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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | 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
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