| import OAI.Geometry.IsometricImmersion.Calculus.StateJetSegment |
| import OAI.Geometry.IsometricImmersion.Darboux.SixVariableLinearization |
| import Mathlib.Analysis.Normed.Group.Bounded |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open Set Filter |
|
|
| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry |
|
|
| def stateFirstCoefficient (g : MetricField) (i : Fin 6) (w : DarbouxState) : ℝ := |
| fderiv ℝ (sixVariableP g) w (Pi.single i 1) |
|
|
| theorem stateFirstCoefficient_contDiffOn {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (i : Fin 6) : |
| ContDiffOn ℝ ∞ (stateFirstCoefficient g i) (darbouxStateDomain g U) := |
| ((sixVariableP_contDiffOn hg hU).fderiv_of_isOpen |
| (darbouxStateDomain_isOpen hg hU) (by simp)).clm_apply contDiffOn_const |
|
|
| theorem stateFirstCoefficient_mixed {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) {w : DarbouxState} |
| (hw : w ∈ darbouxStateDomain g U) : |
| stateFirstCoefficient g 4 w = 2 * (stateMixed g w / stateDenominator g w) := by |
| have hd : DifferentiableAt ℝ (sixVariableP g) w := |
| ((sixVariableP_contDiffOn hg hU).contDiffAt |
| ((darbouxStateDomain_isOpen hg hU).mem_nhds hw)).differentiableAt (by simp) |
| unfold stateFirstCoefficient |
| rw [fderiv_axis_eq_deriv_update hd] |
| have heq : (fun t => sixVariableP g (Function.update w 4 t)) = |
| (fun t => solvedDarboux g (statePoint w) t (w 5) (stateGradient w)) := |
| funext (sixVariableP_update_mixed g w) |
| rw [heq, solvedDarboux_deriv_b g (statePoint w) (stateGradient w) (w 4) (w 5) hw.2] |
| rfl |
|
|
| theorem stateFirstCoefficient_yy {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) {w : DarbouxState} |
| (hw : w ∈ darbouxStateDomain g U) : |
| stateFirstCoefficient g 5 w = |
| -((stateMixed g w / stateDenominator g w) ^ 2 + |
| gaussianCurvature g (statePoint w) * stateEnergy g w / (stateDenominator g w) ^ 2) := by |
| have hd : DifferentiableAt ℝ (sixVariableP g) w := |
| ((sixVariableP_contDiffOn hg hU).contDiffAt |
| ((darbouxStateDomain_isOpen hg hU).mem_nhds hw)).differentiableAt (by simp) |
| unfold stateFirstCoefficient |
| rw [fderiv_axis_eq_deriv_update hd] |
| have heq : (fun t => sixVariableP g (Function.update w 5 t)) = |
| (fun t => solvedDarboux g (statePoint w) (w 4) t (stateGradient w)) := |
| funext (sixVariableP_update_yy g w) |
| rw [heq, solvedDarboux_deriv_c g (statePoint w) (stateGradient w) (w 4) (w 5) hw.2] |
| rfl |
|
|
| theorem stateFirstCoefficient_gradient {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) {w : DarbouxState} |
| (hw : w ∈ darbouxStateDomain g U) (i : Fin 2) : |
| stateFirstCoefficient g (gradientStateIndex i) w = |
| christoffelAlongD g (statePoint w) (stateMixed g w / stateDenominator g w) i + |
| gaussianCurvature g (statePoint w) * |
| (coordPartial i (jetEnergy g (statePoint w)) (stateGradient w) / stateDenominator g w + |
| stateEnergy g w * christoffel g i 1 1 (statePoint w) / (stateDenominator g w)^2) := by |
| rw [stateFirstCoefficient, sixVariableP_gradient_axis hg hU hw i] |
| exact solvedDarboux_firstJet hg hU hw.1 (stateGradient w) (w 4) (w 5) hw.2 i |
|
|
| def stateBaseCompact (S : Set Coord) (M : ℝ) : Set DarbouxState := |
| (statePoint ⁻¹' S) ∩ Icc (fun _ => -M) (fun _ => M) |
|
|
| def stateCompactTube (g : MetricField) (S : Set Coord) (M c : ℝ) : Set DarbouxState := |
| stateBaseCompact S M ∩ (fun w => |stateDenominator g w|) ⁻¹' Ici c |
|
|
| theorem stateBaseCompact_isCompact {S : Set Coord} (hS : IsCompact S) (M : ℝ) : |
| IsCompact (stateBaseCompact S M) := |
| isCompact_Icc.inter_left (hS.isClosed.preimage statePoint_contDiff.continuous) |
|
|
| theorem stateCompactTube_isCompact {g : MetricField} {U S : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hS : IsCompact S) (hSU : S ⊆ U) |
| (M c : ℝ) : IsCompact (stateCompactTube g S M c) := by |
| have hbase := stateBaseCompact_isCompact hS M |
| have hc : ContinuousOn (fun w => |stateDenominator g w|) (stateBaseCompact S M) := |
| (stateDenominator_contDiffOn hg hU).continuousOn.abs.mono (fun w hw => hSU hw.1) |
| have hclosed : IsClosed (stateCompactTube g S M c) := |
| hc.preimage_isClosed_of_isClosed hbase.isClosed isClosed_Ici |
| exact hbase.of_isClosed_subset hclosed inter_subset_left |
|
|
| theorem stateCompactTube_subset_domain {g : MetricField} {U S : Set Coord} |
| (hSU : S ⊆ U) (M : ℝ) {c : ℝ} (hc : 0 < c) : |
| stateCompactTube g S M c ⊆ darbouxStateDomain g U := by |
| intro w hw |
| refine ⟨hSU hw.1.1, ?_⟩ |
| intro hz |
| have hh : c ≤ |stateDenominator g w| := hw.2 |
| rw [hz, abs_zero] at hh |
| linarith |
| |
| theorem stateCompactTube_derivative_bound {g : MetricField} {U S : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hS : IsCompact S) (hSU : S ⊆ U) |
| (M : ℝ) {c : ℝ} (hc : 0 < c) {f : DarbouxState → ℝ} |
| (hf : ContDiffOn ℝ ∞ f (darbouxStateDomain g U)) (k : ℕ) : |
| ∃ C : ℝ, ∀ w ∈ stateCompactTube g S M c, ‖iteratedFDeriv ℝ k f w‖ ≤ C := by |
| have hDo := darbouxStateDomain_isOpen hg hU |
| have hcont : ContinuousOn (iteratedFDeriv ℝ k f) (darbouxStateDomain g U) := by |
| apply (hf.continuousOn_iteratedFDerivWithin (WithTop.coe_le_coe.mpr le_top) |
| hDo.uniqueDiffOn).congr |
| intro w hw |
| exact (iteratedFDerivWithin_eq_iteratedFDeriv hDo.uniqueDiffOn |
| ((hf.contDiffAt (hDo.mem_nhds hw)).of_le (WithTop.coe_le_coe.mpr le_top)) hw).symm |
| exact (stateCompactTube_isCompact hg hU hS hSU M c).exists_bound_of_continuousOn |
| (hcont.mono (stateCompactTube_subset_domain hSU M hc)) |
| |
| theorem stateFirstCoefficient_uniform_derivative_bound {g : MetricField} {U S : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hS : IsCompact S) (hSU : S ⊆ U) |
| (M : ℝ) {c : ℝ} (hc : 0 < c) (i : Fin 6) (k : ℕ) : |
| ∃ C : ℝ, ∀ w ∈ stateCompactTube g S M c, |
| ‖iteratedFDeriv ℝ k (stateFirstCoefficient g i) w‖ ≤ C := |
| stateCompactTube_derivative_bound hg hU hS hSU M hc |
| (stateFirstCoefficient_contDiffOn hg hU i) k |
| |
| theorem stateSegment_coord_bound {w0 w1 : DarbouxState} {sigma M : ℝ} |
| (hs : sigma ∈ Icc (0 : ℝ) 1) |
| (h0 : ∀ i, |w0 i| ≤ M) (h1 : ∀ i, |w1 i| ≤ M) (i : Fin 6) : |
| |stateSegment w0 w1 sigma i| ≤ M := by |
| change |(1 - sigma) * w0 i + sigma * w1 i| ≤ M |
| calc |
| _ ≤ |(1 - sigma) * w0 i| + |sigma * w1 i| := abs_add_le _ _ |
| _ = (1 - sigma) * |w0 i| + sigma * |w1 i| := by |
| rw [abs_mul, abs_mul, abs_of_nonneg (sub_nonneg.mpr hs.2), abs_of_nonneg hs.1] |
| _ ≤ (1 - sigma) * M + sigma * M := |
| add_le_add (mul_le_mul_of_nonneg_left (h0 i) (sub_nonneg.mpr hs.2)) |
| (mul_le_mul_of_nonneg_left (h1 i) hs.1) |
| _ = M := by ring |
| |
| theorem heightJetSegment_mem_compactTube (g : MetricField) (z0 z : Coord → ℝ) |
| {S : Set Coord} {p : Coord} (hp : p ∈ S) {sigma M c : ℝ} |
| (hs : sigma ∈ Icc (0 : ℝ) 1) |
| (h0 : ∀ i, |solutionJet z0 p i| ≤ M) (h1 : ∀ i, |solutionJet z p i| ≤ M) |
| (hden : c ≤ |stateDenominator g (heightJetSegment z0 z sigma p)|) : |
| heightJetSegment z0 z sigma p ∈ stateCompactTube g S M c := by |
| refine ⟨⟨?_, ?_⟩, hden⟩ |
| · simpa only [Set.mem_preimage, statePoint_heightJetSegment] using hp |
| · constructor <;> intro i |
| · exact (abs_le.mp (stateSegment_coord_bound hs h0 h1 i)).1 |
| · exact (abs_le.mp (stateSegment_coord_bound hs h0 h1 i)).2 |
|
|
| end SmoothLocal.HighEquation |
|
|
| end |
|
|
| end OAI |
|
|
