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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | import Mathlib
import OAI.Analysis.BiholderTransport.LinearAlgebra.FirstMatrixLimit
import OAI.Analysis.BiholderTransport.Coordinates.CoordinateMatrices
import OAI.Analysis.BiholderTransport.Coordinates.ChartActiveLimit
namespace OAI
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
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]
structure ChartFixedPrefixLimit (a : M) (v G : M → ℝ) (t : ℝ)
(q : ℕ → subgradientGraph (n := n) (cTransform v))
(q₀ : subgradientGraph (n := n) (cTransform v)) where
σ : ℕ → ℕ
strictMono : StrictMono σ
pj : ℕ → Fin (Module.finrank ℝ (Model n)+1) → Model n
w : ℕ → Fin (Module.finrank ℝ (Model n)+1) → ℝ
pj₀ : Fin (Module.finrank ℝ (Model n)+1) → Model n
w₀ : Fin (Module.finrank ℝ (Model n)+1) → ℝ
H : Model n →L[ℝ] Model n →L[ℝ] ℝ
L : Model n →L[ℝ] Model n →L[ℝ] ℝ
activeLimit : ∀ i,chartFiberInverse a (graphBaseCoordinate a q₀.1) (pj₀ i)∈
activeLogs v ((extChartAt 𝓘(ℝ,Model n) a).symm (graphBaseCoordinate a q₀.1))
nonnegLimit : ∀ i,0≤w₀ i
totalLimit : ∑ i,w₀ i=1
baryLimit : ∑ i,w₀ i • pj₀ i=graphVelocityCoordinate a q₀.1
pjLimit : Tendsto pj atTop (𝓝 pj₀)
wLimit : Tendsto w atTop (𝓝 w₀)
HLimit : Tendsto (fun k=>coordinateCenterMatrix a t G
(graphBaseCoordinate a (q (σ k)).1)
(graphVelocityCoordinate a (q (σ k)).1)) atTop (𝓝 H)
LLimit : Tendsto (fun k=>coordinatePoleMatrix a t v
(graphBaseCoordinate a (q (σ k)).1)
(graphVelocityCoordinate a (q (σ k)).1)) atTop (𝓝 L)
positive : ∀ d:Model n,0≤L d d
symmetric : ∀ d e:Model n,L d e=L e d
kernel : ∀ i,∀ d:Model n,L (pj₀ i-graphVelocityCoordinate a q₀.1) d=0
upper : ∀ d:Model n,H d d≤L d d
samples : ∀ᶠ k in atTop,
let b := graphBaseCoordinate a (q (σ k)).1
let p := graphVelocityCoordinate a (q (σ k)).1
let A := chartFiberInverse a b
(∀ i,0≤w k i) ∧ (∑ i,w k i=1) ∧ (∑ i,w k i • pj k i=p) ∧
(∀ i,A (pj k i)∈activeLogs v ((extChartAt 𝓘(ℝ,Model n) a).symm b)) ∧
(∀ i,0<w k i → HasLowerSecondTaylor
(fun r:Model n=>v (movingNormal a (b,pj k i+r))+‖A (pj k i+r)‖^2/2) 0
(w k i • coordinatePoleMatrix a t v b p))
lemma WeakMTW.first_fixed_prefix_limit_in_chart
(hmtw : WeakMTW (n := n) (M := M)) {v G : M → ℝ} (hv : Continuous v)
{t : ℝ} (ht : 0<t) (ht1 : t<1)
(q : ℕ → subgradientGraph (n := n) (cTransform v))
(q₀ : subgradientGraph (n := n) (cTransform v))
(hz : Tendsto (fun k=>graphProjection (cTransform v) t (q k)) atTop
(𝓝 (graphProjection (cTransform v) t q₀)))
{a c d : M}
(ha0 : q₀.1.1∈(extChartAt 𝓘(ℝ,Model n) a).source)
(hc : graphProjection (cTransform v) t q₀∈(extChartAt 𝓘(ℝ,Model n) c).source)
(hd : q₀.1.1∈(extChartAt 𝓘(ℝ,Model n) d).source) :
let χ := extChartAt 𝓘(ℝ,Model n) c
let F := fun z:Model n=>hopfLax t (cTransform v) (χ.symm z)
let C := fun z:Model n=>G (χ.symm z)
let z := fun k=>χ (graphProjection (cTransform v) t (q k))
let z₀ := χ (graphProjection (cTransform v) t q₀)
∀ Af Ac : Model n →L[ℝ] Model n →L[ℝ] ℝ,
ContinuousAt (fderiv ℝ F) z₀ → ContinuousAt (fderiv ℝ C) z₀ →
(∀ᶠ z in 𝓝 z₀,DifferentiableAt ℝ F z ∧ DifferentiableAt ℝ C z) →
Tendsto (fun k=>fderiv ℝ (fderiv ℝ F) (z k)) atTop (𝓝 Af) →
Tendsto (fun k=>fderiv ℝ (fderiv ℝ C) (z k)) atTop (𝓝 Ac) →
Tendsto (fun k=>fderiv ℝ F (z k)+fderiv ℝ C (z k)) atTop (𝓝 0) →
(∀ r:Model n,Af r r+Ac r r≤0) →
(∀ᶠ k in atTop,DifferentiableAt ℝ (fderiv ℝ F) (z k) ∧
DifferentiableAt ℝ (fderiv ℝ C) (z k) ∧ DifferentiableAt ℝ
(fun w:Model n=>extChartAt 𝓘(ℝ,Model n) d
(hopfPole (n := n) t (cTransform v) (χ.symm w))) (z k)) →
Nonempty (ChartFixedPrefixLimit a v G t q q₀) := by
dsimp only
intro Af Ac hF hC hnear hAf hAc herr hquad hsample
let χ := extChartAt 𝓘(ℝ,Model n) c
let F := fun z:Model n=>hopfLax t (cTransform v) (χ.symm z)
let C := fun z:Model n=>G (χ.symm z)
let z := fun k=>χ (graphProjection (cTransform v) t (q k))
let z₀ := χ (graphProjection (cTransform v) t q₀)
have hzCoord : Tendsto z atTop (𝓝 z₀) :=
((continuousOn_extChartAt _).continuousAt ((isOpen_extChartAt_source _).mem_nhds hc)).tendsto.comp hz
have hjF := hF.tendsto.comp hzCoord
have hjC := hC.tendsto.comp hzCoord
have hgrad : fderiv ℝ F z₀+fderiv ℝ C z₀=0 := tendsto_nhds_unique (hjF.add hjC) herr
obtain ⟨K,hK,hKs,pj,w,pj₀,w₀,σ,hσ,hs,hpj,hw,ha,hn,h1,hbar⟩ :=
hmtw.active_barycentric_tail_limit_in_chart hv ht ht1 q q₀ hz ha0
let b := fun k=>graphBaseCoordinate a (q (σ k)).1
let p := fun k=>graphVelocityCoordinate a (q (σ k)).1
let b₀ := graphBaseCoordinate a q₀.1
let p₀ := graphVelocityCoordinate a q₀.1
have hq := hmtw.graph_points_tendsto hv ht ht1 (hz.comp hσ.tendsto_atTop)
have hbase := (FiberBundle.continuous_proj (Model n) (TangentSpace 𝓘(ℝ,Model n))).tendsto q₀.1 |>.comp hq
have hbsource : ∀ᶠ k in atTop,(q (σ k)).1.1∈(extChartAt 𝓘(ℝ,Model n) a).source :=
hbase.eventually ((isOpen_extChartAt_source _).mem_nhds ha0)
have hdsource : ∀ᶠ k in atTop,(q (σ k)).1.1∈(extChartAt 𝓘(ℝ,Model n) d).source :=
hbase.eventually ((isOpen_extChartAt_source _).mem_nhds hd)
have hzsource : ∀ᶠ k in atTop,graphProjection (cTransform v) t (q (σ k))∈χ.source :=
(hz.comp hσ.tendsto_atTop).eventually ((isOpen_extChartAt_source _).mem_nhds hc)
obtain ⟨hb,hp⟩ := hmtw.graph_coordinate_limit_in_chart hv ht ht1 (hz.comp hσ.tendsto_atTop) ha0
have hb₀ : b₀∈(extChartAt 𝓘(ℝ,Model n) a).target :=
(extChartAt 𝓘(ℝ,Model n) a).map_source ha0
have hp₀ : t • chartFiberInverse a b₀ p₀∈
injectivityDomain ((extChartAt 𝓘(ℝ,Model n) a).symm b₀) := by
exact
hmtw.graphCoordinate_prefix_regular hv ht ht1 q₀ ha0
have he₀ : movingPrefix a t b₀ p₀=graphProjection (cTransform v) t q₀ := by
exact
movingPrefix_graph_coordinates ha0 t (q := q₀.1)
have hg := movingPrefixChart_contDiffAt hb₀ (he₀ ▸ hc)
have henergy := movingPrefixEnergy_contDiffAt hb₀ hp₀
have he : ∀ᶠ k in atTop,movingPrefix a t (b k) (p k)=
graphProjection (cTransform v) t (q (σ k)) :=
hbsource.mono (fun k hk=>movingPrefix_graph_coordinates hk t)
have hzEq : ∀ᶠ k in atTop,movingPrefixChart a c t (b k) (p k)=z (σ k) :=
he.mono (fun k hk=>congrArg χ hk)
have hM : ∀ᶠ k in atTop,
(∀ r:Model n,0≤coordinatePoleMatrix a t v (b k) (p k) r r) ∧
(∀ r s:Model n,coordinatePoleMatrix a t v (b k) (p k) r s=
coordinatePoleMatrix a t v (b k) (p k) s r) ∧
(∀ i,∀ r:Model n,coordinatePoleMatrix a t v (b k) (p k) (pj k i-p k) r=0) ∧
(∀ i,chartFiberInverse a (b k) (pj k i)∈
activeLogs v ((extChartAt 𝓘(ℝ,Model n) a).symm (b k))) ∧
(∀ i,0<w k i → HasLowerSecondTaylor
(fun r:Model n=>v (movingNormal a (b k,pj k i+r))+
‖chartFiberInverse a (b k) (pj k i+r)‖^2/2) 0
(w k i • coordinatePoleMatrix a t v (b k) (p k))) := by
filter_upwards [hbsource,hdsource,hzsource,he,hzEq,hσ.tendsto_atTop.eventually hsample]
with k hbk hdk hzk hek hzEk hsk
have hbtk := (extChartAt 𝓘(ℝ,Model n) a).map_source hbk
have hba : (extChartAt 𝓘(ℝ,Model n) a).symm (b k)=(q (σ k)).1.1 :=
(extChartAt 𝓘(ℝ,Model n) a).left_inv hbk
have hact : ∀ i,chartFiberInverse a (b k) (pj k i)∈
activeLogs v ((extChartAt 𝓘(ℝ,Model n) a).symm (b k)) :=
fun i=>(mem_active_coordinates hKs ((hs k).1 i)).2.1
obtain ⟨hpos,hsym,hker,hjet⟩ := hmtw.coordinate_sample_pole_matrix hv ht ht1 hbtk
(graphCoordinate_normalSubdifferential (q (σ k)) hbk) (hek ▸ hzk) (hba ▸ hdk)
(show DifferentiableAt ℝ _ (movingPrefixChart a c t (b k) (p k)) from hzEk ▸ hsk.2.2)
exact ⟨hpos,hsym,fun i=>hker (pj k i) (hact i),hact,
fun i hi=>hjet _ (pj k) (w k) (hs k).2.1 (hs k).2.2.1 (hs k).2.2.2 hact i hi⟩
have hL_eq : ∀ᶠ k in atTop,coordinatePoleMatrix a t v (b k) (p k)=
fderiv ℝ (fderiv ℝ (movingPrefixEnergy a t (b k))) (p k)-
fderiv ℝ (fderiv ℝ (fun r=>F (movingPrefixChart a c t (b k) r))) (p k) := by
filter_upwards [hbsource,hzsource,he] with k hbk hzk hek
exact coordinatePoleMatrix_chart ((extChartAt 𝓘(ℝ,Model n) a).map_source hbk) (hek ▸ hzk)
have hH_eq : ∀ᶠ k in atTop,coordinateCenterMatrix a t G (b k) (p k)=
fderiv ℝ (fderiv ℝ (movingPrefixEnergy a t (b k))) (p k)+
fderiv ℝ (fderiv ℝ (fun r=>C (movingPrefixChart a c t (b k) r))) (p k) := by
filter_upwards [hbsource,hzsource,he] with k hbk hzk hek
exact coordinateCenterMatrix_chart ((extChartAt 𝓘(ℝ,Model n) a).map_source hbk) (hek ▸ hzk)
have hDz : ∀ᶠ k in atTop,
(∀ᶠ y in 𝓝 (movingPrefixChart a c t (b k) (p k)),DifferentiableAt ℝ F y ∧ DifferentiableAt ℝ C y) ∧
DifferentiableAt ℝ (fderiv ℝ F) (movingPrefixChart a c t (b k) (p k)) ∧
DifferentiableAt ℝ (fderiv ℝ C) (movingPrefixChart a c t (b k) (p k)) := by
filter_upwards [(hzCoord.comp hσ.tendsto_atTop).eventually (eventually_eventually_nhds.mpr hnear),
hσ.tendsto_atTop.eventually hsample,hzEq] with k hnk hsk hek
rw [hek]
exact ⟨hnk,hsk.1,hsk.2.1⟩
have hpos : ∀ᶠ k in atTop,∀ r:Model n,
0≤(fderiv ℝ (fderiv ℝ (movingPrefixEnergy a t (b k))) (p k)-
fderiv ℝ (fderiv ℝ (fun r=>F (movingPrefixChart a c t (b k) r))) (p k)) r r := by
filter_upwards [hM,hL_eq] with k hk he; rw [←he]; exact hk.1
have hker : ∀ᶠ k in atTop,∀ i,∀ r:Model n,
(fderiv ℝ (fderiv ℝ (movingPrefixEnergy a t (b k))) (p k)-
fderiv ℝ (fderiv ℝ (fun r=>F (movingPrefixChart a c t (b k) r))) (p k)) (pj k i-p k) r=0 := by
filter_upwards [hM,hL_eq] with k hk he; rw [←he]; exact hk.2.2.1
obtain ⟨H,L,hH,hL,hpositive,hkernel,hupper⟩ := first_fixed_prefix_matrix_limit
henergy hg hb hp (fun i=>tendsto_pi_nhds.mp hpj i)
((hjF.comp hσ.tendsto_atTop).congr' (hzEq.mono (fun k hk=>congrArg (fderiv ℝ F) hk.symm)))
((hjC.comp hσ.tendsto_atTop).congr' (hzEq.mono (fun k hk=>congrArg (fderiv ℝ C) hk.symm)))
((hAf.comp hσ.tendsto_atTop).congr' (hzEq.mono (fun k hk=>congrArg (fderiv ℝ (fderiv ℝ F)) hk.symm)))
((hAc.comp hσ.tendsto_atTop).congr' (hzEq.mono (fun k hk=>congrArg (fderiv ℝ (fderiv ℝ C)) hk.symm)))
(hDz.mono (fun k hk=>⟨hk.1.mono (fun y hy=>hy.1),hk.2.1⟩))
(hDz.mono (fun k hk=>⟨hk.1.mono (fun y hy=>hy.2),hk.2.2⟩)) hgrad hquad hpos hker
have hH' := hH.congr' (hH_eq.mono (fun k hk=>hk.symm))
have hL' := hL.congr' (hL_eq.mono (fun k hk=>hk.symm))
refine ⟨⟨σ,hσ,pj,w,pj₀,w₀,H,L,(fun i=>(mem_active_coordinates hKs (ha i)).2.1),hn,h1,hbar,hpj,hw,hH',hL',hpositive,
fun r s=>bilinear_limit_symm hL' (hM.mono (fun k hk=>hk.2.1)) r s,?_,hupper,?_⟩⟩
· exact hkernel
· filter_upwards [hM] with k hk
exact ⟨(hs k).2.1,(hs k).2.2.1,(hs k).2.2.2,hk.2.2.2.1,hk.2.2.2.2⟩
end WeakMTWTransport
end
end OAI
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