| import Mathlib |
| import OAI.Analysis.BiholderTransport.Volume.UpperJacobianBarrier |
| import OAI.Analysis.BiholderTransport.Regularity.ActiveDifference |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ChartFirstMatrix |
|
|
| namespace OAI |
|
|
| section |
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| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
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|
| namespace WeakMTWTransport |
| section FirstRank |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty 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] |
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| omit [CompactSpace M] [Nonempty 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 ChartFixedPrefixLimit.positive_rank_direction |
| {a:M} {v G:M → ℝ} {t:ℝ} |
| {q:ℕ → subgradientGraph (n := n) (cTransform v)} |
| {q₀:subgradientGraph (n := n) (cTransform v)} |
| (J:ChartFixedPrefixLimit a v G t q q₀) |
| (hbase:q₀.1.1∈(extChartAt 𝓘(ℝ,Model n) a).source) : |
| let b:=graphBaseCoordinate a q₀.1 |
| let p:=graphVelocityCoordinate a q₀.1 |
| let E:=TangentSpace 𝓘(ℝ,Model n) ((extChartAt 𝓘(ℝ,Model n) a).symm b) |
| let κ:Type:={i:Fin (Module.finrank ℝ (Model n)+1) |
| let pj:κ → E:=fun i=>chartFiberInverse a b (J.pj₀ i) |
| let P:E:=chartFiberInverse a b p |
| let S:=Submodule.span ℝ (range (fun i=>pj i-P)) |
| ∀ (A C:ℝ),0<A → |
| (∀d:Model n,A*‖chartFiberInverse a b d‖^2+ |
| C*(inner ℝ P (chartFiberInverse a b d))^2≤J.H d d) → |
| ∀ (level:E → ℝ) (Bc Bo:ℝ → ℝ), |
| Bc (level P)≤Bo (level P) → |
| 0<Bc (level P)-∑ i,J.w₀ i*Bo (level (chartFiberInverse a b (J.pj₀ i))) → |
| ∃ e∈S,‖e‖=1 ∧ 0 < inner ℝ P e ∧ S=Submodule.span ℝ {e} ∧ A≤(-C)*(inner ℝ P e)^2 := by |
| dsimp only |
| let b:=graphBaseCoordinate a q₀.1 |
| let p:=graphVelocityCoordinate a q₀.1 |
| let x:=(extChartAt 𝓘(ℝ,Model n) a).symm b |
| let E:=TangentSpace 𝓘(ℝ,Model n) x |
| let T:=trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a |
| let U:=chartFiberInverse a b |
| let V:=T.continuousLinearMapAt ℝ x |
| let κ:Type:={i:Fin (Module.finrank ℝ (Model n)+1) |
| let pj:κ → E:=fun i=>U (J.pj₀ i) |
| let P:E:=U p |
| let S:=Submodule.span ℝ (range (fun i=>pj i-P)) |
| have hb : b∈(extChartAt 𝓘(ℝ,Model n) a).target := |
| (extChartAt 𝓘(ℝ,Model n) a).map_source hbase |
| have hx : x∈T.baseSet := by |
| simpa only [x,T,TangentBundle.trivializationAt_baseSet,extChartAt_source] using (extChartAt 𝓘(ℝ,Model n) a).map_target hb |
| have hUV (d:E):U (V d)=d := T.symmL_continuousLinearMapAt (R := ℝ) hx d |
| have hVU (d:Model n):V (U d)=d := T.continuousLinearMapAt_symmL (R := ℝ) hx d |
| intro A C hA hbound level Bc Bo hprofiles hparameter |
| have hres:=positive_barycentric_restriction (fun i=>U (J.pj₀ i)) J.w₀ J.nonnegLimit J.totalLimit |
| let : Nonempty κ:=hres.1 |
| have hparam:0<Bc (level P)-∑ i:κ,J.w₀ i*Bo (level (pj i)) := by |
| rwa [hres.2.2.2 (fun i=>Bo (level (U (J.pj₀ i))))] |
| classical |
| have hS:S≠⊥:=active_difference_ne_bot hres.2.1 hprofiles hparam |
| let L:E →L[ℝ] E →L[ℝ] ℝ:=J.L.bilinearComp V V |
| have hk:S≤LinearMap.ker L.toLinearMap := by |
| apply Submodule.span_le.mpr |
| rintro _ ⟨i,rfl⟩ |
| change L (pj i-P)=0 |
| ext d |
| change J.L (V (U (J.pj₀ i)-U p)) (V d)=0 |
| rw [map_sub,hVU,hVU] |
| exact J.kernel i (V d) |
| apply unit_positive_direction_of_rank_one_semibound S hA hS |
| intro d hd |
| have hl:L d=0 := hk hd |
| have hzero:J.L (V d) (V d)=0 := congrArg (fun f:E →L[ℝ] ℝ=>f d) hl |
| have HH:=hbound (V d) |
| rw [hUV] at HH |
| have HU:=J.upper (V d) |
| rw [hzero] at HU |
| linarith only [HH,HU] |
|
|
| end FirstRank |
| end WeakMTWTransport |
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|
| end |
| end |
|
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| end OAI |
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