| import OAI.Geometry.IsometricImmersion.Metrics.LocalMetricPCoefficients | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function MeasureTheory | |
| open scoped ContDiff Topology BigOperators Matrix ENNReal NNReal | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE | |
| theorem exists_localized_actualHighRemainder_bound | |
| (G Z M : ℝ) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hM : 0 ≤ M) | |
| {d c : ℝ} (hd : 0 < d) (hc : 0 < c) (m : ℕ) (hm : 8 ≤ m + 3) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (g : MetricField) (z : Coord → ℝ) (U V : Set Coord), | |
| SmoothPositiveOn g U → IsOpen U → modelSquare ⊆ U → | |
| (∀ i j : Fin 2, ∀ k ≤ 4, ∀ p ∈ modelSquare, | |
| ‖iteratedFDeriv ℝ k (fun q => g q i j) p‖ ≤ G) → | |
| (∀ p ∈ modelSquare, d ≤ |(g p).det|) → | |
| ContDiffOn ℝ ∞ z U → | |
| (∀ k ≤ 8, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k z p‖ ≤ Z) → | |
| (∀ p ∈ modelSquare, c ≤ |covHessian g z p 1 1|) → | |
| MeasurableSet V → V ⊆ modelSquare → volume V < (⊤ : ℝ≥0∞) → | |
| (∀ i j k, k ≤ m + 5 → ∀ p ∈ V, | |
| ‖iteratedFDeriv ℝ k (fun q => g q i j) p‖ ≤ M) → | |
| ∀ B : ℝ, 1 ≤ B → ∀ H : ℝ≥0, | |
| (∀ n j, n + heightStateBaseOrder j ≤ m + 1 → | |
| ∀ p ∈ V, |heightStateFactor z n j p| ≤ B) → | |
| (∀ n j, n + heightStateBaseOrder j ≤ m + 3 → | |
| eLpNorm (heightStateFactor z n j) 2 (volume.restrict V) ≤ (H : ℝ≥0∞)) → | |
| eLpNorm (actualHighRemainder g z m) 2 (volume.restrict V) ≤ | |
| actualHighRemainderL2Budget C (heightPFirstBound G Z d c) B H V m ∧ | |
| actualHighRemainderL2Budget C (heightPFirstBound G Z d c) B H V m < (⊤ : ℝ≥0∞) ∧ | |
| MemLp (actualHighRemainder g z m) 2 (volume.restrict V) := by | |
| obtain ⟨C, hC, houter⟩ := exists_same_region_remainder_outer_bound M Z hM hd hc m | |
| refine ⟨C, hC, ?_⟩ | |
| intro g z U V hg hU hSU hgB hdet hz hzB hyy hV hVS hVfinite hgHigh B hB H hlow hheight | |
| have hgCoord (i j : Fin 2) : CoordinateBound (fun p => g p i j) modelSquare 4 G := | |
| coordinateBound_of_frechet_bounds (hg.1 i j) hU hSU (hgB i j) | |
| have hzCoord : CoordinateBound z modelSquare 5 Z := | |
| coordinateBound_five_of_frechet_eight hz hU hSU hzB | |
| obtain ⟨W, hW, hSW, hWU, hne, _, _, _, hfirst, _⟩ := | |
| exists_open_low_height_coefficients hg hU hSU hz hG hZ hd hc hgCoord hzCoord hdet hyy | |
| have hgW : SmoothPositiveOn g W := | |
| ⟨fun i j => (hg.1 i j).mono hWU, fun p hp => hg.2 p (hWU hp)⟩ | |
| have hVW : V ⊆ W := hVS.trans hSW | |
| have hCfirst := heightPFirstBound_nonneg hG hZ hd hc | |
| have ho : ∀ w ∈ topResidualWords m, ∀ r, ∀ p ∈ V, | |
| |coordinateChainCoefficient g z w r p| ≤ C := | |
| houter g z U V hg hU (hVS.trans hSU) hVS hgHigh | |
| ((hzCoord.mono (by norm_num) le_rfl).restrict_domain hVS) | |
| (fun p hp => hdet p (hVS hp)) (fun p hp => hyy p (hVS hp)) | |
| have hf : ∀ i : Fin 2, ∀ p ∈ V, | |
| |coordPartial 1 (heightPFirst g z i) p| ≤ heightPFirstBound G Z d c := | |
| fun i p hp => hfirst i [1] (by norm_num) p (hVS hp) | |
| exact ⟨actualHighRemainder_eLpNorm_two hgW hW (hz.mono hWU) hne hV hVW hm hC hCfirst hB | |
| hlow hheight ho hf, | |
| actualHighRemainderL2Budget_lt_top C (heightPFirstBound G Z d c) B H hVfinite m, | |
| actualHighRemainder_memLp_two hgW hW (hz.mono hWU) hne hV hVW hVfinite hm hC hCfirst hB | |
| hlow hheight ho hf⟩ | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |