| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Calculus.PartialDerivatives | |
| import OAI.Analysis.BiholderTransport.Coordinates.NormalCoordinates3 | |
| import OAI.Analysis.BiholderTransport.Coordinates.FixedJoinSmooth | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteProfile (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end WeakMTWTransport | |
| end | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| def movingFixedJoinAction (a : M) (h : ℝ) (q : ℝ×(Model n×Model n)) (z : Model n) : ℝ := | |
| cost (movingNormal a (q.2.1,0)) (movingNormal a (q.2.1,h • (q.2.2+z)))/h+ | |
| cost (movingNormal a (q.2.1,h • (q.2.2+z))) (movingNormal a (q.2.1,q.1 • q.2.2))/(q.1-h) | |
| def movingFixedJoinMiddle (a : M) (h : ℝ) (q : ℝ×(Model n×Model n)) : | |
| Model n →L[ℝ] Model n →L[ℝ] ℝ := | |
| fderiv ℝ (fderiv ℝ (movingFixedJoinAction a h q)) 0 | |
| omit [CompactSpace M] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] in | |
| lemma movingFixedJoinAction_eq {a : M} {b : Model n} | |
| (hb : b∈(extChartAt 𝓘(ℝ,Model n) a).target) (h T : ℝ) (p z : Model n) : | |
| movingFixedJoinAction a h (T,(b,p)) z= | |
| fixedJoinAction ((extChartAt 𝓘(ℝ,Model n) a).symm b) h T | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p) | |
| (0,((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p)+ | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) z)) := by | |
| simp only [movingFixedJoinAction,fixedJoinAction,movingNormal_eq hb,map_smul,map_add,map_zero] | |
| lemma movingFixedJoinAction_contDiffAt {a : M} {b p : Model n} {h T : ℝ} | |
| (hb : b∈(extChartAt 𝓘(ℝ,Model n) a).target) (hTh : T≠h) | |
| (hleft : h • (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p∈ | |
| injectivityDomain ((extChartAt 𝓘(ℝ,Model n) a).symm b)) | |
| (hright : (T-h) • (sprayFlow h | |
| (⟨((extChartAt 𝓘(ℝ,Model n) a).symm b), | |
| (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2∈ | |
| injectivityDomain (sprayFlow h | |
| (⟨((extChartAt 𝓘(ℝ,Model n) a).symm b), | |
| (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1) : | |
| ContDiffAt ℝ ∞ (Function.uncurry (movingFixedJoinAction a h)) ((T,(b,p)),0) := by | |
| let x := (extChartAt 𝓘(ℝ,Model n) a).symm b | |
| let L := (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ x | |
| let q := ((T,(b,p)),(0 : Model n)) | |
| let Q := (ℝ×(Model n×Model n))×Model n | |
| have hA : ContMDiffAt 𝓘(ℝ,Q) 𝓘(ℝ,Model n) ∞ | |
| (fun z : Q => movingNormal a (z.1.2.1,0)) q := | |
| (movingNormal_contMDiffAt (q := (b,0)) hb).comp q | |
| (contDiffAt_fst.snd.fst.prodMk contDiffAt_const).contMDiffAt | |
| have hB : ContMDiffAt 𝓘(ℝ,Q) 𝓘(ℝ,Model n) ∞ | |
| (fun z : Q => movingNormal a (z.1.2.1,h • (z.1.2.2+z.2))) q := | |
| (movingNormal_contMDiffAt (q := (b,h • (p+0))) hb).comp q | |
| (contDiffAt_fst.snd.fst.prodMk | |
| ((contDiffAt_const (c := h)).smul (contDiffAt_fst.snd.snd.add contDiffAt_snd))).contMDiffAt | |
| have hC : ContMDiffAt 𝓘(ℝ,Q) 𝓘(ℝ,Model n) ∞ | |
| (fun z : Q => movingNormal a (z.1.2.1,z.1.1 • z.1.2.2)) q := | |
| (movingNormal_contMDiffAt (q := (b,T • p)) hb).comp q | |
| (contDiffAt_fst.snd.fst.prodMk | |
| (contDiffAt_fst.fst.smul contDiffAt_fst.snd.snd)).contMDiffAt | |
| have hc₁ := cost_contMDiffAt_of_injectivityDomain | |
| (⟨x,h • L p⟩ : TangentBundle 𝓘(ℝ,Model n) M) hleft | |
| have hc₂ := cost_contMDiffAt_of_injectivityDomain | |
| (⟨(sprayFlow h (⟨x,L p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1, | |
| (T-h) • (sprayFlow h (⟨x,L p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2⟩ : | |
| TangentBundle 𝓘(ℝ,Model n) M) hright | |
| have hc₁' : ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) 𝓘(ℝ,ℝ) ∞ | |
| (fun z : M×M => cost z.1 z.2) | |
| (movingNormal a (b,0),movingNormal a (b,h • (p+0))) := by | |
| simpa only [movingNormal_eq hb,map_zero,riemannianExp_zero,add_zero,map_smul] using hc₁ | |
| have hc₂' : ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) 𝓘(ℝ,ℝ) ∞ | |
| (fun z : M×M => cost z.1 z.2) | |
| (movingNormal a (b,h • (p+0)),movingNormal a (b,T • p)) := by | |
| dsimp only at hc₂ | |
| rw [shifted_exp_time] at hc₂ | |
| simpa only [movingNormal_eq hb,add_zero,map_smul,riemannianExp_smul] using hc₂ | |
| exact ((hc₁'.comp q (hA.prodMk hB)).contDiffAt.div_const h).add | |
| (((hc₂'.comp q (hB.prodMk hC)).contDiffAt).div | |
| (contDiffAt_fst.fst.sub contDiffAt_const) (sub_ne_zero.mpr hTh)) | |
| omit [CompactSpace M] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] in | |
| lemma movingFixedJoinMiddle_contDiffAt {a : M} {h T : ℝ} {b p : Model n} | |
| (H : ContDiffAt ℝ ∞ (Function.uncurry (movingFixedJoinAction a h)) ((T,(b,p)),0)) : | |
| ContDiffAt ℝ ∞ (movingFixedJoinMiddle a h) (T,(b,p)) := | |
| (ContDiffAt.partial_snd_fderiv_two H).comp (T,(b,p)) | |
| (contDiffAt_id.prodMk contDiffAt_const) | |
| omit [CompactSpace M] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| lemma movingFixedJoinMiddle_eq {a : M} {b : Model n} | |
| (hb : b∈(extChartAt 𝓘(ℝ,Model n) a).target) (h T : ℝ) (p v w : Model n) | |
| (H : ContDiffAt ℝ 2 (fixedJoinAction ((extChartAt 𝓘(ℝ,Model n) a).symm b) h T | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p)) | |
| (0,((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p))) : | |
| movingFixedJoinMiddle a h (T,(b,p)) v w= | |
| fixedJoinMiddle ((extChartAt 𝓘(ℝ,Model n) a).symm b) h T | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) p) | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) v) | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) w) := by | |
| let x := (extChartAt 𝓘(ℝ,Model n) a).symm b | |
| let L := (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ x | |
| let I := (0 : Model n →L[ℝ] TangentSpace 𝓘(ℝ,Model n) x).prod L | |
| have he : movingFixedJoinAction a h (T,(b,p))= | |
| (fun z => fixedJoinAction x h T (L p) ((0,L p)+I z)) := by | |
| ext z | |
| simpa only [I,ContinuousLinearMap.prod_apply,zero_apply, | |
| Prod.mk_add_mk,add_zero] using movingFixedJoinAction_eq hb h T p z | |
| rw [movingFixedJoinMiddle,he] | |
| have H' := second_fderiv_comp_affine I (0,L p) 0 (by simpa only [map_zero,add_zero] using H) v w | |
| simpa only [I,ContinuousLinearMap.prod_apply,zero_apply,map_zero, | |
| add_zero,fixedJoinMiddle_apply] using H' | |
| lemma movingFixedJoinMiddle_trivialization {a x : M} | |
| (hx : x∈(extChartAt 𝓘(ℝ,Model n) a).source) (h T : ℝ) | |
| (p v w : TangentSpace 𝓘(ℝ,Model n) x) | |
| (H : ContDiffAt ℝ 2 (fixedJoinAction x h T p) (0,p)) : | |
| movingFixedJoinMiddle a h (T,(extChartAt 𝓘(ℝ,Model n) a x, | |
| (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x p)) | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x v) | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x w)= | |
| fixedJoinMiddle x h T p v w := by | |
| let e := trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a | |
| have hx' : x∈e.baseSet := by simpa only [e,TangentBundle.trivializationAt_baseSet,extChartAt_source] using hx | |
| have HL : ∀ v : TangentSpace 𝓘(ℝ,Model n) x, e.symmL ℝ x (e.continuousLinearMapAt ℝ x v)=v := | |
| fun v => e.symmL_continuousLinearMapAt hx' v | |
| have HB : ContDiffAt ℝ 2 | |
| (fixedJoinAction ((extChartAt 𝓘(ℝ,Model n) a).symm (extChartAt 𝓘(ℝ,Model n) a x)) h T | |
| (e.symmL ℝ ((extChartAt 𝓘(ℝ,Model n) a).symm (extChartAt 𝓘(ℝ,Model n) a x)) | |
| (e.continuousLinearMapAt ℝ x p))) | |
| (0,e.symmL ℝ ((extChartAt 𝓘(ℝ,Model n) a).symm (extChartAt 𝓘(ℝ,Model n) a x)) | |
| (e.continuousLinearMapAt ℝ x p)) := by | |
| rw [(extChartAt 𝓘(ℝ,Model n) a).left_inv hx,HL] | |
| exact H | |
| have HE := movingFixedJoinMiddle_eq ((extChartAt 𝓘(ℝ,Model n) a).map_source hx) h T | |
| (e.continuousLinearMapAt ℝ x p) (e.continuousLinearMapAt ℝ x v) | |
| (e.continuousLinearMapAt ℝ x w) HB | |
| dsimp only [e] at HL HE | |
| rw [(extChartAt 𝓘(ℝ,Model n) a).left_inv hx] at HE | |
| simpa only [HL] using HE | |
| end WeakMTWTransport | |
| end | |
| end OAI | |