let delta := modelPrincipalRadius cG kappa rhoMax epsilon 2 MG ell |
| let c2 := min (lambda / 8) (margin / 4) |
| have hmargin : 0 < margin := modelCoercivityMargin_pos ell hcG hk hrhoMax heps |
| have hdelta : 0 < delta := modelPrincipalRadius_pos ell hcG hk hrhoMax heps (by norm_num) hMG |
| have hc2 : 0 < c2 := by dsimp [c2]; positivity |
| obtain ⟨Y, hchart, hYclosed, hYs, hvar, hstart, hode, hconf, hdisp50, hdisp, |
| hrho, hG1s, hG1B, hGfloor, hall⟩ := |
| exists_height_flow_uniform_coefficients hg hU hSU hz hG hZ hd hc he0 hgB hzB hdet hyy hEfloor hsmall hD |
| have hmapO : MapsTo (capChart Y) capChartDomain modelOpenSquare := |
| fun p hp => capChart_mem_modelOpenSquare hp (hdisp p hp) |
| have hmapS : MapsTo (capChart Y) capChartDomain modelSquare := |
| fun p hp => modelOpenSquare_subset (hmapO hp) |
| have hAB := heightChartA_bound_one hg hU hSU hG hZ hd hc hgB hdet hYs hvar hode hmapS hsmall |
| (fun p hp => (hrho p hp).2.1) hG1s hG1B |
| obtain ⟨hBs, hBB, hrs, hrB⟩ := hall ell |
| have hOU : modelOpenSquare ⊆ U := modelOpenSquare_subset.trans hSU |
| have hgO : SmoothPositiveOn (g0 + eta) modelOpenSquare := |
| ⟨fun i j => (hg.1 i j).mono hOU, fun p hp => hg.2 p (hOU hp)⟩ |
| have hzO := hz.mono hOU |
| have hyyn := LowQuotient.denominator_ne_zero hc hyy |
| have hyyO := fun p (hp : p ∈ modelOpenSquare) => hyyn p (modelOpenSquare_subset hp) |
| have hEO := fun p (hp : p ∈ modelOpenSquare) => he0.trans_le (hEfloor p (modelOpenSquare_subset hp)) |
| have hDO := fun p (hp : p ∈ modelOpenSquare) => hD p (modelOpenSquare_subset hp) |
| have hbackgroundO := fun p (hp : p ∈ modelOpenSquare) => hbackground p (hOU hp) |
| have hAs := heightChartA_contDiffOn hgO modelOpenSquare_isOpen hzO hyyO hYs hvar hmapO |
| have hKB := heightCurvaturePullback_bound_one hg hU hSU hG hZ hd hc hgB hdet |
| hYs hvar hode hmapS hsmall (fun p hp => (hrho p hp).2.1) |
| obtain ⟨hMF,hMFA,hMFI,hMFB,hMFC⟩ := heightCapCoefficientBound_dominates hG hZ hd hc he0 ell |
| have hfixed (p : Coord) (hp : p ∈ capChartDomain) : |
| heightG1Lower G Z d c e0 ≤ heightChartG1 (g0 + eta) z Y p ∧ |
| |coordinatePrimitive (heightChartB (g0 + eta) z Y ell) p| ≤ 2*heightBJetBound G Z d c ell ∧ |
| (∀ i : Fin 2, |coordPartial i (coordinatePrimitive (heightChartB (g0 + eta) z Y ell)) p| ≤ |
| heightCapCoefficientBound G Z d c e0 ell) ∧ |
| (∀ i : Fin 2, |coordPartial i (capPullback Y (gaussianCurvature (g0 + eta))) p| ≤ |
| heightCapCurvatureBound G Z d c) ∧ |
| |heightChartA (g0 + eta) z Y p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧ |
| (∀ i : Fin 2, |coordPartial i (heightChartA (g0 + eta) z Y) p| ≤ |
| heightCapCoefficientBound G Z d c e0 ell) ∧ |
| |heightChartB (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧ |
| |heightChartC (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell := by |
| obtain ⟨hIv,hIi⟩ := heightChartB_primitive_first_bounds ell hMB hBs hBB hp |
| exact ⟨hGfloor p hp,hIv,fun i => (hIi i).trans hMFI, |
| fun i => hKB [i] (by norm_num) p hp, |
| (hAB [] (by norm_num) p hp).trans hMFA, |
| fun i => (hAB [i] (by norm_num) p hp).trans hMFA, |
| (hBB [] (by norm_num) p hp).trans hMFB, |
| (heightChartC_value_bound hG hZ hd hc he0 ell hAB hrB hp).trans hMFC⟩ |
| refine ⟨Y, heps, heps1, hlambda, hc1, hc2, hchart, hYs, hvar, hstart, hode, hdisp, |
| hAs, hBs, heightChartC_contDiffOn hgO modelOpenSquare_isOpen hzO hDO hyyO hEO hYs hvar hmapO ell, |
| coordinatePrimitive_contDiffOn hBs, |
| capPullback_contDiffOn hYs (gaussianCurvature_contDiffOn hg hU) (fun p hp => hSU (hmapS hp)), |
| hG1s,hfixed, ?_⟩ |
| intro edge hedge0 p hp hedge hKstrip x y |
| have hs : p 1 ≤ 0 := (le_of_lt hedge).trans hedge0 |
| have hdist : edge - p 1 ≤ 2 := by linarith [hp.2.1] |
| have hI := (heightChartB_primitive_bounds ell hBs hBB hp).2 |
| have hBpoint : |heightChartB (g0 + eta) z Y ell p| ≤ MB := hBB [] (by norm_num) p hp |
| have hrpoint : |heightChartRemainder (g0 + eta) z Y ell p| ≤ Mr := (hrB p hp).1 |
| have hApoint : |heightChartA (g0 + eta) z Y p| ≤ MA := hAB [] (by norm_num) p hp |
| have hAspoint : |coordPartial 1 (heightChartA (g0 + eta) z Y) p| ≤ MA := hAB [1] (by norm_num) p hp |
| have hAtpoint : |coordPartial 0 (heightChartA (g0 + eta) z Y) p| ≤ MA := hAB [0] (by norm_num) p hp |
| have hGspoint : |coordPartial 1 (heightChartG1 (g0 + eta) z Y) p| ≤ MG := hG1B [1] (by norm_num) p hp |
| have hGtpoint : |coordPartial 0 (heightChartG1 (g0 + eta) z Y) p| ≤ MG := hG1B [0] (by norm_num) p hp |
| have hGmax : heightChartG1 (g0 + eta) z Y p ≤ MG := (le_abs_self _).trans (hG1B [] (by norm_num) p hp) |
| by_cases hnear : |capPullback Y (gaussianCurvature (g0 + eta)) p| ≤ delta |
| · exact patch_actual_directed_near_coercivity hgO modelOpenSquare_isOpen hzO hDO hyyO hEO |
| hYs hode hvar hdisp50 hsmall hmapO hbackgroundO hsupport hcentral hp hs ell |
| cG rhoMax epsilon MG (2 * MB) MB Mr MA MA MG c1 2 edge lambda hk hcG (hGfloor p hp) |
| (hrho p hp).2.2 heps heps1 hqbudget hGspoint hGtpoint hnear hedge hlambda hI hBpoint hrpoint |
| hApoint hAspoint hTbudget hEbudget hcross hloss hGmax hc1.le hdist hKstrip hstripbudget x y |
| · have hKupper : capPullback Y (gaussianCurvature (g0 + eta)) p ≤ delta := |
| hKstrip.trans ((mul_le_mul_of_nonneg_left hdist hc1.le).trans hnearstrip) |
| have hKnegative : capPullback Y (gaussianCurvature (g0 + eta)) p ≤ -delta := by |
| by_contra hneg |
| exact hnear (abs_le.mpr ⟨(lt_of_not_ge hneg).le, hKupper⟩) |
| have hAfar := heightChartA_negative_of_curvature (g0 + eta) z Y p hcG hdelta (hGfloor p hp) hKnegative |
| have hloss' : |
| (4 * (2 * ((ell : ℝ) * MA + MA * Mr + MA * (2 * MB) + 1))^2 * epsilon^2) / lambda ≤ |
| (margin / 2) / 4 := by |
| calc |
| _ ≤ margin / 8 := hloss |
| _ = _ := by ring |
| have hh := primitive_directed_farnegative_coercivity hBs (by norm_num : (0 : ℝ) < 2) hp |
| (heightChartA (g0 + eta) z Y) (heightChartC (g0 + eta) z Y ell) |
| (heightChartRemainder (g0 + eta) z Y ell) |
| ((hAs.contDiffAt (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp)) |
| edge lambda epsilon ell hedge hlambda heps.le heps1 (hrB p hp).2 |
| (2 * MB) MB Mr MA MA MA (margin / 2) (cG * delta) (by positivity) |
| hI hBpoint hrpoint hApoint hAspoint hTbudget hEbudget hcross hloss' hAtpoint |
| (mul_pos hcG hdelta) hAfar hfarbudget x y |
| have hhalf : (margin / 2) / 2 = margin / 4 := by ring |
| rw [hhalf] at hh |
| exact hh |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
|
|
