| import OAI.Geometry.IsometricImmersion.Darboux.HeightPrincipalBounds | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped ContDiff Topology | |
| namespace SmoothLocal.Flow | |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted | |
| def heightCapCurvatureBound (G Z d c : ℝ) : ℝ := | |
| curvatureFirstBound G d*(1+flowCoordinateBound (heightQuotientJetBound G Z d c)) | |
| def heightCapCValueBound (G Z d c e0 : ℝ) (ell : ℕ) : ℝ := | |
| ((ell : ℝ)+1)*heightAJetBound G Z d c + | |
| heightAJetBound G Z d c*heightRemainderBound G Z d c e0 ell | |
| def heightCapCoefficientBound (G Z d c e0 : ℝ) (ell : ℕ) : ℝ := | |
| heightAJetBound G Z d c + 2*heightBJetBound G Z d c ell + heightCapCValueBound G Z d c e0 ell | |
| theorem heightCapCurvatureBound_nonneg {G Z d c : ℝ} | |
| (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) : | |
| 0 ≤ heightCapCurvatureBound G Z d c := by | |
| have hK := curvatureFirstBound_nonneg hG hd | |
| have hF := flowCoordinateBound_nonneg (heightQuotientJetBound_nonneg hG hZ hd hc) | |
| unfold heightCapCurvatureBound | |
| positivity | |
| theorem heightCapCoefficientBound_dominates {G Z d c e0 : ℝ} | |
| (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0) (ell : ℕ) : | |
| 0 ≤ heightCapCoefficientBound G Z d c e0 ell ∧ | |
| heightAJetBound G Z d c ≤ heightCapCoefficientBound G Z d c e0 ell ∧ | |
| 2*heightBJetBound G Z d c ell ≤ heightCapCoefficientBound G Z d c e0 ell ∧ | |
| heightBJetBound G Z d c ell ≤ heightCapCoefficientBound G Z d c e0 ell ∧ | |
| heightCapCValueBound G Z d c e0 ell ≤ heightCapCoefficientBound G Z d c e0 ell := by | |
| have hA := heightAJetBound_nonneg hG hZ hd hc | |
| have hB := heightBJetBound_nonneg hG hZ hd hc ell | |
| have hr := heightRemainderBound_nonneg hG hZ hd hc he0 ell | |
| have hC : 0 ≤ heightCapCValueBound G Z d c e0 ell := by | |
| unfold heightCapCValueBound | |
| positivity | |
| unfold heightCapCoefficientBound | |
| constructor | |
| · positivity | |
| constructor | |
| · linarith only [hB,hC] | |
| constructor | |
| · linarith only [hA,hC] | |
| constructor | |
| · linarith only [hA,hB,hC] | |
| · linarith only [hA,hB] | |
| theorem heightChartC_value_bound | |
| {g : MetricField} {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ} {G Z d c e0 : ℝ} | |
| (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0) (ell : ℕ) | |
| (hAB : CoordinateBound (heightChartA g z Y) capChartDomain 1 (heightAJetBound G Z d c)) | |
| (hrB : ∀ p ∈ capChartDomain, | |
| |heightChartRemainder g z Y ell p| ≤ heightRemainderBound G Z d c e0 ell ∧ | |
| heightChartC g z Y ell p = ((ell : ℝ)+1)*coordPartial 1 (heightChartA g z Y) p + | |
| heightChartA g z Y p*heightChartRemainder g z Y ell p) | |
| {p : Coord} (hp : p ∈ capChartDomain) : | |
| |heightChartC g z Y ell p| ≤ heightCapCValueBound G Z d c e0 ell := by | |
| have hel : 0 ≤ (ell : ℝ)+1 := by positivity | |
| have hfirst := mul_le_mul_of_nonneg_left (hAB [1] (by norm_num) p hp) hel | |
| have hsecond := | |
| (mul_le_mul_of_nonneg_left (hrB p hp).1 (abs_nonneg (heightChartA g z Y p))).trans | |
| (mul_le_mul_of_nonneg_right (hAB [] (by norm_num) p hp) | |
| (heightRemainderBound_nonneg hG hZ hd hc he0 ell)) | |
| rw [(hrB p hp).2] | |
| calc | |
| _ ≤ |((ell : ℝ)+1)*coordPartial 1 (heightChartA g z Y) p| + | |
| |heightChartA g z Y p*heightChartRemainder g z Y ell p| := abs_add_le _ _ | |
| _ ≤ _ := by | |
| rw [abs_mul,abs_of_nonneg hel,abs_mul] | |
| exact add_le_add hfirst hsecond | |
| theorem heightChartB_primitive_first_bounds | |
| {g : MetricField} {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ} {MB : ℝ} (ell : ℕ) | |
| (hMB : 0 ≤ MB) (hB : ContDiffOn ℝ ∞ (heightChartB g z Y ell) capChartDomain) | |
| (hBB : CoordinateBound (heightChartB g z Y ell) capChartDomain 1 MB) | |
| {p : Coord} (hp : p ∈ capChartDomain) : | |
| |coordinatePrimitive (heightChartB g z Y ell) p| ≤ 2*MB ∧ | |
| ∀ i : Fin 2, |coordPartial i (coordinatePrimitive (heightChartB g z Y ell)) p| ≤ 2*MB := by | |
| obtain ⟨hIv,hIs⟩ := heightChartB_primitive_bounds ell hB hBB hp | |
| refine ⟨hIv,?_⟩ | |
| intro i | |
| fin_cases i | |
| · simp only [Fin.mk_zero] | |
| rw [coordinatePrimitive_partial_t hB (by norm_num : (0 : ℝ) < 2) hp] | |
| exact (hBB [] (by norm_num) p hp).trans (by linarith only [hMB]) | |
| · simpa only [Fin.mk_one] using hIs | |
| theorem heightCurvaturePullback_bound_one | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {G Z d c : ℝ} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) | |
| (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) | |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) modelSquare 4 G) | |
| (hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|) | |
| (hY : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2)) | |
| (hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s) | |
| (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, | |
| HasDerivWithinAt (Y s) (-hessianQuotient g z (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) | |
| (hmapS : MapsTo (capChart Y) capChartDomain modelSquare) | |
| (hsmall : ∀ p ∈ modelSquare, |hessianQuotient g z p| ≤ (1 : ℝ)/100) | |
| (hrho : ∀ p ∈ capChartDomain, 1/Real.exp (2*heightQuotientJetBound G Z d c) ≤ capChartRho Y p) : | |
| CoordinateBound (capPullback Y (gaussianCurvature g)) capChartDomain 1 (heightCapCurvatureBound G Z d c) := by | |
| have hM := heightQuotientJetBound_nonneg hG hZ hd hc | |
| have hF := flowCoordinateBound_nonneg hM | |
| have hK := curvatureFirstBound_nonneg hG hd | |
| exact capPullback_bound_one_from_partials hY (gaussianCurvature_contDiffOn hg hU) hU | |
| (fun p hp => hSU (hmapS hp)) hmapS hK hF | |
| (gaussianCurvature_coordinate_bound_one_closed hg hU hSU hG hd hgB hdet) | |
| (fun i p hp => capFlowHeight_partial_bound_of_rho hY hvar hode hmapS hsmall hM hrho hp i) | |
| end SmoothLocal.Flow | |
| end | |
| end OAI | |