openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Metrics /MetricHorizontalRequests.lean
| 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 | |