| import OAI.Geometry.IsometricImmersion.Comparison.QFirstJetComparison | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry SmoothLocal.HighEquation | |
| theorem exists_uniform_qMetricFirstError_bound (B D : ℝ) {d c : ℝ} | |
| (hD : 0 ≤ D) (hd : 0 < d) (hc : 0 < c) : | |
| ∃ H C : ℝ, 0 ≤ H ∧ 0 ≤ C ∧ | |
| ∀ (gTau gStar : MetricField) (w : DarbouxState) (epsilon : ℝ), | |
| ‖qFirstBundle gStar w‖ ≤ B → d ≤ (gStar (statePoint w)).det → | |
| c ≤ |stateQDenominator gStar w| → | |
| ‖actualCurvatureFirstInput gTau (statePoint w)- | |
| actualCurvatureFirstInput gStar (statePoint w)‖ ≤ epsilon → | |
| epsilon ≤ 1 → (4*max B 0+2)*epsilon ≤ d/2 → H*epsilon ≤ c/2 → | |
| |curvatureDensity gTau (statePoint w)| ≤ D → | |
| c/2 ≤ |stateQDenominator gTau w| ∧ | |
| |qMetricFirstError gTau gStar w| ≤ C*epsilon := by | |
| obtain ⟨H,F,L,hH,hF,hL,hcompare⟩ := exists_uniform_qFirst_comparison B hd hc | |
| refine ⟨H,D*F+L,hH,add_nonneg (mul_nonneg hD hF) hL,?_⟩ | |
| intro gTau gStar w epsilon hB hdet hxx hdist he1 hsmall hxxsmall hdensity | |
| have hscale : 0 ≤ 4*max B 0+2 := by linarith [le_max_right B 0] | |
| have hb : ‖qFirstBundle gTau w-qFirstBundle gStar w‖ ≤ epsilon := by | |
| simpa only [qFirstBundle_distance] using hdist | |
| obtain ⟨hfloor,hfactor,hlower⟩ := hcompare (qFirstBundle gStar w) (qFirstBundle gTau w) | |
| hB hdet hxx (hb.trans he1) | |
| ((mul_le_mul_of_nonneg_left hb hscale).trans hsmall) | |
| ((mul_le_mul_of_nonneg_left hb hH).trans hxxsmall) | |
| have hxxTau : c/2 ≤ |stateQDenominator gTau w| := by | |
| simpa only [qFirstXX_bundle] using | |
| hfloor (qFirstBundle gTau w) (right_mem_segment ℝ _ _) | |
| have hfactorBase : |qMetricDensityFactor gTau w-qMetricDensityFactor gStar w| ≤ | |
| F*‖qFirstBundle gTau w-qFirstBundle gStar w‖ := by | |
| simpa only [qFirstDensityFactor_bundle] using hfactor | |
| have hlowerBase : |qMetricFirstLower gTau w-qMetricFirstLower gStar w| ≤ | |
| L*‖qFirstBundle gTau w-qFirstBundle gStar w‖ := by | |
| simpa only [qFirstLower_bundle] using hlower | |
| have hfactor' : |qMetricDensityFactor gTau w-qMetricDensityFactor gStar w| ≤ F*epsilon := | |
| hfactorBase.trans (mul_le_mul_of_nonneg_left hb hF) | |
| have hlower' : |qMetricFirstLower gTau w-qMetricFirstLower gStar w| ≤ L*epsilon := | |
| hlowerBase.trans (mul_le_mul_of_nonneg_left hb hL) | |
| refine ⟨hxxTau,?_⟩ | |
| unfold qMetricFirstError | |
| calc | |
| _ ≤ |curvatureDensity gTau (statePoint w)| * | |
| |qMetricDensityFactor gTau w-qMetricDensityFactor gStar w|+ | |
| |qMetricFirstLower gTau w-qMetricFirstLower gStar w| := by | |
| simpa only [abs_mul] using abs_add_le | |
| (curvatureDensity gTau (statePoint w)*(qMetricDensityFactor gTau w-qMetricDensityFactor gStar w)) | |
| (qMetricFirstLower gTau w-qMetricFirstLower gStar w) | |
| _ ≤ D*(F*epsilon)+L*epsilon := add_le_add | |
| (mul_le_mul hdensity hfactor' (abs_nonneg _) hD) hlower' | |
| _ = _ := by ring | |
| theorem exists_actual_Q_forcing_error_bound (B D A : ℝ) {d c : ℝ} | |
| (hD : 0 ≤ D) (hA : 0 ≤ A) (hd : 0 < d) (hc : 0 < c) : | |
| ∃ H C : ℝ, 0 ≤ H ∧ 0 ≤ C ∧ | |
| ∀ (gTau gStar : MetricField) (U : Set Coord), | |
| SmoothPositiveOn gTau U → SmoothPositiveOn gStar U → IsOpen U → | |
| ∀ (z : Coord → ℝ) (p : Coord), p ∈ U → ∀ epsilon leading : ℝ, | |
| ‖qFirstBundle gStar (qSolutionJet z p)‖ ≤ B → d ≤ (gStar p).det → | |
| c ≤ |covHessian gStar z p 0 0| → | |
| ‖actualCurvatureFirstInput gTau p-actualCurvatureFirstInput gStar p‖ ≤ epsilon → | |
| epsilon ≤ 1 → (4*max B 0+2)*epsilon ≤ d/2 → H*epsilon ≤ c/2 → | |
| |curvatureDensity gTau p| ≤ D → |forcingCoefficient gStar z p| ≤ A → | |
| |(sixVariableQ gTau (qSolutionJet z p)-sixVariableQ gStar (qSolutionJet z p))- | |
| 2*forcingCoefficient gStar z p*leading| ≤ | |
| 2*A*|(curvatureDensity gTau p-curvatureDensity gStar p)-leading|+C*epsilon := by | |
| obtain ⟨H,C,hH,hC,herror⟩ := exists_uniform_qMetricFirstError_bound B D hD hd hc | |
| refine ⟨H,C,hH,hC,?_⟩ | |
| intro gTau gStar U hgTau hgStar hU z p hp epsilon leading hB hdet hxx hdist | |
| he1 hsmall hxxsmall hdensity hforce | |
| obtain ⟨hfloor,herr⟩ := herror gTau gStar (qSolutionJet z p) epsilon hB | |
| (by simpa only [statePoint_qSolutionJet] using hdet) | |
| (by simpa only [stateQDenominator_qSolutionJet] using hxx) | |
| (by simpa only [statePoint_qSolutionJet] using hdist) he1 hsmall hxxsmall | |
| (by simpa only [statePoint_qSolutionJet] using hdensity) | |
| have hxxStar : covHessian gStar z p 0 0 ≠ 0 := by | |
| intro hz | |
| rw [hz,abs_zero] at hxx | |
| linarith | |
| have hxxTau : covHessian gTau z p 0 0 ≠ 0 := by | |
| rw [stateQDenominator_qSolutionJet] at hfloor | |
| intro hz | |
| rw [hz,abs_zero] at hfloor | |
| linarith | |
| rw [actual_Q_metric_forcing_leading_identity hgTau hgStar hU z hp hxxTau hxxStar] | |
| have hfactor : |2*forcingCoefficient gStar z p| ≤ 2*A := by | |
| simpa only [abs_mul,abs_of_pos (by norm_num : (0 : ℝ) < 2)] using | |
| mul_le_mul_of_nonneg_left hforce (by norm_num : (0 : ℝ) ≤ 2) | |
| calc | |
| _ ≤ |2*forcingCoefficient gStar z p| * | |
| |(curvatureDensity gTau p-curvatureDensity gStar p)-leading|+ | |
| |qMetricFirstError gTau gStar (qSolutionJet z p)| := by | |
| simpa only [abs_mul] using abs_add_le | |
| (2*forcingCoefficient gStar z p*((curvatureDensity gTau p-curvatureDensity gStar p)-leading)) | |
| (qMetricFirstError gTau gStar (qSolutionJet z p)) | |
| _ ≤ _ := add_le_add | |
| (mul_le_mul hfactor le_rfl (abs_nonneg _) (mul_nonneg (by norm_num) hA)) herr | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |