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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | import OAI.Geometry.IsometricImmersion.Metrics.UniversalMetricQ
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic
def HorizontalMetricTwoJetBound (g : MetricField) (p : Coord) (N : ℕ) (G : ℝ) : Prop :=
(∀ i j n, n ≤ N → |horizontalJet (fun x => g x i j) n p| ≤ G) ∧
(∀ d i j n, n ≤ N → |horizontalJet (coordPartial d (fun x => g x i j)) n p| ≤ G) ∧
(∀ d e i j n, n ≤ N →
|horizontalJet (coordPartial d (coordPartial e (fun x => g x i j))) n p| ≤ G)
def qFixedTimeBaseDomain (U : Set Coord) (theta : ℝ) : Set QSpatialState :=
{v | coordinatePoint (v 0) theta ∈ U}
def metricQSpatialBundle (g : MetricField) (theta : ℝ) : QSpatialState → MetricPInput :=
metricPBundle g ∘ qFixedTimeState theta
theorem statePoint_qFixedTimeState (theta : ℝ) (v : QSpatialState) :
statePoint (qFixedTimeState theta v) = coordinatePoint (v 0) theta := by
ext i
fin_cases i <;> simp [statePoint, qFixedTimeState, qSpatialCLM, qSpatialLinear,
coordinatePoint]
theorem qFixedTimeBaseDomain_isOpen {U : Set Coord} (hU : IsOpen U) (theta : ℝ) :
IsOpen (qFixedTimeBaseDomain U theta) :=
hU.preimage (by unfold coordinatePoint; fun_prop)
theorem metricQSpatialBundle_contDiffOn {g : MetricField} {U : Set Coord}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (theta : ℝ) :
ContDiffOn ℝ ∞ (metricQSpatialBundle g theta) (qFixedTimeBaseDomain U theta) := by
apply (metricPBundle_contDiffOn hg hU).comp (qFixedTimeState_contDiff theta).contDiffOn
intro v hv
change statePoint (qFixedTimeState theta v) ∈ U
simpa only [statePoint_qFixedTimeState, qFixedTimeBaseDomain, Set.mem_ofPred_eq] using hv
theorem norm_iteratedFDeriv_fixedTime_coordinate
{f : Coord → ℝ} {U : Set Coord} (hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U)
{theta : ℝ} {v : QSpatialState} (hp : coordinatePoint (v 0) theta ∈ U) (n : ℕ) :
‖iteratedFDeriv ℝ n (fun w : QSpatialState => f (coordinatePoint (w 0) theta)) v‖ ≤
|horizontalJet f n (coordinatePoint (v 0) theta)| := by
let I : Set ℝ := {x | coordinatePoint x theta ∈ U}
let L : QSpatialState →L[ℝ] ℝ := ContinuousLinearMap.proj 0
have hI : IsOpen I := hU.preimage (horizontalPoint_contDiff theta).continuous
have hs : ContDiffOn ℝ ∞ (fun x : ℝ => f (coordinatePoint x theta)) I :=
hf.comp (horizontalPoint_contDiff theta).contDiffOn (fun x hx => hx)
have hL : ‖L‖ ≤ 1 := by
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro w
simpa only [one_mul, L, ContinuousLinearMap.proj_apply] using norm_le_pi_norm w 0
have h := norm_iteratedFDeriv_comp_contraction L hs hI hL hp n
rw [norm_iteratedFDeriv_eq_norm_iteratedDeriv] at h
change ‖iteratedFDeriv ℝ n (fun w : QSpatialState => f (coordinatePoint (w 0) theta)) v‖ ≤
‖iteratedDeriv n (fun x => f (coordinatePoint x theta)) (v 0)‖ at h
rw [horizontalJet_slice hU hf theta n (v 0) hp, Real.norm_eq_abs] at h
exact h
theorem qSpatialCLM_norm_le : ‖qSpatialCLM‖ ≤ 1 := by
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro v
rw [one_mul]
apply (pi_norm_le_iff_of_nonneg (norm_nonneg v)).mpr
intro i
fin_cases i
· exact norm_le_pi_norm v 0
· simp [qSpatialCLM, qSpatialLinear]
· exact norm_le_pi_norm v 1
· exact norm_le_pi_norm v 2
· exact norm_le_pi_norm v 3
· exact norm_le_pi_norm v 4
theorem qFixedTimeState_positive_jet_bound (theta : ℝ) (v : QSpatialState)
{n : ℕ} (hn : 1 ≤ n) : ‖iteratedFDeriv ℝ n (qFixedTimeState theta) v‖ ≤ 1 := by
cases n with
| zero => omega
| succ n =>
rw [← norm_iteratedFDeriv_fderiv]
have he : fderiv ℝ (qFixedTimeState theta) = (fun _ : QSpatialState => qSpatialCLM) := by
funext w
exact (qSpatialCLM.hasFDerivAt.add_const (Pi.single 1 theta)).fderiv
rw [he]
cases n with
| zero => simpa only [norm_iteratedFDeriv_zero] using qSpatialCLM_norm_le
| succ _ => simp only [iteratedFDeriv_succ_const]; norm_num
theorem metricQSpatialBundle_norm_bound
{g : MetricField} {theta G W : ℝ} {v : QSpatialState} {N : ℕ}
(hG : 0 ≤ G) (hgB : HorizontalMetricTwoJetBound g (coordinatePoint (v 0) theta) N G)
(hwB : ‖qFixedTimeState theta v‖ ≤ W) :
‖metricQSpatialBundle g theta v‖ ≤ max G W := by
have h0 : ‖g (coordinatePoint (v 0) theta)‖ ≤ G :=
(pi_norm_le_iff_of_nonneg hG).mpr (fun i =>
(pi_norm_le_iff_of_nonneg hG).mpr (fun j => hgB.1 i j 0 (by omega)))
have h1 : ‖(fun d i j => coordPartial d (fun p => g p i j) (coordinatePoint (v 0) theta))‖ ≤ G :=
(pi_norm_le_iff_of_nonneg hG).mpr (fun d =>
(pi_norm_le_iff_of_nonneg hG).mpr (fun i =>
(pi_norm_le_iff_of_nonneg hG).mpr (fun j => hgB.2.1 d i j 0 (by omega))))
have h2 : ‖(fun d e i j => coordPartial d (coordPartial e (fun p => g p i j))
(coordinatePoint (v 0) theta))‖ ≤ G :=
(pi_norm_le_iff_of_nonneg hG).mpr (fun d =>
(pi_norm_le_iff_of_nonneg hG).mpr (fun e =>
(pi_norm_le_iff_of_nonneg hG).mpr (fun i =>
(pi_norm_le_iff_of_nonneg hG).mpr (fun j => hgB.2.2 d e i j 0 (by omega)))))
unfold metricQSpatialBundle metricPBundle
simp only [Function.comp_apply, statePoint_qFixedTimeState, Prod.norm_def]
exact max_le (h0.trans (le_max_left _ _))
(max_le (h1.trans (le_max_left _ _))
(max_le (h2.trans (le_max_left _ _)) (hwB.trans (le_max_right _ _))))
theorem metricQSpatialBundle_positive_jet_bound
{g : MetricField} {U : Set Coord} {theta G : ℝ} {v : QSpatialState} {N : ℕ}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : coordinatePoint (v 0) theta ∈ U)
(hG : 0 ≤ G) (hgB : HorizontalMetricTwoJetBound g (coordinatePoint (v 0) theta) N G)
{n : ℕ} (hn : 1 ≤ n) (hnN : n ≤ N) :
‖iteratedFDeriv ℝ n (metricQSpatialBundle g theta) v‖ ≤ max 1 G := by
have hB := (metricQSpatialBundle_contDiffOn hg hU theta).contDiffAt
((qFixedTimeBaseDomain_isOpen hU theta).mem_nhds hp)
have h0 := hB.fst
have h1 := hB.snd.fst
have h2 := hB.snd.snd.fst
have hId := hB.snd.snd.snd
change ContDiffAt ℝ ∞ (fun w : QSpatialState => (metricQSpatialBundle g theta w).1) v at h0
change ContDiffAt ℝ ∞ (fun w : QSpatialState => (metricQSpatialBundle g theta w).2.1) v at h1
change ContDiffAt ℝ ∞ (fun w : QSpatialState => (metricQSpatialBundle g theta w).2.2.1) v at h2
have he0 : (fun w : QSpatialState => (metricQSpatialBundle g theta w).1) =
(fun w => g (coordinatePoint (w 0) theta)) := by
funext w
exact congrArg g (statePoint_qFixedTimeState theta w)
have he1 : (fun w : QSpatialState => (metricQSpatialBundle g theta w).2.1) =
(fun w d i j => coordPartial d (fun p => g p i j) (coordinatePoint (w 0) theta)) := by
funext w d i j
exact congrArg (coordPartial d (fun p => g p i j)) (statePoint_qFixedTimeState theta w)
have he2 : (fun w : QSpatialState => (metricQSpatialBundle g theta w).2.2.1) =
(fun w d e i j => coordPartial d (coordPartial e (fun p => g p i j)) (coordinatePoint (w 0) theta)) := by
funext w d e i j
exact congrArg (coordPartial d (coordPartial e (fun p => g p i j))) (statePoint_qFixedTimeState theta w)
rw [he0] at h0
rw [he1] at h1
rw [he2] at h2
have hb0 : ‖iteratedFDeriv ℝ n (fun w : QSpatialState => g (coordinatePoint (w 0) theta)) v‖ ≤ G := by
apply norm_iteratedFDeriv_pi_le h0 n hG
intro i
apply norm_iteratedFDeriv_pi_le ((contDiffAt_pi.mp h0) i) n hG
intro j
exact (norm_iteratedFDeriv_fixedTime_coordinate (hg.1 i j) hU hp n).trans (hgB.1 i j n hnN)
have hb1 : ‖iteratedFDeriv ℝ n
(fun w : QSpatialState => fun d i j => coordPartial d (fun p => g p i j) (coordinatePoint (w 0) theta)) v‖ ≤ G := by
apply norm_iteratedFDeriv_pi_le (F := Matrix (Fin 2) (Fin 2) ℝ) h1 n hG
intro d
have h1d := (contDiffAt_pi (F' := fun _ : Fin 2 => Matrix (Fin 2) (Fin 2) ℝ)).mp h1 d
apply norm_iteratedFDeriv_pi_le h1d n hG
intro i
apply norm_iteratedFDeriv_pi_le ((contDiffAt_pi.mp h1d) i) n hG
intro j
exact (norm_iteratedFDeriv_fixedTime_coordinate (partial_contDiffOn (hg.1 i j) hU d) hU hp n).trans
(hgB.2.1 d i j n hnN)
have hb2 : ‖iteratedFDeriv ℝ n
(fun w : QSpatialState => fun d e i j => coordPartial d (coordPartial e (fun p => g p i j))
(coordinatePoint (w 0) theta)) v‖ ≤ G := by
apply norm_iteratedFDeriv_pi_le (F := Fin 2 → Matrix (Fin 2) (Fin 2) ℝ) h2 n hG
intro d
have h2d := (contDiffAt_pi (F' := fun _ : Fin 2 => Fin 2 → Matrix (Fin 2) (Fin 2) ℝ)).mp h2 d
apply norm_iteratedFDeriv_pi_le (F := Matrix (Fin 2) (Fin 2) ℝ) h2d n hG
intro e
have h2de := (contDiffAt_pi (F' := fun _ : Fin 2 => Matrix (Fin 2) (Fin 2) ℝ)).mp h2d e
apply norm_iteratedFDeriv_pi_le h2de n hG
intro i
apply norm_iteratedFDeriv_pi_le ((contDiffAt_pi.mp h2de) i) n hG
intro j
exact (norm_iteratedFDeriv_fixedTime_coordinate
(partial_contDiffOn (partial_contDiffOn (hg.1 i j) hU e) hU d) hU hp n).trans
(hgB.2.2 d e i j n hnN)
have hb0' : ‖iteratedFDeriv ℝ n (fun w => (metricQSpatialBundle g theta w).1) v‖ ≤ G := by
rw [he0]
exact hb0
have hb1' : ‖iteratedFDeriv ℝ n (fun w => (metricQSpatialBundle g theta w).2.1) v‖ ≤ G := by
rw [he1]
exact hb1
have hb2' : ‖iteratedFDeriv ℝ n (fun w => (metricQSpatialBundle g theta w).2.2.1) v‖ ≤ G := by
rw [he2]
exact hb2
exact norm_iteratedFDeriv_prod_le hB.fst hB.snd n (hb0'.trans (le_max_right _ _))
(norm_iteratedFDeriv_prod_le hB.snd.fst hB.snd.snd n (hb1'.trans (le_max_right _ _))
(norm_iteratedFDeriv_prod_le hB.snd.snd.fst hId n (hb2'.trans (le_max_right _ _))
((qFixedTimeState_positive_jet_bound theta v hn).trans (le_max_left _ _))))
end SmoothLocal.HighEquation
end
end OAI
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