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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 | import OAI.Geometry.IsometricImmersion.Metrics.MetricQSpatialBundle
namespace OAI
noncomputable section
open Set
open scoped ContDiff
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry
theorem horizontalJet_coordinate_word_prefix (f : Coord → ℝ) (ds : List (Fin 2)) (n : ℕ) :
horizontalJet (iteratedCoordPartial ds f) n =
iteratedCoordPartial (List.replicate n 0 ++ ds) f := by
induction n with
| zero => rfl
| succ n hn =>
simp only [horizontalJet, List.replicate_succ, List.cons_append, iteratedCoordPartial, hn]
theorem coordinateMetricBound_to_horizontalTwoJet
{g : MetricField} {S : Set Coord} {N : ℕ} {G : ℝ}
(hgB : ∀ i j, CoordinateBound (fun p => g p i j) S (N + 2) G)
{p : Coord} (hp : p ∈ S) : HorizontalMetricTwoJetBound g p N G := by
have hb (i j : Fin 2) (ds : List (Fin 2)) (hds : ds.length ≤ 2) (n : ℕ) (hn : n ≤ N) :
|horizontalJet (iteratedCoordPartial ds (fun x => g x i j)) n p| ≤ G := by
rw [horizontalJet_coordinate_word_prefix]
apply hgB i j _ _ p hp
simp only [List.length_append, List.length_replicate]
omega
refine ⟨?_, ?_, ?_⟩
· intro i j n hn
exact hb i j [] (by simp) n hn
· intro d i j n hn
exact hb i j [d] (by simp) n hn
· intro d e i j n hn
exact hb i j [d, e] (by simp) n hn
end SmoothLocal.HighEquation
end
end OAI
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