openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Calculus /MetricInverseJetBounds.lean
| import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetBounds | |
| import OAI.Geometry.IsometricImmersion.Metrics.MetricSmoothness | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology Matrix | |
| namespace SmoothLocal.Geometry | |
| def metricQuotientNumeratorBound (G : ℝ) : ℝ := G + 16 * G^2 | |
| def metricInverseJetBound (G d : ℝ) : ℝ := | |
| LowQuotient.boundThroughThree (metricQuotientNumeratorBound G) d | |
| theorem metricInverseJetBound_nonneg {G d : ℝ} (hG : 0 ≤ G) (hd : 0 < d) : | |
| 0 ≤ metricInverseJetBound G d := by | |
| have hN : 0 ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; positivity | |
| obtain ⟨h0, h1, h2, h3⟩ := LowQuotient.bounds_nonneg hN hd | |
| dsimp [metricInverseJetBound, LowQuotient.boundThroughThree] | |
| positivity | |
| variable {g : MetricField} {U : Set Coord} {G d : ℝ} | |
| theorem metricDet_coordinate_bound | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hG : 0 ≤ G) | |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) U 3 G) : | |
| CoordinateBound (fun p => (g p).det) U 3 (16 * G^2) := by | |
| have hdiag := CoordinateBound.mul_through_three (hg.1 0 0) (hg.1 1 1) hU hG hG (hgB 0 0) (hgB 1 1) | |
| have hoff := CoordinateBound.mul_through_three (hg.1 0 1) (hg.1 1 0) hU hG hG (hgB 0 1) (hgB 1 0) | |
| have hdet := CoordinateBound.sub ((hg.1 0 0).mul (hg.1 1 1)) | |
| ((hg.1 0 1).mul (hg.1 1 0)) hU hdiag hoff | |
| have hc : 8 * G * G + 8 * G * G = 16 * G^2 := by ring | |
| simpa only [Matrix.det_fin_two, hc] using hdet | |
| theorem metricAdjugate_coordinate_bound | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) | |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) U 3 G) (i j : Fin 2) : | |
| CoordinateBound (fun p => (g p).adjugate i j) U 3 G := by | |
| fin_cases i <;> fin_cases j | |
| · simpa [Matrix.adjugate_fin_two] using hgB 1 1 | |
| · simpa [Matrix.adjugate_fin_two] using CoordinateBound.neg (hg.1 0 1) hU (hgB 0 1) | |
| · simpa [Matrix.adjugate_fin_two] using CoordinateBound.neg (hg.1 1 0) hU (hgB 1 0) | |
| · simpa [Matrix.adjugate_fin_two] using hgB 0 0 | |
| theorem inverseMetric_eq_adjugate_div (g : MetricField) (i j : Fin 2) : | |
| (fun p => inverseMetric g p i j) = (fun p => (g p).adjugate i j / (g p).det) := by | |
| funext p | |
| simp [inverseMetric, Matrix.inv_def, Ring.inverse_eq_inv, div_eq_mul_inv, smul_eq_mul, mul_comm] | |
| theorem inverseMetric_coordinate_bound | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hG : 0 ≤ G) (hd : 0 < d) | |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) U 3 G) | |
| (hdet : ∀ p ∈ U, d ≤ |(g p).det|) (i j : Fin 2) : | |
| CoordinateBound (fun p => inverseMetric g p i j) U 3 (metricInverseJetBound G d) := by | |
| have hN : 0 ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; positivity | |
| have hGN : G ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; nlinarith [sq_nonneg G] | |
| have hDN : 16 * G^2 ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; linarith | |
| have hadj := (metricAdjugate_coordinate_bound hg hU hgB i j).mono le_rfl hGN | |
| have hdetB := (metricDet_coordinate_bound hg hU hG hgB).mono le_rfl hDN | |
| rw [inverseMetric_eq_adjugate_div] | |
| intro ds hds p hp | |
| exact LowQuotient.quotient_bound_through_three (metricAdjugate_contDiffOn hg i j) | |
| (metricDet_contDiffOn hg) hU hN hd hadj hdetB hdet ds hds hp | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |