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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | 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
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