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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 | import OAI.Geometry.IsometricImmersion.Coordinates.BoundedPatchSlabGeometry
import OAI.Geometry.IsometricImmersion.Comparison.ShearFirstJetComparison
import OAI.Geometry.IsometricImmersion.Darboux.QReferenceInitialTolerance
namespace OAI
noncomputable section
open Set
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.HighEquation SmoothLocal.Flow SmoothLocal.Perturbation
theorem actual_first_input_norm_le {g : MetricField} {p : Coord} {G : ℝ} (hG : 0 ≤ G)
(hvalue : ∀ i j, |g p i j| ≤ G)
(hpartial : ∀ a i j, |coordPartial a (fun q => g q i j) p| ≤ G) :
‖actualCurvatureFirstInput g p‖ ≤ G := by
change max ‖(fun i j => g p i j)‖
‖(fun a i j => coordPartial a (fun q => g q i j) p)‖ ≤ G
apply max_le
· apply (pi_norm_le_iff_of_nonneg hG).mpr
intro i
apply (pi_norm_le_iff_of_nonneg hG).mpr
intro j
exact hvalue i j
· apply (pi_norm_le_iff_of_nonneg hG).mpr
intro a
apply (pi_norm_le_iff_of_nonneg hG).mpr
intro i
apply (pi_norm_le_iff_of_nonneg hG).mpr
intro j
exact hpartial a i j
theorem sheared_first_input_norm_le_of_original_jets
{g : MetricField} {U : Set Coord} {p : Coord} {q0 G : ℝ}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : inverseShearCoordinates q0 p ∈ U)
(hG : 0 ≤ G) (hq0 : |q0| ≤ 1)
(hgB : ∀ i j k, k ≤ 1 →
‖iteratedFDeriv ℝ k (fun a => g a i j) (inverseShearCoordinates q0 p)‖ ≤ G) :
‖actualCurvatureFirstInput (metricInShearCoordinates g q0) p‖ ≤ 8 * G := by
have hf : ‖actualCurvatureFirstInput g (inverseShearCoordinates q0 p)‖ ≤ G := by
apply actual_first_input_norm_le hG
· intro i j
simpa only [norm_iteratedFDeriv_zero, Real.norm_eq_abs] using hgB i j 0 (by omega)
· intro a i j
exact (norm_iteratedCoordPartial_le_jet (hg.1 i j) hU [a] hp).trans
(hgB i j 1 (by omega))
have hp' : affineCoordinates 0 (inverseShearMatrix q0) p ∈ U := by
simpa only [← inverseShearCoordinates_eq_affine] using hp
unfold metricInShearCoordinates
rw [actualCurvatureFirstInput_affinePullback hg hU 0 (inverseShearMatrix q0) p hp']
exact affineFirstJetMap_norm_le_eight _ (inverseShearMatrix_entry_le_one hq0) _ hG
(by simpa only [← inverseShearCoordinates_eq_affine] using hf)
def boundedClassShearedStateBudget (q0 : ℝ) (M : ℕ) : ℝ :=
max 1 ((2 * (1 + |q0|))^2 * (M : ℝ))
theorem boundedClassShearedStateBudget_pos (q0 : ℝ) (M : ℕ) :
0 < boundedClassShearedStateBudget q0 M :=
zero_lt_one.trans_le (le_max_left _ _)
theorem bounded_class_sheared_solution_state_bound
{g : MetricField} {z : Coord → ℝ} {M : ℕ} (h : BoundedAdmissibleHeight g M z)
(q0 : ℝ) {p : Coord} (hp : inverseShearCoordinates q0 p ∈ SmoothLocal.Flow.modelSquare)
(hcoords : |p 0| ≤ 1 ∧ |p 1| ≤ 1) :
‖qSolutionJet (heightInShearCoordinates z q0) p‖ ≤ boundedClassShearedStateBudget q0 M := by
obtain ⟨_, hadm, hjet, _, _⟩ := h
obtain ⟨U, hU, hSU, _, hz, _, _, _, _⟩ := hadm
have hzB : CoordinateBound z SmoothLocal.Flow.modelSquare 2 (M : ℝ) := by
intro ds hds p hp
exact (norm_iteratedCoordPartial_le_jet hz hU ds (hSU hp)).trans
(hjet ds.length (by omega) p hp)
have hshear := SmoothLocal.Pulse.CoordinateBound.inverse_shear hzB hz hU hSU (Nat.cast_nonneg M) q0
have hstate : ∀ i : Fin 6, |qSolutionJet (heightInShearCoordinates z q0) p i| ≤
boundedClassShearedStateBudget q0 M := by
intro i
fin_cases i
· exact hcoords.1.trans (le_max_left _ _)
· exact hcoords.2.trans (le_max_left _ _)
· exact (hshear [0] (by norm_num) p hp).trans (le_max_right _ _)
· exact (hshear [1] (by norm_num) p hp).trans (le_max_right _ _)
· exact (hshear [0, 1] (by norm_num) p hp).trans (le_max_right _ _)
· exact (hshear [0, 0] (by norm_num) p hp).trans (le_max_right _ _)
exact (pi_norm_le_iff_of_nonneg (boundedClassShearedStateBudget_pos q0 M).le).mpr hstate
end SmoothLocal.Pulse
end
end OAI
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