| import OAI.Geometry.IsometricImmersion.Darboux.QMetricDensityDecomposition |
| import OAI.Geometry.IsometricImmersion.Calculus.SmoothMetricJet |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.HighEquation |
|
|
| abbrev QFirstInput := CurvatureFirstInput × DarbouxState |
|
|
| def qFirstDet (a : QFirstInput) : ℝ := (Matrix.of a.1.1).det |
| def qFirstConnection (a : QFirstInput) (i j : Fin 2) : ℝ := |
| ∑ k, christoffelJet (Matrix.of a.1.1) a.1.2 k i j * stateGradient a.2 k |
| def qFirstXX (a : QFirstInput) : ℝ := a.2 5-qFirstConnection a 0 0 |
| def qFirstMixed (a : QFirstInput) : ℝ := a.2 4-qFirstConnection a 0 1 |
| def qFirstEnergy (a : QFirstInput) : ℝ := |
| qFirstDet a-(a.1.1 1 1*(stateGradient a.2 0)^2- |
| (a.1.1 0 1+a.1.1 1 0)*stateGradient a.2 0*stateGradient a.2 1+ |
| a.1.1 0 0*(stateGradient a.2 1)^2) |
| def qFirstDensityFactor (a : QFirstInput) : ℝ := |
| qFirstEnergy a/(qFirstDet a*qFirstXX a) |
| def qFirstLower (a : QFirstInput) : ℝ := |
| qFirstConnection a 1 1+(qFirstMixed a)^2/qFirstXX a |
|
|
| def qFirstBundle (g : MetricField) (w : DarbouxState) : QFirstInput := |
| (actualCurvatureFirstInput g (statePoint w),w) |
|
|
| theorem qFirstDet_bundle (g : MetricField) (w : DarbouxState) : |
| qFirstDet (qFirstBundle g w) = (g (statePoint w)).det := rfl |
| theorem qFirstConnection_bundle (g : MetricField) (w : DarbouxState) (i j : Fin 2) : |
| qFirstConnection (qFirstBundle g w) i j = stateConnection g i j w := rfl |
| theorem qFirstXX_bundle (g : MetricField) (w : DarbouxState) : |
| qFirstXX (qFirstBundle g w) = stateQDenominator g w := rfl |
| theorem qFirstDensityFactor_bundle (g : MetricField) (w : DarbouxState) : |
| qFirstDensityFactor (qFirstBundle g w) = qMetricDensityFactor g w := rfl |
| theorem qFirstLower_bundle (g : MetricField) (w : DarbouxState) : |
| qFirstLower (qFirstBundle g w) = qMetricFirstLower g w := rfl |
|
|
| theorem qFirstBundle_distance (g h : MetricField) (w : DarbouxState) : |
| ‖qFirstBundle g w-qFirstBundle h w‖ = |
| ‖actualCurvatureFirstInput g (statePoint w)-actualCurvatureFirstInput h (statePoint w)‖ := by |
| simp [qFirstBundle,Prod.norm_def] |
|
|
| def qFirstBase : Set QFirstInput := {a | qFirstDet a ≠ 0} |
| def qFirstDomain : Set QFirstInput := {a | qFirstDet a ≠ 0 ∧ qFirstXX a ≠ 0} |
| def qFirstBaseTube (M d : ℝ) : Set QFirstInput := |
| Metric.closedBall 0 M ∩ qFirstDet ⁻¹' Ici d |
| def qFirstTube (M d c : ℝ) : Set QFirstInput := |
| qFirstBaseTube M d ∩ qFirstXX ⁻¹' {v : ℝ | c ≤ |v|} |
|
|
| theorem qFirstDet_contDiff : ContDiff ℝ ∞ qFirstDet := by |
| unfold qFirstDet |
| simp only [Matrix.det_fin_two, Matrix.of_apply] |
| fun_prop |
|
|
| theorem qFirstBase_isOpen : IsOpen qFirstBase := |
| isOpen_ne_fun qFirstDet_contDiff.continuous continuous_const |
|
|
| theorem qFirstConnection_contDiffAt {a : QFirstInput} (ha : a ∈ qFirstBase) |
| (i j : Fin 2) : ContDiffAt ℝ ∞ (fun b => qFirstConnection b i j) a := by |
| have hΓ (k : Fin 2) : ContDiffAt ℝ ∞ |
| (fun b : QFirstInput => christoffelJet (Matrix.of b.1.1) b.1.2 k i j) a := |
| christoffelJet_contDiffAt (G := fun b : QFirstInput => Matrix.of b.1.1) |
| (D := fun b : QFirstInput => b.1.2) (fun row column => by |
| simp only [Matrix.of_apply] |
| fun_prop) |
| (fun d i j => by fun_prop) ha k i j |
| have hv (k : Fin 2) : ContDiffAt ℝ ∞ |
| (fun b : QFirstInput => stateGradient b.2 k) a := |
| ((contDiff_apply ℝ ℝ k).comp (stateGradient_contDiff.comp (by fun_prop))).contDiffAt |
| simpa only [qFirstConnection,Fin.sum_univ_two] using |
| ((hΓ 0).mul (hv 0)).add ((hΓ 1).mul (hv 1)) |
|
|
| theorem qFirstXX_contDiffOn : ContDiffOn ℝ ∞ qFirstXX qFirstBase := by |
| intro a ha |
| exact ((show ContDiffAt ℝ ∞ (fun b : QFirstInput => b.2 5) a by fun_prop).sub |
| (qFirstConnection_contDiffAt ha 0 0)).contDiffWithinAt |
|
|
| theorem qFirstDomain_isOpen : IsOpen qFirstDomain := |
| qFirstXX_contDiffOn.continuousOn.isOpen_inter_preimage qFirstBase_isOpen |
| isClosed_singleton.isOpen_compl |
|
|
| theorem qFirstEnergy_contDiff : ContDiff ℝ ∞ qFirstEnergy := by |
| unfold qFirstEnergy |
| have hv (k : Fin 2) : ContDiff ℝ ∞ (fun a : QFirstInput => stateGradient a.2 k) := |
| (contDiff_apply ℝ ℝ k).comp (stateGradient_contDiff.comp (by fun_prop)) |
| exact qFirstDet_contDiff.sub |
| ((((show ContDiff ℝ ∞ (fun a : QFirstInput => a.1.1 1 1) by fun_prop).mul |
| ((hv 0).pow 2)).sub |
| ((((show ContDiff ℝ ∞ (fun a : QFirstInput => a.1.1 0 1) by fun_prop).add |
| (show ContDiff ℝ ∞ (fun a : QFirstInput => a.1.1 1 0) by fun_prop)).mul |
| (hv 0)).mul (hv 1))).add |
| ((show ContDiff ℝ ∞ (fun a : QFirstInput => a.1.1 0 0) by fun_prop).mul |
| ((hv 1).pow 2))) |
|
|
| theorem qFirstDensityFactor_contDiffOn : |
| ContDiffOn ℝ ∞ qFirstDensityFactor qFirstDomain := by |
| intro a ha |
| have hx := qFirstXX_contDiffOn.contDiffAt (qFirstBase_isOpen.mem_nhds ha.1) |
| exact (qFirstEnergy_contDiff.contDiffAt.div |
| (qFirstDet_contDiff.contDiffAt.mul hx) (mul_ne_zero ha.1 ha.2)).contDiffWithinAt |
|
|
| theorem qFirstLower_contDiffOn : ContDiffOn ℝ ∞ qFirstLower qFirstDomain := by |
| intro a ha |
| have hΓ := qFirstConnection_contDiffAt ha.1 |
| have hx := qFirstXX_contDiffOn.contDiffAt (qFirstBase_isOpen.mem_nhds ha.1) |
| have hm : ContDiffAt ℝ ∞ qFirstMixed a := |
| (show ContDiffAt ℝ ∞ (fun b : QFirstInput => b.2 4) a by fun_prop).sub (hΓ 0 1) |
| exact ((hΓ 1 1).add ((hm.pow 2).div hx ha.2)).contDiffWithinAt |
|
|
| theorem qFirstBaseTube_isCompact (M d : ℝ) : IsCompact (qFirstBaseTube M d) := |
| (isCompact_closedBall (0 : QFirstInput) M).inter_right |
| (isClosed_Ici.preimage qFirstDet_contDiff.continuous) |
|
|
| theorem qFirstBaseTube_subset_base (M : ℝ) {d : ℝ} (hd : 0 < d) : |
| qFirstBaseTube M d ⊆ qFirstBase := by |
| intro a ha |
| exact ne_of_gt (hd.trans_le ha.2) |
|
|
| theorem qFirstTube_isCompact (M : ℝ) {d : ℝ} (hd : 0 < d) (c : ℝ) : |
| IsCompact (qFirstTube M d c) := by |
| have hbase := qFirstBaseTube_isCompact M d |
| have hc : ContinuousOn qFirstXX (qFirstBaseTube M d) := |
| qFirstXX_contDiffOn.continuousOn.mono (qFirstBaseTube_subset_base M hd) |
| have hclosed : IsClosed (qFirstTube M d c) := |
| hc.preimage_isClosed_of_isClosed hbase.isClosed (isClosed_le continuous_const continuous_abs) |
| exact hbase.of_isClosed_subset hclosed inter_subset_left |
|
|
| theorem qFirstTube_subset_domain (M : ℝ) {d c : ℝ} (hd : 0 < d) (hc : 0 < c) : |
| qFirstTube M d c ⊆ qFirstDomain := by |
| intro a ha |
| refine ⟨qFirstBaseTube_subset_base M hd ha.1,?_⟩ |
| intro hz |
| have hh : c ≤ |qFirstXX a| := ha.2 |
| rw [hz,abs_zero] at hh |
| linarith |
|
|
| theorem qFirstXX_fderiv_bound (M : ℝ) {d : ℝ} (hd : 0 < d) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ a ∈ qFirstBaseTube M d, ‖fderiv ℝ qFirstXX a‖ ≤ C := by |
| have hcont : ContinuousOn (fderiv ℝ qFirstXX) qFirstBase := |
| (qFirstXX_contDiffOn.fderiv_of_isOpen (m := 0) qFirstBase_isOpen (by simp)).continuousOn |
| obtain ⟨C,hC⟩ := (qFirstBaseTube_isCompact M d).exists_bound_of_continuousOn |
| (hcont.mono (qFirstBaseTube_subset_base M hd)) |
| exact ⟨max C 0,le_max_right _ _,fun a ha => (hC a ha).trans (le_max_left _ _)⟩ |
|
|
| theorem qFirstTube_fderiv_bound (f : QFirstInput → ℝ) |
| (hf : ContDiffOn ℝ ∞ f qFirstDomain) (M : ℝ) {d c : ℝ} |
| (hd : 0 < d) (hc : 0 < c) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ a ∈ qFirstTube M d c, ‖fderiv ℝ f a‖ ≤ C := by |
| have hcont : ContinuousOn (fderiv ℝ f) qFirstDomain := |
| (hf.fderiv_of_isOpen (m := 0) qFirstDomain_isOpen (by simp)).continuousOn |
| obtain ⟨C,hC⟩ := (qFirstTube_isCompact M hd c).exists_bound_of_continuousOn |
| (hcont.mono (qFirstTube_subset_domain M hd hc)) |
| exact ⟨max C 0,le_max_right _ _,fun a ha => (hC a ha).trans (le_max_left _ _)⟩ |
|
|
| end SmoothLocal.Pulse |
|
|
| end |
|
|
| end OAI |
|
|
