openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Comparison /ComparisonCoefficientDerivative.lean
| import OAI.Geometry.IsometricImmersion.Comparison.ComparisonCoefficientSmooth | |
| import OAI.Geometry.IsometricImmersion.Darboux.QLowActualBounds | |
| import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped ContDiff Topology | |
| namespace SmoothLocal.Analytic | |
| open SmoothLocal.Geometry | |
| theorem compactParameterIntegral_coordPartial | |
| {H : Coord × ℝ → ℝ} {U : Set Coord} {D : Set (Coord × ℝ)} | |
| (hH : ContDiffOn ℝ ∞ H D) (hD : IsOpen D) (hU : IsOpen U) | |
| (hUD : U ×ˢ Icc (0 : ℝ) 1 ⊆ D) {p : Coord} (hp : p ∈ U) (j : Fin 2) : | |
| coordPartial j (compactParameterIntegral H) p = | |
| ∫ sigma in Icc (0 : ℝ) 1, compactParameterPartial H (p,sigma) (Pi.single j 1) := by | |
| have hH1 : ContDiffOn ℝ 1 H D := hH.of_le (by simp) | |
| have hd := compactParameterIntegral_hasFDerivAt hH1 hD hU hUD hp | |
| have hpart : ContinuousOn (compactParameterPartial H) D := | |
| (compactParameterPartial_contDiffOn hD 0 hH1).continuousOn | |
| have hint : Integrable (fun sigma => compactParameterPartial H (p,sigma)) | |
| (volume.restrict (Icc (0 : ℝ) 1)) := | |
| (parameter_slice_continuousOn hpart hUD hp).integrableOn_compact isCompact_Icc | |
| change fderiv ℝ (compactParameterIntegral H) p (Pi.single j 1) = _ | |
| rw [hd.fderiv] | |
| exact ContinuousLinearMap.integral_apply hint (Pi.single j 1) | |
| end SmoothLocal.Analytic | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Analytic | |
| theorem comparisonIntegrand_parameter_partial | |
| {g : MetricField} {P z : Coord → ℝ} {U : Set Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) | |
| (hP : ContDiffOn ℝ ∞ P U) (hz : ContDiffOn ℝ ∞ z U) | |
| {p : Coord} {sigma : ℝ} (hp : p ∈ U) | |
| (hsegment : qHeightJetSegment P z sigma p ∈ darbouxQStateDomain g U) | |
| (i : Fin 6) (j : Fin 2) : | |
| compactParameterPartial (comparisonIntegrand g P z i) (p,sigma) (Pi.single j 1) = | |
| fderiv ℝ (qFirstCoefficient g i) (qHeightJetSegment P z sigma p) | |
| ((1-sigma) • coordPartial j (qSolutionJet P) p + | |
| sigma • coordPartial j (qSolutionJet z) p) := by | |
| have hJP := ((qSolutionJet_contDiffOn hU hP).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) | |
| have hJz := ((qSolutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) | |
| have hQ := ((qFirstCoefficient_contDiffOn hg hU i).contDiffAt | |
| ((darbouxQStateDomain_isOpen hg hU).mem_nhds hsegment)).differentiableAt (by simp) | |
| have hstate : HasFDerivAt (fun x => qHeightJetSegment P z sigma x) | |
| ((1-sigma) • fderiv ℝ (qSolutionJet P) p + sigma • fderiv ℝ (qSolutionJet z) p) p := | |
| (hJP.hasFDerivAt.const_smul (1-sigma)).add (hJz.hasFDerivAt.const_smul sigma) | |
| have hchain := hQ.hasFDerivAt.comp p hstate | |
| have hparam := compactParameter_slice_hasFDerivAt | |
| ((comparisonIntegrand_contDiffOn hg hU hP hz i).of_le (by simp)) | |
| (comparisonParameterDomain_isOpen hg hU hP hz) | |
| (show (p,sigma) ∈ comparisonParameterDomain g P z U from | |
| ⟨⟨hp,mem_univ _⟩,hsegment⟩) | |
| have he := hparam.unique hchain | |
| simpa only [ContinuousLinearMap.comp_apply,add_apply, | |
| smul_apply,coordPartial] using | |
| congrArg (fun A : Coord →L[ℝ] ℝ => A (Pi.single j 1)) he | |
| theorem comparisonCoefficient_partial | |
| {g : MetricField} {P z : Coord → ℝ} {U W : Set Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) | |
| (hP : ContDiffOn ℝ ∞ P U) (hz : ContDiffOn ℝ ∞ z U) | |
| (hW : IsOpen W) (hWU : W ⊆ U) | |
| (hsegment : ∀ p ∈ W, ∀ sigma ∈ Icc (0 : ℝ) 1, | |
| qHeightJetSegment P z sigma p ∈ darbouxQStateDomain g U) | |
| {p : Coord} (hp : p ∈ W) (i : Fin 6) (j : Fin 2) : | |
| coordPartial j (comparisonCoefficient g P z i) p = | |
| ∫ sigma in (0 : ℝ)..1, | |
| fderiv ℝ (qFirstCoefficient g i) (qHeightJetSegment P z sigma p) | |
| ((1-sigma) • coordPartial j (qSolutionJet P) p + | |
| sigma • coordPartial j (qSolutionJet z) p) := by | |
| rw [comparisonCoefficient_eq_parameterIntegral] | |
| have hUD : W ×ˢ Icc (0 : ℝ) 1 ⊆ comparisonParameterDomain g P z U := by | |
| intro q hq | |
| exact ⟨⟨hWU hq.1,mem_univ _⟩,hsegment q.1 hq.1 q.2 hq.2⟩ | |
| rw [compactParameterIntegral_coordPartial (comparisonIntegrand_contDiffOn hg hU hP hz i) | |
| (comparisonParameterDomain_isOpen hg hU hP hz) hW hUD hp j, | |
| intervalIntegral.integral_of_le zero_le_one,← integral_Icc_eq_integral_Ioc] | |
| apply setIntegral_congr_fun measurableSet_Icc | |
| intro sigma hs | |
| exact comparisonIntegrand_parameter_partial hg hU hP hz (hWU hp) (hsegment p hp sigma hs) i j | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |