| import OAI.Geometry.IsometricImmersion.Darboux.QResidualActive |
| import OAI.Geometry.IsometricImmersion.Metrics.MetricPBundleBounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology BigOperators Matrix |
|
|
| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic |
|
|
| abbrev QSpatialState := Fin 5 → ℝ |
|
|
| def qSpatialAxis : Fin 5 → Fin 6 := ![0, 2, 3, 4, 5] |
| def qSpatialIndex : Fin 6 → Fin 5 := ![0, 0, 1, 2, 3, 4] |
|
|
| def qSpatialLinear : QSpatialState →ₗ[ℝ] DarbouxState where |
| toFun v := ![v 0, 0, v 1, v 2, v 3, v 4] |
| map_add' v w := by ext i; fin_cases i <;> simp |
| map_smul' c v := by ext i; fin_cases i <;> simp |
|
|
| def qSpatialCLM : QSpatialState →L[ℝ] DarbouxState := |
| LinearMap.toContinuousLinearMap qSpatialLinear |
|
|
| def qFixedTimeState (theta : ℝ) (v : QSpatialState) : DarbouxState := |
| qSpatialCLM v + Pi.single 1 theta |
|
|
| def qSpatialProjection (w : DarbouxState) : QSpatialState := ![w 0, w 2, w 3, w 4, w 5] |
|
|
| def qFixedTimeQ (g : MetricField) (theta : ℝ) : QSpatialState → ℝ := |
| sixVariableQ g ∘ qFixedTimeState theta |
|
|
| def qFixedTimeDomain (g : MetricField) (U : Set Coord) (theta : ℝ) : Set QSpatialState := |
| qFixedTimeState theta ⁻¹' darbouxQStateDomain g U |
|
|
| theorem qSpatialAxis_index {i : Fin 6} (hi : i ≠ 1) : qSpatialAxis (qSpatialIndex i) = i := by |
| fin_cases i <;> first | rfl | exact (hi rfl).elim |
|
|
| theorem qSpatialCLM_single (i : Fin 5) : |
| qSpatialCLM (Pi.single i 1) = Pi.single (qSpatialAxis i) 1 := by |
| ext j |
| fin_cases i <;> fin_cases j <;> |
| simp [qSpatialCLM, qSpatialLinear, qSpatialAxis] |
|
|
| theorem qFixedTimeState_projection (w : DarbouxState) : |
| qFixedTimeState (w 1) (qSpatialProjection w) = w := by |
| ext i |
| fin_cases i <;> simp [qFixedTimeState, qSpatialProjection, qSpatialCLM, |
| qSpatialLinear] |
|
|
| theorem qFixedTimeState_contDiff (theta : ℝ) : ContDiff ℝ ∞ (qFixedTimeState theta) := |
| qSpatialCLM.contDiff.add contDiff_const |
|
|
| theorem qFixedTimeDomain_isOpen {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (theta : ℝ) : |
| IsOpen (qFixedTimeDomain g U theta) := |
| (darbouxQStateDomain_isOpen hg hU).preimage (qFixedTimeState_contDiff theta).continuous |
|
|
| theorem qFixedTimeQ_contDiffOn {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (theta : ℝ) : |
| ContDiffOn ℝ ∞ (qFixedTimeQ g theta) (qFixedTimeDomain g U theta) := |
| (sixVariableQ_contDiffOn hg hU).comp (qFixedTimeState_contDiff theta).contDiffOn |
| (fun _ hv => hv) |
|
|
| theorem qFixedTimeQ_iteratedFDeriv |
| {g : MetricField} {U : Set Coord} {theta : ℝ} {v : QSpatialState} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (hv : v ∈ qFixedTimeDomain g U theta) (n : ℕ) : |
| iteratedFDeriv ℝ n (qFixedTimeQ g theta) v = |
| (iteratedFDeriv ℝ n (sixVariableQ g) (qFixedTimeState theta v)).compContinuousLinearMap |
| (fun _ => qSpatialCLM) := by |
| let c : DarbouxState := Pi.single 1 theta |
| let O : Set DarbouxState := (fun w : DarbouxState => w + c) ⁻¹' darbouxQStateDomain g U |
| have hO : IsOpen O := (darbouxQStateDomain_isOpen hg hU).preimage (by fun_prop) |
| have hF : ContDiffOn ℝ ∞ (fun w : DarbouxState => sixVariableQ g (w + c)) O := |
| (sixVariableQ_contDiffOn hg hU).comp (by fun_prop) (fun _ hw => hw) |
| have hpre : IsOpen (qSpatialCLM ⁻¹' O) := hO.preimage qSpatialCLM.continuous |
| have hp : qSpatialCLM v ∈ O := hv |
| have he := qSpatialCLM.iteratedFDerivWithin_comp_right hF hO.uniqueDiffOn hpre.uniqueDiffOn hp |
| (i := n) (WithTop.coe_le_coe.mpr le_top) |
| rw [iteratedFDerivWithin_of_isOpen n hpre hp, |
| iteratedFDerivWithin_of_isOpen n hO hp, |
| iteratedFDeriv_comp_add_right n c (qSpatialCLM v)] at he |
| exact he |
| |
| theorem qCoordinateChainCoefficient_eq_fixedTime |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U) |
| (hxx : covHessian g z p 0 0 ≠ 0) (w : ChainWord) (r : Fin w.arity → Fin 6) |
| (hactive : QActiveComponents w r) : |
| qCoordinateChainCoefficient g z w r p = |
| iteratedFDeriv ℝ w.arity (qFixedTimeQ g (p 1)) (qSpatialProjection (qSolutionJet z p)) |
| (fun i => Pi.single (qSpatialIndex (r i)) 1) := by |
| have he : qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) = qSolutionJet z p := |
| qFixedTimeState_projection (qSolutionJet z p) |
| have hv : qSpatialProjection (qSolutionJet z p) ∈ qFixedTimeDomain g U (p 1) := by |
| change qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) ∈ darbouxQStateDomain g U |
| rw [he] |
| exact qSolutionJet_mem_domain hp hxx |
| rw [qFixedTimeQ_iteratedFDeriv hg hU hv w.arity, |
| ContinuousMultilinearMap.compContinuousLinearMap_apply, he] |
| unfold qCoordinateChainCoefficient |
| congr 1 |
| funext i |
| rw [qSpatialCLM_single, qSpatialAxis_index (hactive i).1] |
|
|
| theorem qCoordinateChainCoefficient_abs_le_fixedTime_derivative |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U) |
| (hxx : covHessian g z p 0 0 ≠ 0) (w : ChainWord) (r : Fin w.arity → Fin 6) |
| (hactive : QActiveComponents w r) : |
| |qCoordinateChainCoefficient g z w r p| ≤ |
| ‖iteratedFDeriv ℝ w.arity (qFixedTimeQ g (p 1)) (qSpatialProjection (qSolutionJet z p))‖ := by |
| rw [qCoordinateChainCoefficient_eq_fixedTime hg hU hp hxx w r hactive, ← Real.norm_eq_abs] |
| have hb := (iteratedFDeriv ℝ w.arity (qFixedTimeQ g (p 1)) |
| (qSpatialProjection (qSolutionJet z p))).le_opNorm |
| (fun i => Pi.single (qSpatialIndex (r i)) (1 : ℝ)) |
| simpa only [Pi.norm_single, norm_one, Finset.prod_const_one, mul_one] using hb |
| |
| end SmoothLocal.HighEquation |
| |
| end |
| |
| end OAI |
| |
