| 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 |
|
|
