| import OAI.Dynamics.SmoothObstruction.SpectralBins |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory Filter Topology TopologicalSpace Set Finset |
| open scoped ENNReal NNReal BoundedContinuousFunction |
| open Finset MeasureTheory Filter Topology |
| open scoped BigOperators |
| open Finset Matrix |
| open scoped Matrix.Norms.L2Operator |
| open Matrix |
|
|
| namespace SmoothObstruction.CoordinateTubes |
| open Set Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] |
|
|
| / |
| It does not assert a smooth extension across a manifold boundary. -/ |
| def carrier (a : M) (U : Set M) : Set (EuclideanSpace ℝ (Fin d)) := |
| I.symm ⁻¹' ((chartAt H a).target∩(chartAt H a).symm ⁻¹' U) |
|
|
| lemma isOpen_carrier (a : M) {U : Set M} (hU : IsOpen U) : IsOpen (carrier I a U) := |
| ((chartAt H a).isOpen_inter_preimage_symm hU).preimage I.continuous_symm |
|
|
| lemma coord_mem_carrier (a x : M) {U : Set M} |
| (hx : x∈(chartAt H a).source) (hU : x∈U) : |
| (extChartAt I a) x∈carrier I a U := by |
| have ht := (extChartAt I a).map_source (by simpa only [extChartAt_source] using hx) |
| have htar : I.symm ((extChartAt I a) x)∈(chartAt H a).target := by |
| rw [extChartAt,OpenPartialHomeomorph.extend_target] at ht |
| exact ht.1 |
| refine ⟨htar,?_⟩ |
| change (extChartAt I a).symm ((extChartAt I a) x)∈U |
| rwa [(extChartAt I a).left_inv (by simpa only [extChartAt_source] using hx)] |
|
|
| lemma mem_of_coord_mem_carrier (a y : M) {U : Set M} |
| (hy : y∈(chartAt H a).source) (hu : (extChartAt I a) y∈carrier I a U) : y∈U := by |
| have hh := hu.2 |
| change (extChartAt I a).symm ((extChartAt I a) y)∈U at hh |
| rwa [(extChartAt I a).left_inv (by simpa only [extChartAt_source] using hy)] at hh |
|
|
| / |
| specified finite open enlargement, also in half-space models. -/ |
| theorem uniform_margin {ι : Type*} [Fintype ι] (a : ι → M) (K U : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hU : ∀ i,IsOpen (U i)) |
| (hKU : ∀ i,K i⊆U i) (hKs : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ δ : ℝ,0<δ ∧ ∀ i x,x∈K i → ∀ y,y∈(chartAt H (a i)).source → |
| ‖(extChartAt I (a i)) y-(extChartAt I (a i)) x‖≤δ → y∈U i := by |
| let L (i : ι) := (extChartAt I (a i)) '' K i |
| have hL (i : ι) : IsCompact (L i) := |
| (hK i).image_of_continuousOn ((continuousOn_extChartAt (I:=I) (a i)).mono |
| (fun _ hx => by simpa only [extChartAt_source] using hKs i hx)) |
| have hex (i : ι) : ∃ r : ℝ,0<r ∧ cthickening r (L i)⊆carrier I (a i) (U i) := by |
| apply (hL i).exists_cthickening_subset_open (isOpen_carrier I _ (hU i)) |
| rintro z ⟨x,hx,rfl⟩ |
| exact coord_mem_carrier I _ _ (hKs i hx) (hKU i hx) |
| choose r hr hu using hex |
| obtain ⟨δ,hδ,hδr⟩ := CarriedCharts.finite_positive_lower r hr |
| refine ⟨δ,hδ,fun i x hx y hy hxy => ?_⟩ |
| apply mem_of_coord_mem_carrier I _ _ hy |
| apply hu i |
| apply closedBall_subset_cthickening (E:=L i) ⟨x,hx,rfl⟩ (r i) |
| simpa only [mem_closedBall,dist_eq_norm] using hxy.trans (hδr i) |
|
|
| end SmoothObstruction.CoordinateTubes |
|
|
| namespace SmoothObstruction.CoordinateTubes |
| open Set Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] |
|
|
| / |
| carried points in any prescribed geometric atlas tube. -/ |
| theorem uniform_metric_tube {ι : Type*} [Fintype ι] |
| (a : ι → M) (K : ι → Set M) (hK : ∀ i,IsCompact (K i)) |
| (hKs : ∀ i,K i⊆(chartAt H (a i)).source) {r : ℝ} (hr : 0<r) : |
| ∃ δ : ℝ,0<δ ∧ ∀ i x,x∈K i → ∀ y,y∈(chartAt H (a i)).source → |
| ‖(extChartAt I (a i)) y-(extChartAt I (a i)) x‖≤δ → |
| y∈cthickening r (K i) := by |
| obtain ⟨δ,hδ,hh⟩ := uniform_margin I a K (fun i => thickening r (K i)) hK |
| (fun i => isOpen_thickening) (fun i => self_subset_thickening hr _) hKs |
| exact ⟨δ,hδ,fun i x hx y hy hxy => thickening_subset_cthickening r _ (hh i x hx y hy hxy)⟩ |
| end SmoothObstruction.CoordinateTubes |
|
|
| namespace SmoothObstruction.CoordinateTubes |
| open Set Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] |
|
|
| / |
| works with a closed model range, so includes the closed half-space without |
| postulating an ambient extension of an inverse chart. -/ |
| theorem uniform_inverse_modulus {ι : Type*} [Fintype ι] |
| (hclosed : IsClosed (range I)) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKs : ∀ i,K i⊆(chartAt H (a i)).source) |
| {ε : ℝ} (hε : 0<ε) : |
| ∃ δ : ℝ,0<δ ∧ ∀ i x,x∈K i → ∀ y,y∈(chartAt H (a i)).source → |
| ‖(extChartAt I (a i)) y-(extChartAt I (a i)) x‖≤δ → dist y x<ε := by |
| let L (i : ι) := (extChartAt I (a i)) '' K i |
| have hL (i : ι) : IsCompact (L i) := |
| (hK i).image_of_continuousOn ((continuousOn_extChartAt (I:=I) (a i)).mono |
| (fun _ hx => by simpa only [extChartAt_source] using hKs i hx)) |
| have hex (i : ι) : ∃ r : ℝ,0<r ∧ cthickening r (L i)⊆carrier I (a i) univ := by |
| apply (hL i).exists_cthickening_subset_open (isOpen_carrier I _ isOpen_univ) |
| rintro z ⟨x,hx,rfl⟩ |
| exact coord_mem_carrier I _ _ (hKs i hx) (mem_univ _) |
| choose r hr hu using hex |
| let Q (i : ι) := cthickening (r i) (L i)∩range I |
| have hQt (i : ι) : Q i⊆(extChartAt I (a i)).target := by |
| intro z hz |
| rw [extChartAt,OpenPartialHomeomorph.extend_target] |
| exact ⟨(hu i hz.1).1,hz.2⟩ |
| have hQ (i : ι) : IsCompact (Q i) := (hL i).cthickening.inter_right hclosed |
| have hm (i : ι) : ∃ s : ℝ,0<s ∧ ∀ u∈Q i,∀ v∈Q i,dist u v<s → |
| dist ((extChartAt I (a i)).symm u) ((extChartAt I (a i)).symm v)<ε := |
| Metric.uniformContinuousOn_iff.mp ((hQ i).uniformContinuousOn_of_continuous |
| ((continuousOn_extChartAt_symm (I:=I) (a i)).mono (hQt i))) ε hε |
| choose s hs hm using hm |
| obtain ⟨δ,hδ,hδi⟩ := CarriedCharts.finite_positive_lower |
| (fun i => min (r i) (s i/2)) (fun i => lt_min (hr i) (div_pos (hs i) (by norm_num))) |
| refine ⟨δ,hδ,fun i x hx y hy hxy => ?_⟩ |
| have hxt : x∈(extChartAt I (a i)).source := by simpa only [extChartAt_source] using hKs i hx |
| have hyt : y∈(extChartAt I (a i)).source := by simpa only [extChartAt_source] using hy |
| have hxQ : (extChartAt I (a i)) x∈Q i := ⟨self_subset_cthickening (L i) ⟨x,hx,rfl⟩, |
| by exact ⟨(chartAt H (a i)) x,rfl⟩⟩ |
| have hyQ : (extChartAt I (a i)) y∈Q i := ⟨ |
| mem_cthickening_of_dist_le _ _ _ _ ⟨x,hx,rfl⟩ |
| (by simpa only [dist_eq_norm] using hxy.trans ((hδi i).trans (min_le_left _ _))), |
| ⟨(chartAt H (a i)) y,rfl⟩⟩ |
| have hh := hm i _ hyQ _ hxQ (show dist ((extChartAt I (a i)) y) ((extChartAt I (a i)) x)<s i from |
| (by rw [dist_eq_norm]; exact lt_of_le_of_lt (hxy.trans ((hδi i).trans (min_le_right _ _))) (by linarith [hs i]))) |
| simpa only [(extChartAt I (a i)).left_inv hyt,(extChartAt I (a i)).left_inv hxt] using hh |
| end SmoothObstruction.CoordinateTubes |
|
|
| namespace SmoothObstruction.CoordinateTubes |
| open Set Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] |
|
|
| / |
| actual inverse-chart domain. This is a target statement, not an assertion |
| about the total value assigned to a partial chart outside its source. -/ |
| theorem uniform_target_tube {ι : Type*} [Fintype ι] |
| (a : ι → M) (K : ι → Set M) (hK : ∀ i,IsCompact (K i)) |
| (hKs : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r : ℝ,0<r ∧ ∀ i x,x∈K i → |
| closedBall ((extChartAt I (a i)) x) r∩range I⊆(extChartAt I (a i)).target := by |
| let L (i : ι) := (extChartAt I (a i)) '' K i |
| have hL (i : ι) : IsCompact (L i) := |
| (hK i).image_of_continuousOn ((continuousOn_extChartAt (I:=I) (a i)).mono |
| (fun _ hx => by simpa only [extChartAt_source] using hKs i hx)) |
| have hex (i : ι) : ∃ r : ℝ,0<r ∧ cthickening r (L i)⊆carrier I (a i) univ := by |
| apply (hL i).exists_cthickening_subset_open (isOpen_carrier I _ isOpen_univ) |
| rintro z ⟨x,hx,rfl⟩ |
| exact coord_mem_carrier I _ _ (hKs i hx) (mem_univ _) |
| choose r hr hu using hex |
| obtain ⟨δ,hδ,hδr⟩ := CarriedCharts.finite_positive_lower r hr |
| refine ⟨δ,hδ,fun i x hx z hz => ?_⟩ |
| rw [extChartAt,OpenPartialHomeomorph.extend_target] |
| exact ⟨(hu i ((closedBall_subset_cthickening (E:=L i) ⟨x,hx,rfl⟩ (r i)) |
| (closedBall_subset_closedBall (hδr i) hz.1))).1,hz.2⟩ |
| end SmoothObstruction.CoordinateTubes |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology Metric MeasureTheory |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| not a recomputed QR path or a newly chosen stationary witness. -/ |
| def BilateralGauge.restrict (S : Diffeomorph I I M M ∞) (a : ι → M) (c : M → ι) |
| {n N : ℕ} {B R : ℝ} {x : M} (A : BilateralGauge I S a c n B R x) (hN : N≤n) : |
| BilateralGauge I S a c N B R x where |
| lam := A.lam |
| rate_bound := A.rate_bound |
| UF := A.UF |
| VF := A.VF |
| UB := A.UB |
| VB := A.VB |
| invF := A.invF |
| invB := A.invB |
| centerU := A.centerU |
| centerV := A.centerV |
| normF t ht := A.normF t (ht.trans hN) |
| normB t ht := A.normB t (ht.trans hN) |
| stepF t ht := A.stepF t (ht.trans_le hN) |
| stepB t ht := A.stepB t (ht.trans_le hN) |
|
|
| omit [ProperSpace M] in |
| / |
| Membership of the nearby orbit at ALL intermediate times is a conclusion. -/ |
| theorem actual_tube_comparison (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) |
| {τ : ℝ} (hτ : 0<τ) : |
| ∃ r C : ℝ,0<r ∧ 1≤C ∧ ∀ (c : ℕ → ι) (x y : M) (N : ℕ), |
| (∀ t≤N,S^[t] x∈K (c t)) → I.IsInteriorPoint x → I.IsInteriorPoint y → |
| y∈(chartAt H (a (c 0))).source → |
| C ^ N*‖(extChartAt I (a (c 0))) y-(extChartAt I (a (c 0))) x‖≤r → |
| (∀ t≤N,S^[t] y∈(chartAt H (a (c t))).source ∧ |
| S^[t] y∈cthickening τ (K (c t))) ∧ |
| ‖fderiv ℝ (ShortDerivative.iterate (fun t => localMap I S (a (c t)) (a (c (t+1)))) N) |
| ((extChartAt I (a (c 0))) y)- |
| fderiv ℝ (ShortDerivative.iterate (fun t => localMap I S (a (c t)) (a (c (t+1)))) N) |
| ((extChartAt I (a (c 0))) x)‖≤ |
| N*C*C ^ (2*N)*‖(extChartAt I (a (c 0))) y-(extChartAt I (a (c 0))) x‖ := by |
| obtain ⟨r,C,hr,hC,hh⟩ := actual_short_comparison I hclosed S a K hK hKU |
| obtain ⟨δ,hδ,hmargin⟩ := CoordinateTubes.uniform_metric_tube I a K hK hKU hτ |
| refine ⟨min r δ,C,lt_min hr hδ,hC,fun c x y N hx hxi hyi hys hsmall => ?_⟩ |
| obtain ⟨ht,hd⟩ := hh c x y N hx hxi hyi hys (hsmall.trans (min_le_left _ _)) |
| refine ⟨fun t htn => ⟨(ht t htn).1,?_⟩,hd⟩ |
| apply hmargin (c t) (S^[t] x) (hx t htn) _ (ht t htn).1 |
| apply (ht t htn).2.trans |
| apply le_trans _ (hsmall.trans (min_le_right _ _)) |
| exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ hC htn) (norm_nonneg _) |
|
|
| / |
| reference chart codes of a nearby orbit. No continued chart membership, |
| frame continuity, or derivative comparison is an extra hypothesis. -/ |
| theorem uniform_genuine_bin_gauges (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ rF CF rB CB D : ℝ,0<rF ∧ 1≤CF ∧ 0<rB ∧ 1≤CB ∧ 1≤D ∧ |
| ∀ (c : M → ι) (J : Type*) (x : J → M) (e : J) (cF cB : ℕ → ι) |
| (n N : ℕ) (B R h : ℝ) (A : ∀ i,BilateralGauge I S a c n B R (x i)), |
| (∀ y,y∈K (c y)) → N≤n → cB 0=cF 0 → |
| (∀ t≤N,S^[t] (x e)∈K (cF t)) → |
| (∀ t≤N,S.symm^[t] (x e)∈K (cB t)) → |
| (∀ i,I.IsInteriorPoint (x i)) → |
| (∀ i,x i∈(chartAt H (a (cF 0))).source) → |
| (∀ i,‖(extChartAt I (a (cF 0))) (x i)-(extChartAt I (a (cF 0))) (x e)‖≤h) → |
| CF^N*h≤rF → CB^N*h≤rB → |
| ∃ F Z : J → ShortBin.Gauge d N (B+Real.log D), |
| (∀ i k,(F i).lam k=(A i).lam k ∧ (Z i).lam k= -(A i).lam k) ∧ |
| (∀ i,(Z i).U 0=(F i).U 0 ∧ (Z i).V 0=(F i).V 0) ∧ |
| (∀ i,∀ t<N,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (cF t)) (a (cF (t+1))) (S^[t] (x i)))*(F i).U t= |
| (F i).U (t+1)*EuclideanSplitting.rate (F i).lam 1) ∧ |
| (∀ i,∀ t<N,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S.symm (a (cB t)) (a (cB (t+1))) (S.symm^[t] (x i)))*(Z i).U t= |
| (Z i).U (t+1)*EuclideanSplitting.rate (Z i).lam 1) ∧ |
| (∀ i,‖(F i).product-(F e).product‖≤N*CF*CF^(2*N)*h) ∧ |
| (∀ i,‖(Z i).product-(Z e).product‖≤N*CB*CB^(2*N)*h) ∧ |
| ∀ i t,t≤N → |
| (F i).U t=(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (c (S^[t] (x i)))) (a (cF t)) (S^[t] (x i)))*(A i).UF t ∧ |
| (F i).V t=(A i).VF t*(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (cF t)) (a (c (S^[t] (x i)))) (S^[t] (x i))) ∧ |
| (Z i).U t=(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (c (S.symm^[t] (x i)))) (a (cB t)) (S.symm^[t] (x i)))*(A i).UB t ∧ |
| (Z i).V t=(A i).VB t*(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (cB t)) (a (c (S.symm^[t] (x i)))) (S.symm^[t] (x i))) := by |
| classical |
| obtain ⟨τ,D,hτ,hD,hcarry⟩ := uniform_carried_bilateral_explicit I a K hK hKU |
| obtain ⟨rF,CF,hrF,hCF,hhF⟩ := actual_tube_comparison I hclosed S a K hK hKU hτ |
| obtain ⟨rB,CB,hrB,hCB,hhB⟩ := actual_tube_comparison I hclosed S.symm a K hK hKU hτ |
| refine ⟨rF,CF,rB,CB,D,hrF,hCF,hrB,hCB,hD, |
| fun c J x e cF cB n N B R h A hc hN h0 hxF hxB hint hsource hdist hsF hsB => ?_⟩ |
| have hdistB (i : J) : ‖(extChartAt I (a (cB 0))) (x i)-(extChartAt I (a (cB 0))) (x e)‖≤h := by |
| simpa only [h0] using hdist i |
| have hsourceB (i : J) : x i∈(chartAt H (a (cB 0))).source := by |
| simpa only [h0] using hsource i |
| have htF (i : J) := hhF cF (x e) (x i) N hxF (hint e) (hint i) (hsource i) |
| ((mul_le_mul_of_nonneg_left (hdist i) (pow_nonneg (zero_le_one.trans hCF) N)).trans hsF) |
| have htB (i : J) := hhB cB (x e) (x i) N hxB (hint e) (hint i) (hsourceB i) |
| ((mul_le_mul_of_nonneg_left (hdistB i) (pow_nonneg (zero_le_one.trans hCB) N)).trans hsB) |
| choose F Z hFlam hZlam hU hV hstepF hstepB hform using fun i => |
| hcarry S c (x i) N B R ((A i).restrict I S a c hN) cF cB hc h0 |
| (fun t ht => ((htF i).1 t ht).2) (fun t ht => ((htB i).1 t ht).2) |
| refine ⟨F,Z,fun i k => ⟨hFlam i k,hZlam i k⟩,fun i => ⟨hU i,hV i⟩,hstepF,hstepB,?_,?_,?_⟩ |
| · intro i |
| have hp (j : J) := actual_gauge_product I S (fun t => a (cF t)) (x j) (F j) |
| (fun t ht => ((htF j).1 t ht).1) (hint j) (hstepF j) |
| rw [←hp i,←hp e] |
| exact ((htF i).2).trans (mul_le_mul_of_nonneg_left (hdist i) (by positivity)) |
| · intro i |
| have hp (j : J) := actual_gauge_product I S.symm (fun t => a (cB t)) (x j) (Z j) |
| (fun t ht => ((htB j).1 t ht).1) (hint j) (hstepB j) |
| rw [←hp i,←hp e] |
| exact ((htB i).2).trans (mul_le_mul_of_nonneg_left (hdistB i) (by positivity)) |
| · exact hform |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
