openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Comparison /ComparisonCoefficientSmooth.lean
| import OAI.Geometry.IsometricImmersion.Comparison.ActualComparisonEquation | |
| import OAI.Geometry.IsometricImmersion.Flows.CompactParameterIntegral | |
| import OAI.Geometry.IsometricImmersion.Darboux.QSpatialJets | |
| import Mathlib.Topology.Compactness.Compact | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped ContDiff Topology | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Analytic | |
| def comparisonStateMap (P z : Coord → ℝ) (q : Coord × ℝ) : DarbouxState := | |
| qHeightJetSegment P z q.2 q.1 | |
| def comparisonIntegrand (g : MetricField) (P z : Coord → ℝ) (i : Fin 6) | |
| (q : Coord × ℝ) : ℝ := qFirstCoefficient g i (comparisonStateMap P z q) | |
| def comparisonParameterDomain (g : MetricField) (P z : Coord → ℝ) (U : Set Coord) : | |
| Set (Coord × ℝ) := | |
| (U ×ˢ (univ : Set ℝ)) ∩ comparisonStateMap P z ⁻¹' darbouxQStateDomain g U | |
| theorem comparisonStateMap_contDiffOn {P z : Coord → ℝ} {U : Set Coord} | |
| (hU : IsOpen U) (hP : ContDiffOn ℝ ∞ P U) (hz : ContDiffOn ℝ ∞ z U) : | |
| ContDiffOn ℝ ∞ (comparisonStateMap P z) (U ×ˢ (univ : Set ℝ)) := by | |
| have h0 : ContDiffOn ℝ ∞ (fun q : Coord × ℝ => qSolutionJet P q.1) | |
| (U ×ˢ (univ : Set ℝ)) := (qSolutionJet_contDiffOn hU hP).comp contDiff_fst.contDiffOn | |
| (fun _ hq => hq.1) | |
| have h1 : ContDiffOn ℝ ∞ (fun q : Coord × ℝ => qSolutionJet z q.1) | |
| (U ×ˢ (univ : Set ℝ)) := (qSolutionJet_contDiffOn hU hz).comp contDiff_fst.contDiffOn | |
| (fun _ hq => hq.1) | |
| exact ((contDiffOn_const.sub contDiff_snd.contDiffOn).smul h0).add | |
| (contDiff_snd.contDiffOn.smul h1) | |
| theorem comparisonParameterDomain_isOpen {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) : | |
| IsOpen (comparisonParameterDomain g P z U) := | |
| (comparisonStateMap_contDiffOn hU hP hz).continuousOn.isOpen_inter_preimage | |
| (hU.prod isOpen_univ) (darbouxQStateDomain_isOpen hg hU) | |
| theorem comparisonIntegrand_contDiffOn {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) (i : Fin 6) : | |
| ContDiffOn ℝ ∞ (comparisonIntegrand g P z i) (comparisonParameterDomain g P z U) := | |
| (qFirstCoefficient_contDiffOn hg hU i).comp | |
| ((comparisonStateMap_contDiffOn hU hP hz).mono inter_subset_left) (fun _ hq => hq.2) | |
| theorem comparisonCoefficient_eq_parameterIntegral (g : MetricField) (P z : Coord → ℝ) | |
| (i : Fin 6) : | |
| comparisonCoefficient g P z i = compactParameterIntegral (comparisonIntegrand g P z i) := by | |
| funext p | |
| change (∫ sigma in (0 : ℝ)..1, _) = ∫ sigma in Icc (0 : ℝ) 1, _ | |
| simp only [intervalIntegral.integral_of_le zero_le_one, integral_Icc_eq_integral_Ioc] | |
| rfl | |
| theorem comparisonCoefficient_contDiffOn {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) (i : Fin 6) : | |
| ContDiffOn ℝ ∞ (comparisonCoefficient g P z i) W := by | |
| rw [comparisonCoefficient_eq_parameterIntegral] | |
| apply compactParameterIntegral_contDiffOn (comparisonIntegrand_contDiffOn hg hU hP hz i) | |
| (comparisonParameterDomain_isOpen hg hU hP hz) hW | |
| intro q hq | |
| exact ⟨⟨hWU hq.1,mem_univ _⟩,hsegment q.1 hq.1 q.2 hq.2⟩ | |
| theorem exists_comparison_coefficient_neighborhood | |
| {g : MetricField} {P z : Coord → ℝ} {U S : Set Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) | |
| (hP : ContDiffOn ℝ ∞ P U) (hz : ContDiffOn ℝ ∞ z U) | |
| (hS : IsCompact S) (hSU : S ⊆ U) | |
| (hsegment : ∀ p ∈ S, ∀ sigma ∈ Icc (0 : ℝ) 1, | |
| qHeightJetSegment P z sigma p ∈ darbouxQStateDomain g U) : | |
| ∃ W : Set Coord, IsOpen W ∧ S ⊆ W ∧ W ⊆ U ∧ | |
| (∀ p ∈ W, ∀ sigma ∈ Icc (0 : ℝ) 1, | |
| qHeightJetSegment P z sigma p ∈ darbouxQStateDomain g U) ∧ | |
| ∀ i : Fin 6, ContDiffOn ℝ ∞ (comparisonCoefficient g P z i) W := by | |
| have hprod : S ×ˢ Icc (0 : ℝ) 1 ⊆ comparisonParameterDomain g P z U := by | |
| intro q hq | |
| exact ⟨⟨hSU hq.1,mem_univ _⟩,hsegment q.1 hq.1 q.2 hq.2⟩ | |
| obtain ⟨W,J,hW,_,hSW,hIJ,hWJ⟩ := generalized_tube_lemma hS isCompact_Icc | |
| (comparisonParameterDomain_isOpen hg hU hP hz) hprod | |
| have hWU : W ⊆ U := by | |
| intro p hp | |
| have hj : (0 : ℝ) ∈ J := hIJ (by norm_num) | |
| exact (hWJ (show (p, (0 : ℝ)) ∈ W ×ˢ J from ⟨hp,hj⟩)).1.1 | |
| have hseg : ∀ p ∈ W, ∀ sigma ∈ Icc (0 : ℝ) 1, | |
| qHeightJetSegment P z sigma p ∈ darbouxQStateDomain g U := by | |
| intro p hp sigma hs | |
| exact (hWJ (show (p, sigma) ∈ W ×ˢ J from ⟨hp,hIJ hs⟩)).2 | |
| exact ⟨W,hW,hSW,hWU,hseg,fun i => comparisonCoefficient_contDiffOn hg hU hP hz hW hWU hseg i⟩ | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |