File size: 4,032 Bytes
110f4f3 | 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 | import Mathlib
import OAI.Analysis.BiholderTransport.Volume.UpperJacobianBarrier
import OAI.Analysis.BiholderTransport.Regularity.ActiveDifference
import OAI.Analysis.BiholderTransport.LinearAlgebra.ChartFirstMatrix
namespace OAI
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
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]
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) // 0<J.w₀ i}
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) // 0<J.w₀ i}
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
end
end
end OAI
|