| import Mathlib |
| import OAI.Analysis.BiholderTransport.LocalFlow.CotangentFlow |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| section PhaseJetAlgebra |
| variable {E:Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| variable {P:Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] |
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| lemma limit_from_comp_near_id {Aj Hj:ℕ → E →L[ℝ] P} {Rj:ℕ → E →L[ℝ] E} |
| {H:E →L[ℝ] P} (hR:Tendsto Rj atTop (𝓝 1)) |
| (hH:Tendsto Hj atTop (𝓝 H)) |
| (he:∀ᶠ j in atTop,(Aj j).comp (Rj j)=Hj j) : Tendsto Aj atTop (𝓝 H) := by |
| have hi:Tendsto (fun j=>Ring.inverse (Rj j)) atTop (𝓝 (1:E →L[ℝ] E)) := by |
| simpa only [Function.comp_def, Units.val_one, Ring.inverse_one] using (NormedRing.inverse_continuousAt (1:(E →L[ℝ] E)ˣ)).tendsto.comp hR |
| have hu:∀ᶠ j in atTop,IsUnit (Rj j) := hR.eventually (Units.isOpen.mem_nhds isUnit_one) |
| have hl:= (ContinuousLinearMap.compL ℝ E E P).continuous₂.tendsto (H,1) |>.comp (hH.prodMk_nhds hi) |
| change Tendsto (fun j=>(Hj j).comp (Ring.inverse (Rj j))) atTop (𝓝 H) at hl |
| apply hl.congr' |
| filter_upwards [he,hu] with j hej huj |
| ext v |
| have hh:=congrArg (fun T:E →L[ℝ] P=>T (Ring.inverse (Rj j) v)) hej |
| have hr:Rj j (Ring.inverse (Rj j) v)=v := |
| congrArg (fun T:E →L[ℝ] E=>T v) (Ring.mul_inverse_cancel (Rj j) huj) |
| simpa only [ContinuousLinearMap.comp_apply,hr] using hh.symm |
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| lemma phase_graph_derivative_limit |
| {Dj:ℕ → (E×P) →L[ℝ] (E×P)} {Hj Aj:ℕ → E →L[ℝ] P} |
| {Rj:ℕ → E →L[ℝ] E} {H:E →L[ℝ] P} |
| (hD:Tendsto Dj atTop (𝓝 (ContinuousLinearMap.id ℝ (E×P)))) |
| (hH:Tendsto Hj atTop (𝓝 H)) |
| (he:∀ᶠ j in atTop,(Dj j).comp ((ContinuousLinearMap.id ℝ E).prod (Hj j))= |
| (Rj j).prod ((Aj j).comp (Rj j))) : |
| Tendsto Rj atTop (𝓝 1) ∧ Tendsto Aj atTop (𝓝 H) := by |
| have hpair:Tendsto (fun j=>(ContinuousLinearMap.id ℝ E).prod (Hj j)) atTop |
| (𝓝 ((ContinuousLinearMap.id ℝ E).prod H)) := (ContinuousLinearMap.prodₗᵢ ℝ).continuous.tendsto _ |>.comp (tendsto_const_nhds.prodMk_nhds hH) |
| have hd:= (ContinuousLinearMap.compL ℝ E (E×P) (E×P)).continuous₂.tendsto _ |>.comp (hD.prodMk_nhds hpair) |
| change Tendsto (fun j=>(Dj j).comp ((ContinuousLinearMap.id ℝ E).prod (Hj j))) atTop |
| (𝓝 ((ContinuousLinearMap.id ℝ (E×P)).comp ((ContinuousLinearMap.id ℝ E).prod H))) at hd |
| rw [ContinuousLinearMap.id_comp] at hd |
| have hd':Tendsto (fun j=>(Rj j).prod ((Aj j).comp (Rj j))) atTop |
| (𝓝 ((ContinuousLinearMap.id ℝ E).prod H)) := hd.congr' he |
| have hR:= (ContinuousLinearMap.compL ℝ E (E×P) E (ContinuousLinearMap.fst ℝ E P)).continuous.tendsto _ |>.comp hd' |
| have hA:= (ContinuousLinearMap.compL ℝ E (E×P) P (ContinuousLinearMap.snd ℝ E P)).continuous.tendsto _ |>.comp hd' |
| change Tendsto (fun j=>(ContinuousLinearMap.fst ℝ E P).comp ((Rj j).prod ((Aj j).comp (Rj j)))) atTop |
| (𝓝 ((ContinuousLinearMap.fst ℝ E P).comp ((ContinuousLinearMap.id ℝ E).prod H))) at hR |
| simp only [ContinuousLinearMap.fst_comp_prod] at hR |
| change Tendsto (fun j=>(ContinuousLinearMap.snd ℝ E P).comp ((Rj j).prod ((Aj j).comp (Rj j)))) atTop |
| (𝓝 ((ContinuousLinearMap.snd ℝ E P).comp ((ContinuousLinearMap.id ℝ E).prod H))) at hA |
| simp only [ContinuousLinearMap.snd_comp_prod] at hA |
| exact ⟨hR,limit_from_comp_near_id hR hA (Eventually.of_forall (fun _=>rfl))⟩ |
| end PhaseJetAlgebra |
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| 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 [Nonempty M] [IsRiemannianManifold 𝓘(ℝ,Model n) M] in |
| lemma coordinate_phase_jet_limit {a:M} {q:Phase n} |
| (hq:q.1∈(extChartAt 𝓘(ℝ,Model n) a).target) |
| {τj:ℕ → ℝ} {qj:ℕ → Phase n} {Hj Aj:ℕ → Model n →L[ℝ] Model n →L[ℝ] ℝ} |
| {Rj:ℕ → Model n →L[ℝ] Model n} {H:Model n →L[ℝ] Model n →L[ℝ] ℝ} |
| (ht:Tendsto τj atTop (𝓝 0)) (hqj:Tendsto qj atTop (𝓝 q)) |
| (hH:Tendsto Hj atTop (𝓝 H)) |
| (he:∀ᶠ j in atTop, |
| (fderiv ℝ (fun v=>coordinatePhaseFlow a (τj j,v)) (qj j)).comp |
| ((ContinuousLinearMap.id ℝ (Model n)).prod (Hj j))= |
| (Rj j).prod ((Aj j).comp (Rj j))) : |
| Tendsto Rj atTop (𝓝 1) ∧ Tendsto Aj atTop (𝓝 H) := by |
| have hd := (coordinatePhaseFlow_derivative_limit hq).comp (ht.prodMk_nhds hqj) |
| exact phase_graph_derivative_limit hd hH he |
| end WeakMTWTransport |
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| end |
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| end OAI |
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