| import OAI.Dynamics.SmoothObstruction.SwitchRecovery |
| import OAI.Dynamics.SmoothObstruction.EndpointFrames |
|
|
| namespace OAI |
|
|
| section |
|
|
| open scoped BigOperators |
| open scoped ContDiff BigOperators |
| open Module |
| open Filter Topology |
| open scoped Real |
|
|
| namespace SmoothObstruction.SmoothScales |
| lemma tendsto_zero_of_square_summable {b : ℕ → ℝ} (hb : ∀ k, 0 ≤ b k) |
| (hs : Summable (fun k => (b k)^2)) : Tendsto b atTop (𝓝 0) := by |
| have h := Real.continuous_sqrt.continuousAt.tendsto.comp hs.tendsto_atTop_zero |
| simpa only [Function.comp_def, Real.sqrt_sq (hb _), Real.sqrt_zero] using h |
|
|
| lemma summable_B_sixteen {b : ℕ → ℝ} (hb : ∀ k, 0 ≤ b k) |
| (hs : Summable (fun k => (b k)^2)) : Summable (fun k => (B (b k))^16) := by |
| exact hs.congr (fun k => (root_sixteen (hb k)).symm) |
|
|
| lemma dyadic_floor (k : ℕ) : |
| ((2 : ℝ)^k) ^ (-1 / 10 : ℝ) = (2 : ℝ)^(-(k : ℝ) / 10) := by |
| rw [← Real.rpow_natCast, ← Real.rpow_mul (show (0 : ℝ) ≤ 2 by norm_num)] |
| congr 1 |
| ring |
| end SmoothObstruction.SmoothScales |
|
|
| namespace SmoothObstruction.ExteriorStability |
| open RelativeProduct Asymptotics |
|
|
| /-- Full asymptotic conclusion in source Lemma `lem:relative-exterior`, at the |
| literal dyadic scales and with the literal summable chord sequence. The same |
| proof supplies one rate of convergence for every matrix family satisfying the |
| fixed displayed constants, since the bounding sequence in |
| `uniform_relative_small` depends only on those constants and `b`. -/ |
| theorem relative_exterior_dyadic (d : ℕ) (_hd : 1 ≤ d) (K : ℝ) (_hK : 1 ≤ K) |
| (p : ℕ) (hp : p ≤ d) |
| (b : ℕ → ℝ) (hbpos : ∀ k, 0 < b k) (hbsum : Summable (fun k => (b k)^2)) |
| (hbfloor : ∀ k : ℕ, (2 : ℝ)^(-(k : ℝ) / 10) ≤ b k) |
| (m : ℕ → ℕ) |
| (A B : ℕ → ℕ → EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)) |
| (σ a ε : ℕ → ℝ) (C₀ C₁ C₂ C₃ : ℝ) (hC₀ : 0 < C₀) (hC₃ : 0 ≤ C₃) |
| (hdata : ∀ᶠ k in atTop, |
| 1 ≤ m k ∧ (m k : ℝ) ≤ 2 * (2 : ℝ)^k ∧ |
| 0 ≤ a k ∧ a k ≤ C₁ * b k * (2 : ℝ)^k ∧ |σ k| ≤ C₂ ∧ |
| 0 ≤ ε k ∧ ε k ≤ 1 ∧ |
| ε k ≤ C₃ * Real.exp (-(3 / 2 : ℝ) * SmoothScales.B (b k) * (2 : ℝ)^k) ∧ |
| (∀ t < m k, ‖A k t‖ ≤ K) ∧ |
| (∀ t < m k, ‖B k t - A k t‖ ≤ ε k) ∧ |
| (∀ s q : ℕ, s + q ≤ m k → |
| ‖exterior p (segment (A k) s q)‖ ≤ Real.exp (σ k * q + a k))) : |
| (fun k => exterior p (segment (B k) 0 (m k)) - exterior p (segment (A k) 0 (m k))) |
| =o[atTop] (fun k => Real.exp (σ k * m k - C₀ * b k * (2 : ℝ)^k)) := by |
| obtain ⟨C, hC, hbound⟩ := relative_exterior_core (E := EuclideanSpace ℝ (Fin d)) d K |
| have hsmall : (fun k => ‖exterior p (segment (B k) 0 (m k)) - |
| exterior p (segment (A k) 0 (m k))‖) =o[atTop] |
| (fun k => Real.exp (σ k * m k - C₀ * b k * (2 : ℝ)^k)) := by |
| apply (Asymptotics.isLittleO_iff_tendsto (fun k hk => (Real.exp_ne_zero _ hk).elim)).mpr |
| apply RelativeAsymptotics.uniform_relative_small (Filter.Eventually.of_forall hbpos) |
| (SmoothScales.tendsto_zero_of_square_summable (fun k => (hbpos k).le) hbsum) |
| (tendsto_pow_atTop_atTop_of_one_lt (show (1 : ℝ) < 2 by norm_num)) |
| (Filter.Eventually.of_forall (fun k => by simpa only [SmoothScales.dyadic_floor] using hbfloor k)) |
| C C₀ C₁ C₂ C₃ hC hC₀.le hC₃ |
| filter_upwards [hdata] with k hk |
| rcases hk with ⟨_, hm, ha, hale, hσ, hε, hε₁, hεle, hA, hBA, hseg⟩ |
| exact ⟨ContinuousLinearMap.opNorm_nonneg _, ha, hε, hm, hale, hσ, hεle, |
| hbound p hp (A k) (B k) (m k) (σ k) (a k) (ε k) hε hε₁ hA hBA hseg⟩ |
| apply Asymptotics.isLittleO_iff.mpr |
| intro c hc |
| filter_upwards [hsmall.bound hc] with k hk |
| simp only [Real.norm_eq_abs] at hk ⊢ |
| exact (le_abs_self _).trans hk |
| end SmoothObstruction.ExteriorStability |
|
|
| namespace SmoothObstruction.RelativeAsymptotics |
| open Filter Topology |
|
|
| /-- Exterior relative errors absorb every fixed exponential gauge cost. |
| The multiplicative nonlinear error remains the authentic envelope times |
| its exponential, not a dimension- or bin-dependent replacement. -/ |
| theorem weighted_envelope_tendsto {α : Type*} {l : Filter α} {b n : α → ℝ} |
| (hbpos : ∀ᶠ k in l,0<b k) (hb : Tendsto b l (𝓝 0)) |
| (hn : Tendsto n l atTop) (hfloor : ∀ᶠ k in l,n k^(-1/10:ℝ)≤b k) |
| (C C₁ C₂ C₃ A K : ℝ) : |
| Tendsto (fun k => A*Real.exp (K*b k*n k)* |
| (envelope C 0 C₁ C₂ C₃ (b k) (n k)* |
| Real.exp (envelope C 0 C₁ C₂ C₃ (b k) (n k)))) l (𝓝 0) := by |
| have hv (J : ℝ) : Tendsto (fun k => envelope C J C₁ C₂ C₃ (b k) (n k)) l (𝓝 0) := by |
| simpa only [envelope,mul_zero] using |
| (SmoothScales.decisive_decay hbpos hb hn hfloor (2*C₁+J)).const_mul |
| (2*C*C₃*Real.exp C₂) |
| have hh := ((hv K).mul ((Real.continuous_exp.tendsto 0).comp (hv 0))).const_mul A |
| simp only [Real.exp_zero,mul_one,mul_zero] at hh |
| apply hh.congr |
| intro k |
| dsimp only [envelope,Function.comp_apply] |
| rw [show -(3/2:ℝ)*SmoothScales.B (b k)*n k+(2*C₁+K)*b k*n k= |
| K*b k*n k+(-(3/2:ℝ)*SmoothScales.B (b k)*n k+(2*C₁+0)*b k*n k) by ring, |
| Real.exp_add] |
| ring |
|
|
| /-- Literal leading-minor relative perturbation parameter tends to zero. -/ |
| theorem common_relative_budget {α : Type*} {l : Filter α} {b n : α → ℝ} |
| (hbpos : ∀ᶠ k in l,0<b k) (hb : Tendsto b l (𝓝 0)) |
| (hn : Tendsto n l atTop) (hfloor : ∀ᶠ k in l,n k^(-1/10:ℝ)≤b k) |
| (C Crel C₁ C₂ C₃ D W : ℝ) (p : ℕ) : |
| Tendsto (fun k => 2*((C*(Real.exp (D*b k*n k))^p)^2*(C*2^p))*C* |
| (envelope Crel 0 C₁ C₂ C₃ (b k) (n k)* |
| Real.exp (envelope Crel 0 C₁ C₂ C₃ (b k) (n k)))*W) l (𝓝 0) := by |
| apply (weighted_envelope_tendsto hbpos hb hn hfloor Crel C₁ C₂ C₃ |
| (2*C^4*2^p*W) ((2*p:ℕ)*D)).congr |
| intro k |
| have he : Real.exp (((2*p:ℕ):ℝ)*D*b k*n k)=(Real.exp (D*b k*n k))^(2*p) := by |
| rw [←Real.exp_nat_mul] |
| congr 1 |
| ring |
| rw [he] |
| simp only [Nat.mul_comm 2 p,pow_mul] |
| ring |
| end SmoothObstruction.RelativeAsymptotics |
|
|
| namespace SmoothObstruction.EndpointBudget |
| open Filter Topology |
| noncomputable def leading (C K : ℝ) (p : ℕ) : ℝ := (C*(Real.exp K)^p)^2*(C*2^p) |
| noncomputable def normCost (C K N : ℝ) (p : ℕ) : ℝ := C*(C^2*(Real.exp K)^(2*p)*N) |
| noncomputable def mixedCost (C K N V : ℝ) (p : ℕ) : ℝ := C*(C*(Real.exp K)^p)^2*N*V |
| noncomputable def relative (C Crel C₁ C₂ C₃ D V b n : ℝ) (p : ℕ) : ℝ := |
| 2*leading C (D*b*n) p*C* |
| (RelativeAsymptotics.envelope Crel 0 C₁ C₂ C₃ b n* |
| Real.exp (RelativeAsymptotics.envelope Crel 0 C₁ C₂ C₃ b n))*V |
| noncomputable def graph (C D N V W b n δ : ℝ) (p : ℕ) : ℝ := |
| 2*leading C (D*b*n) p* |
| (mixedCost C (D*b*n) N V p*Real.exp (-2*b^(1/2:ℝ)*n)+normCost C (D*b*n) N p*W*δ) |
|
|
| /-- One eventual scalar budget works at ALL ranks, before choosing any |
| starting cell, endpoint label, good gauge, or input frame. -/ |
| theorem all_ranks {α : Type*} {l : Filter α} {b n δ : α → ℝ} |
| (hbpos : ∀ᶠ k in l,0<b k) (hb : Tendsto b l (𝓝 0)) |
| (hn : Tendsto n l atTop) (hfloor : ∀ᶠ k in l,n k^(-1/10:ℝ)≤b k) |
| (d : ℕ) (C Crel C₁ C₂ C₃ D c : ℝ) (hc : 0<c) |
| (F N V W β : Fin (d+1) → ℝ) |
| (hF : ∀ p,0≤F p) (hN : ∀ p,0≤N p) (hV : ∀ p,0≤V p) |
| (hW : ∀ p,0≤W p) (hβ : ∀ p,0<β p) |
| (hδ : ∀ᶠ k in l,0≤δ k ∧ δ k≤Real.exp (-c*b k^(3/4:ℝ)*n k)) : |
| ∀ᶠ k in l,∀ p : Fin (d+1), |
| (F p*δ k)*(1+leading C (D*b k*n k) p.val*normCost C (D*b k*n k) (N p) p.val)≤1/2 ∧ |
| relative C Crel C₁ C₂ C₃ D (V p) (b k) (n k) p.val≤1/100 ∧ |
| (graph C D (N p) (V p) (W p) (b k) (n k) (δ k) p.val+ |
| relative C Crel C₁ C₂ C₃ D (V p) (b k) (n k) p.val)/ |
| (1-relative C Crel C₁ C₂ C₃ D (V p) (b k) (n k) p.val)≤β p := by |
| apply Filter.eventually_all.mpr |
| intro p |
| have hf := SmoothScales.common_frame_budget (C:=C) hbpos hb hn hfloor hc (hF p) (hN p) D p.val hδ |
| have hr := RelativeAsymptotics.common_relative_budget hbpos hb hn hfloor C Crel C₁ C₂ C₃ D (V p) p.val |
| have hg := SmoothScales.common_graph_budget (C:=C) hbpos hb hn hfloor hc (hN p) (hV p) (hW p) D p.val hδ |
| change Tendsto (fun k => relative C Crel C₁ C₂ C₃ D (V p) (b k) (n k) p.val) l (𝓝 0) at hr |
| change Tendsto (fun k => graph C D (N p) (V p) (W p) (b k) (n k) (δ k) p.val) l (𝓝 0) at hg |
| filter_upwards [hf,hr.eventually_le_const (by norm_num : (0:ℝ)<1/100), |
| (SmoothScales.perturbed_graph_decay hg hr).eventually_le_const (hβ p)] with k hk hrk hgk |
| exact ⟨hk,hrk,hgk⟩ |
| end SmoothObstruction.EndpointBudget |
|
|
| end |
|
|
| section |
|
|
| namespace SmoothObstruction.AtlasCores |
| open Set Filter Topology |
| variable {X Y ι : Type*} [TopologicalSpace X] [TopologicalSpace Y] [Fintype ι] |
|
|
| /-- A finite closed atlas selector can only approach charts whose compact |
| carrier contains the limiting point. It need not be a continuous selector. -/ |
| theorem closed_selector_germ (K : ι → Set Y) (hK : ∀ i,IsClosed (K i)) |
| (q : Y → ι) (hq : ∀ y,y∈K (q y)) (y : Y) : |
| ∀ᶠ z in 𝓝 y,y∈K (q z) := by |
| have hall : ∀ i,∀ᶠ z in 𝓝 y,q z=i → y∈K i := by |
| intro i |
| by_cases hy : y∈K i |
| · exact Eventually.of_forall (fun _ _ => hy) |
| · filter_upwards [(hK i).isOpen_compl.mem_nhds hy] with z hz he |
| exact (hz (he ▸ hq z)).elim |
| filter_upwards [Filter.eventually_all.mpr hall] with z hz |
| exact hz (q z) rfl |
|
|
| /-- Local analytic branches may be checked simultaneously in finitely many |
| possible chart codes, without adding those codes to any global partition. -/ |
| theorem eventually_selected (K : ι → Set Y) (hK : ∀ i,IsClosed (K i)) |
| (q : Y → ι) (hq : ∀ y,y∈K (q y)) (f : X → Y) (x : X) |
| (hf : ContinuousAt f x) (V : ι → Set X) |
| (hV : ∀ i,f x∈K i → V i∈𝓝 x) : |
| ∀ᶠ u in 𝓝 x,u∈V (q (f u)) := by |
| have hcodes := hf.eventually (closed_selector_germ K hK q hq (f x)) |
| have hall : ∀ i,∀ᶠ u in 𝓝 x,f x∈K i → u∈V i := by |
| intro i |
| by_cases hi : f x∈K i |
| · filter_upwards [hV i hi] with u hu using fun _ => hu |
| · exact Eventually.of_forall (fun _ h => (hi h).elim) |
| filter_upwards [hcodes,Filter.eventually_all.mpr hall] with u hu hv |
| exact hv (q (f u)) hu |
| end SmoothObstruction.AtlasCores |
|
|
| namespace SmoothObstruction.AtlasCores |
| open Set Filter Topology |
| variable {X Y E ι : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
| [NormedAddCommGroup E] [Fintype ι] |
|
|
| /-- A smooth local branch in all possible old charts satisfies the selected |
| canonical tube relation on a whole relative neighborhood. No regularity of |
| the finite chart selector is asserted and no intermediate chart pattern is |
| recorded in the partition. -/ |
| theorem selected_tube_germ (K U : ι → Set Y) (hK : ∀ i,IsClosed (K i)) |
| (hU : ∀ i,IsOpen (U i)) (hKU : ∀ i,K i⊆U i) |
| (q : Y → ι) (hq : ∀ y,y∈K (q y)) |
| (φ : ι → Y → E) (hφ : ∀ i,ContinuousOn (φ i) (U i)) |
| (f g : X → Y) (x : X) (hf : ContinuousAt f x) (hg : ContinuousAt g x) |
| (δ : ℝ) (hbase : ∀ i,f x∈K i → |
| g x∈U i ∧ ‖φ i (g x)-φ i (f x)‖<δ) : |
| ∀ᶠ u in 𝓝 x,g u∈U (q (f u)) ∧ |
| ‖φ (q (f u)) (g u)-φ (q (f u)) (f u)‖≤δ := by |
| apply eventually_selected K hK q hq f x hf |
| (fun i => {u | g u∈U i ∧ ‖φ i (g u)-φ i (f u)‖≤δ}) |
| intro i hi |
| have hgf := hbase i hi |
| have hcf : ContinuousAt (fun u => φ i (f u)) x := |
| ((hφ i).continuousAt ((hU i).mem_nhds (hKU i hi))).comp hf |
| have hcg : ContinuousAt (fun u => φ i (g u)) x := |
| ((hφ i).continuousAt ((hU i).mem_nhds hgf.1)).comp hg |
| filter_upwards [hg.eventually ((hU i).mem_nhds hgf.1), |
| (hcg.sub hcf).norm.eventually (gt_mem_nhds hgf.2)] with u hu hv |
| exact ⟨hu,hv.le⟩ |
|
|
| /-- Finitely many positive and negative iterate charts may be handled at |
| once. The number of cases affects only a local neighborhood, never the |
| number of partition labels or an error/mass bound. -/ |
| theorem selected_path_tube_germ {T : Type*} [Finite T] |
| (K U : ι → Set Y) (hK : ∀ i,IsClosed (K i)) |
| (hU : ∀ i,IsOpen (U i)) (hKU : ∀ i,K i⊆U i) |
| (q : Y → ι) (hq : ∀ y,y∈K (q y)) |
| (φ : ι → Y → E) (hφ : ∀ i,ContinuousOn (φ i) (U i)) |
| (f g : T → X → Y) (x : X) |
| (hf : ∀ t,ContinuousAt (f t) x) (hg : ∀ t,ContinuousAt (g t) x) |
| (δ : ℝ) (hbase : ∀ t i,f t x∈K i → |
| g t x∈U i ∧ ‖φ i (g t x)-φ i (f t x)‖<δ) : |
| ∀ᶠ u in 𝓝 x,∀ t,g t u∈U (q (f t u)) ∧ |
| ‖φ (q (f t u)) (g t u)-φ (q (f t u)) (f t u)‖≤δ := by |
| exact Filter.eventually_all.mpr (fun t => selected_tube_germ K U hK hU hKU |
| q hq φ hφ (f t) (g t) x (hf t) (hg t) δ (hbase t)) |
| end SmoothObstruction.AtlasCores |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| /-- A genuine sharp orbit tube persists in all selected old charts along a |
| continuous local branch. Intermediate chart selectors need not be locally |
| constant; their finite compact cores suffice. -/ |
| theorem actual_selected_tube_germ (hclosed : IsClosed (range I)) |
| (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C : ℝ,0<r ∧ 1≤C ∧ |
| ∀ {X T : Type*} [TopologicalSpace X] [Finite T] |
| (c : M → ι) (_ : ∀ z,z∈K (c z)) |
| (f g : T → X → M) (x : X) (ρ δ : ℝ), |
| (∀ t,ContinuousAt (f t) x) → (∀ t,ContinuousAt (g t) x) → |
| 0≤ρ → ρ≤r → C*ρ<δ → |
| (∀ t,closedBall ((extChartAt I (a (c (f t x)))) (f t x)) ρ⊆interior (range I)) → |
| (∀ t,g t x∈(extChartAt I (a (c (f t x)))).source ∧ |
| ‖(extChartAt I (a (c (f t x)))) (g t x)- |
| (extChartAt I (a (c (f t x)))) (f t x)‖≤ρ) → |
| ∀ᶠ u in 𝓝 x,∀ t,g t u∈(extChartAt I (a (c (f t u)))).source ∧ |
| ‖(extChartAt I (a (c (f t u)))) (g t u)- |
| (extChartAt I (a (c (f t u)))) (f t u)‖≤δ := by |
| obtain ⟨r,C,hr,hC,ht⟩ := uniform_chart_transfer I hclosed a K hK hKU |
| refine ⟨r,C,hr,hC,?_⟩ |
| intro X T _ _ c hc f g x ρ δ hf hg hρ hρr hδ hsafe hbase |
| apply AtlasCores.selected_path_tube_germ K |
| (fun i => (extChartAt I (a i)).source) (fun i => (hK i).isClosed) |
| (fun i => by simpa only [extChartAt_source] using (chartAt H (a i)).open_source) |
| (fun i => by simpa only [extChartAt_source] using hKU i) |
| c hc (fun i => extChartAt I (a i)) (fun i => continuousOn_extChartAt (I := I) (a i)) |
| f g x hf hg δ |
| intro t i hi |
| obtain ⟨hs,hb⟩ := ht (c (f t x)) i (f t x) (g t x) ρ (hc _) hi hρ hρr (hsafe t) |
| (by simpa only [extChartAt_source] using (hbase t).1) (hbase t).2 |
| exact ⟨by simpa only [extChartAt_source] using hs, |
| lt_of_le_of_lt (hb.trans (mul_le_mul_of_nonneg_left (hbase t).2 (by linarith))) hδ⟩ |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology |
| 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] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| /-- Actual orbit tubes persist locally under continuous input and output |
| branches without continuity of the atlas selector. -/ |
| theorem original_tube_germ (hclosed : IsClosed (range I)) |
| (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C : ℝ,0<r ∧ 1≤C ∧ |
| ∀ {X : Type*} [TopologicalSpace X] (S : Diffeomorph I I M M ∞) |
| (c : M → ι) (_ : ∀ z,z∈K (c z)) |
| (f g : X → M) (x : X) (n : ℕ) (ρ δ : ℝ), |
| ContinuousAt f x → ContinuousAt g x → |
| 0≤ρ → ρ≤r → C*ρ<δ → |
| (∀ t : Fin (n+1),closedBall ((extChartAt I (a (c (S^[t.val] (f x))))) (S^[t.val] (f x))) ρ |
| ⊆interior (range I)) → |
| originalTube I S a c (n:=n) ρ (f x) (g x) → |
| ∀ᶠ u in 𝓝 x,originalTube I S a c (n:=n) δ (f u) (g u) := by |
| obtain ⟨r,C,hr,hC,hgerm⟩ := actual_selected_tube_germ I hclosed a K hK hKU |
| refine ⟨r,C,hr,hC,?_⟩ |
| intro X _ S c hc f g x n ρ δ hf hg hρ hρr hδ hsafe hbase |
| have hh := hgerm c hc (fun t : Fin (n+1) => S^[t.val] ∘ f) |
| (fun t : Fin (n+1) => S^[t.val] ∘ g) x ρ δ |
| (fun t => (S.continuous.iterate t.val).continuousAt.comp hf) |
| (fun t => (S.continuous.iterate t.val).continuousAt.comp hg) |
| hρ hρr hδ hsafe (fun t => ⟨by simpa only [extChartAt_source,Function.comp_apply] using (hbase t).1,(hbase t).2⟩) |
| simpa only [originalTube,Function.comp_apply,extChartAt_source] using hh |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology |
| 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] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| /-- The authentic smooth local pair branch agrees with the canonical switch |
| on a relative neighborhood of the retained set. No regularity of good gauge |
| witnesses or intermediate chart labels enters this assertion. -/ |
| theorem original_pair_germ (hclosed : IsClosed (range I)) |
| (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C : ℝ,0<r ∧ 1≤C ∧ |
| ∀ {X J A : Type*} [TopologicalSpace X] |
| (S : Diffeomorph I I M M ∞) (c : M → ι) (_ : ∀ z,z∈K (c z)) |
| (P : M → A) (L : M → J) (R : Set M) |
| (Wf Wb : A×A → Submodule ℝ (EuclideanSpace ℝ (Fin d))) |
| (n : ℕ) (ρ δ : ℝ) (Φ : M×M → M×M) |
| (e Ψ : X → M×M) (x : X), |
| ContinuousAt e x → ContinuousAt Ψ x → 0≤ρ → ρ≤r → C*ρ<δ → |
| (∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] (e x).1)))) (S^[t.val] (e x).1)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] (e x).1)))) (S.symm^[t.val] (e x).1)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S^[t.val] (e x).2)))) (S^[t.val] (e x).2)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] (e x).2)))) (S.symm^[t.val] (e x).2)) ρ⊆interior (range I)) → |
| originalTube I S a c (n:=n) ρ (e x).1 (Ψ x).1 → |
| originalTube I S.symm a c (n:=n) ρ (e x).2 (Ψ x).1 → |
| originalTube I S a c (n:=n) ρ (e x).2 (Ψ x).2 → |
| originalTube I S.symm a c (n:=n) ρ (e x).1 (Ψ x).2 → |
| (∀ u∈R,∀ v∈R,L u=L v → ∀ z, |
| originalTube I S a c (n:=n) δ u z → originalTube I S.symm a c (n:=n) δ v z → |
| originalEndpoint I S a c (n:=n) (Wf (P u,P (S^[n] u))) u z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P v,P (S.symm^[n] v))) v z → |
| z=(Φ (u,v)).1) → |
| (∀ u∈R,∀ v∈R,L u=L v → |
| originalTube I S a c (n:=n) δ v (Φ (u,v)).2 ∧ |
| originalTube I S.symm a c (n:=n) δ u (Φ (u,v)).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P v,P (S^[n] v))) v (Φ (u,v)).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u,P (S.symm^[n] u))) u (Φ (u,v)).2) → |
| (∀ᶠ t in 𝓝 x, |
| originalEndpoint I S a c (n:=n) (Wf (P (e t).1,P (S^[n] (e t).1))) (e t).1 (Ψ t).1 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P (e t).2,P (S.symm^[n] (e t).2))) (e t).2 (Ψ t).1 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P (e t).2,P (S^[n] (e t).2))) (e t).2 (Ψ t).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P (e t).1,P (S.symm^[n] (e t).1))) (e t).1 (Ψ t).2) → |
| ∀ᶠ t in 𝓝 x,(e t).1∈R → (e t).2∈R → L (e t).1=L (e t).2 → Ψ t=Φ (e t) := by |
| obtain ⟨r,C,hr,hC,hgerm⟩ := original_tube_germ I hclosed a K hK hKU |
| refine ⟨r,C,hr,hC,?_⟩ |
| intro X J A _ S c hc P L R Wf Wb n ρ δ Φ e Ψ x he hΨ hρ hρr hδ |
| hsafe h1 h2 h3 h4 huniq hsecond heq |
| have ht1 := hgerm S c hc (fun t => (e t).1) (fun t => (Ψ t).1) x n ρ δ |
| he.fst hΨ.fst hρ hρr hδ (fun t => (hsafe t).1) h1 |
| have ht2 := hgerm S.symm c hc (fun t => (e t).2) (fun t => (Ψ t).1) x n ρ δ |
| he.snd hΨ.fst hρ hρr hδ (fun t => (hsafe t).2.2.2) h2 |
| have ht3 := hgerm S c hc (fun t => (e t).2) (fun t => (Ψ t).2) x n ρ δ |
| he.snd hΨ.snd hρ hρr hδ (fun t => (hsafe t).2.2.1) h3 |
| have ht4 := hgerm S.symm c hc (fun t => (e t).1) (fun t => (Ψ t).2) x n ρ δ |
| he.fst hΨ.snd hρ hρr hδ (fun t => (hsafe t).2.1) h4 |
| filter_upwards [ht1,ht2,ht3,ht4,heq] with t h1t h2t h3t h4t ht |
| intro hu hv hL |
| apply Prod.ext |
| · exact huniq _ hu _ hv hL _ h1t h2t ht.1 ht.2.1 |
| · obtain ⟨h1',h2',h3',h4'⟩ := hsecond _ hu _ hv hL |
| exact (huniq _ hv _ hu hL.symm _ h3t h4t ht.2.2.1 ht.2.2.2).trans |
| (huniq _ hv _ hu hL.symm _ h1' h2' h3' h4').symm |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ChartDensity |
| open Set Filter Topology |
| 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] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
|
|
| /-- Measurable total chart extensions agree with the authentic inverse chart |
| on a neighborhood of every interior coordinate point. -/ |
| lemma inverse_germ (a : M) (u : E₀) (hu : u∈(extChartAt I a).target) |
| (hi : u∈interior (range I)) : |
| ContinuousAt (extChartAt I a).symm u ∧ |
| ∀ᶠ v in 𝓝 u,inverse I a v=(extChartAt I a).symm v ∧ |
| coordinates I a ((extChartAt I a).symm v)=v := by |
| have ht := extChartAt_target_mem_nhdsWithin_of_mem hu |
| rw [nhdsWithin_eq_nhds.mpr (mem_interior_iff_mem_nhds.mp hi)] at ht |
| refine ⟨(continuousOn_extChartAt_symm a u hu).continuousAt ht,?_⟩ |
| filter_upwards [ht] with v hv |
| refine ⟨inverse_eq I a v hv,?_⟩ |
| rw [coordinates_eq I a _ ((extChartAt I a).map_target hv)] |
| exact (extChartAt I a).right_inv hv |
|
|
| lemma pair_inverse_germ (a : M) (u : E₀×E₀) |
| (h₁ : u.1∈(extChartAt I a).target) (h₂ : u.2∈(extChartAt I a).target) |
| (i₁ : u.1∈interior (range I)) (i₂ : u.2∈interior (range I)) : |
| let r₀ := Prod.map (extChartAt I a).symm (extChartAt I a).symm |
| let r := Prod.map (inverse I a) (inverse I a) |
| let e := Prod.map (coordinates I a) (coordinates I a) |
| ContinuousAt r₀ u ∧ ∀ᶠ v in 𝓝 u,r v=r₀ v ∧ e (r₀ v)=v := by |
| intro r₀ r e |
| obtain ⟨hc₁,hg₁⟩ := inverse_germ I a u.1 h₁ i₁ |
| obtain ⟨hc₂,hg₂⟩ := inverse_germ I a u.2 h₂ i₂ |
| refine ⟨(hc₁.comp continuousAt_fst).prodMk (hc₂.comp continuousAt_snd),?_⟩ |
| filter_upwards [continuousAt_fst.eventually hg₁,continuousAt_snd.eventually hg₂] with v hv₁ hv₂ |
| exact ⟨Prod.ext hv₁.1 hv₂.1,Prod.ext hv₁.2 hv₂.2⟩ |
|
|
| /-- Physical agreement on retained inputs gives the exact coordinate germ |
| required for change of variables, despite arbitrary chart extensions away |
| from their sources. Only the original retained set is used. -/ |
| theorem retained_pair_germ (a : M) (G : Set (M×M)) (Φ : M×M → M×M) |
| (Ψ : E₀×E₀ → E₀×E₀) (u : E₀×E₀) (hΨ : ContinuousAt Ψ u) |
| (hu₁ : u.1∈(extChartAt I a).target) (hu₂ : u.2∈(extChartAt I a).target) |
| (iu₁ : u.1∈interior (range I)) (iu₂ : u.2∈interior (range I)) |
| (hv₁ : (Ψ u).1∈(extChartAt I a).target) (hv₂ : (Ψ u).2∈(extChartAt I a).target) |
| (iv₁ : (Ψ u).1∈interior (range I)) (iv₂ : (Ψ u).2∈interior (range I)) |
| (hag : let r₀ := Prod.map (extChartAt I a).symm (extChartAt I a).symm |
| ∀ᶠ v in 𝓝 u,r₀ v∈G → r₀ (Ψ v)=Φ (r₀ v)) : |
| let e := Prod.map (coordinates I a) (coordinates I a) |
| let r := Prod.map (inverse I a) (inverse I a) |
| Ψ =ᶠ[𝓝[ChartDomination.coordinateSet e r G] u] (e ∘ Φ ∘ r) := by |
| intro e r |
| let r₀ := Prod.map (extChartAt I a).symm (extChartAt I a).symm |
| have hinput := (pair_inverse_germ I a u hu₁ hu₂ iu₁ iu₂).2 |
| have hout := hΨ.eventually ((pair_inverse_germ I a (Ψ u) hv₁ hv₂ iv₁ iv₂).2) |
| have hall : ∀ᶠ v in 𝓝 u,v∈ChartDomination.coordinateSet e r G → Ψ v=(e ∘ Φ ∘ r) v := by |
| filter_upwards [hinput,hout,hag] with v hv hv' hag' |
| intro hvG |
| have hrv : r v=r₀ v := hv.1 |
| have hg : r₀ v∈G := by rw [←hrv]; exact hvG.1 |
| have heq : r₀ (Ψ v)=Φ (r₀ v) := hag' hg |
| change Ψ v=e (Φ (r v)) |
| rw [hrv,←heq] |
| exact hv'.2.symm |
| filter_upwards [hall.filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] with v hv hvG |
| exact hv hvG |
| end SmoothObstruction.ChartDensity |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Filter Topology EndpointFrame |
| open scoped Manifold ContDiff |
| variable {d p : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] |
| {ι A : Type*} |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
|
|
| /-- Endpoint rows in a fixed old chart are exactly the canonical projected |
| equation when the ORIGINAL endpoint label is retained. -/ |
| lemma original_endpoint_iff_coordinates (S : Diffeomorph I I M M ∞) |
| (a : ι → M) (c : M → ι) (P : M → A) |
| (hPc : ∀ x y,P x=P y → c x=c y) (W : A×A → Submodule ℝ E₀) |
| (a₀ x : M) (n : ℕ) (u v : E₀) |
| (b : OrthonormalBasis (Fin d) ℝ E₀) (r : powersetCard (Fin d) p) |
| (hr : ∀ j∈r,b j∈W (P x,P (S^[n] x))) |
| (hz : ∀ z∈W (P x,P (S^[n] x)),∀ j∉r,b.repr z j=0) |
| (hP₀ : P ((extChartAt I a₀).symm u)=P x) |
| (hP₁ : P (S^[n] ((extChartAt I a₀).symm u))=P (S^[n] x)) : |
| originalEndpoint I S a c (n:=n) |
| (W (P ((extChartAt I a₀).symm u),P (S^[n] ((extChartAt I a₀).symm u)))) |
| ((extChartAt I a₀).symm u) ((extChartAt I a₀).symm v) ↔ |
| EndpointFrame.coordinates b r |
| ((extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm) v)= |
| EndpointFrame.coordinates b r |
| ((extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm) u) := by |
| rw [originalEndpoint,hP₀,hP₁,hPc _ _ hP₁] |
| exact (EndpointFrame.coordinates_eq_iff _ b r hr hz _ _).symm |
|
|
| lemma inverse_continuousAt_interior (a : M) (u : E₀) |
| (hu : u∈(extChartAt I a).target) (hi : u∈interior (range I)) : |
| ContinuousAt (extChartAt I a).symm u := by |
| have ht := extChartAt_target_mem_nhdsWithin_of_mem hu |
| rw [nhdsWithin_eq_nhds.mpr (mem_interior_iff_mem_nhds.mp hi)] at ht |
| exact (continuousOn_extChartAt_symm a u hu).continuousAt ht |
|
|
| /-- The four original endpoint equations of a smooth pair branch follow |
| from the four IFT row equations, using only the OLD starting and endpoint |
| labels. No intermediate orbit label is frozen or added to the cell. -/ |
| theorem original_equations_germ {q : ℕ} |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (c : M → ι) (P : M → A) |
| (hPc : ∀ x y,P x=P y → c x=c y) |
| (Wf Wb : A×A → Submodule ℝ E₀) (a₀ x y : M) (n : ℕ) |
| (b₀ b₁ c₀ c₁ : OrthonormalBasis (Fin d) ℝ E₀) |
| (r₀ r₁ : powersetCard (Fin d) p) (s₀ s₁ : powersetCard (Fin d) q) |
| (hb₀ : ∀ j∈r₀,b₀ j∈Wf (P x,P (S^[n] x))) |
| (hz₀ : ∀ z∈Wf (P x,P (S^[n] x)),∀ j∉r₀,b₀.repr z j=0) |
| (hb₁ : ∀ j∈r₁,b₁ j∈Wf (P y,P (S^[n] y))) |
| (hz₁ : ∀ z∈Wf (P y,P (S^[n] y)),∀ j∉r₁,b₁.repr z j=0) |
| (hc₀ : ∀ j∈s₀,c₀ j∈Wb (P x,P (S.symm^[n] x))) |
| (hv₀ : ∀ z∈Wb (P x,P (S.symm^[n] x)),∀ j∉s₀,c₀.repr z j=0) |
| (hc₁ : ∀ j∈s₁,c₁ j∈Wb (P y,P (S.symm^[n] y))) |
| (hv₁ : ∀ z∈Wb (P y,P (S.symm^[n] y)),∀ j∉s₁,c₁.repr z j=0) |
| (Ψ : E₀×E₀ → E₀×E₀) (u : E₀×E₀) |
| (hlabels : ∀ᶠ t in 𝓝 u, |
| P ((extChartAt I a₀).symm t.1)=P x ∧ |
| P ((extChartAt I a₀).symm t.2)=P y ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.1))=P (S^[n] x) ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.2))=P (S^[n] y) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.1))=P (S.symm^[n] x) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.2))=P (S.symm^[n] y)) |
| (heq : ∀ᶠ t in 𝓝 u, |
| EndpointFrame.coordinates b₀ r₀ ((extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm) (Ψ t).1)= |
| EndpointFrame.coordinates b₀ r₀ ((extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm) t.1) ∧ |
| EndpointFrame.coordinates b₁ r₁ ((extChartAt I (a (c (S^[n] y))) ∘ S^[n] ∘ (extChartAt I a₀).symm) (Ψ t).2)= |
| EndpointFrame.coordinates b₁ r₁ ((extChartAt I (a (c (S^[n] y))) ∘ S^[n] ∘ (extChartAt I a₀).symm) t.2) ∧ |
| EndpointFrame.coordinates c₀ s₀ ((extChartAt I (a (c (S.symm^[n] x))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm) (Ψ t).2)= |
| EndpointFrame.coordinates c₀ s₀ ((extChartAt I (a (c (S.symm^[n] x))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm) t.1) ∧ |
| EndpointFrame.coordinates c₁ s₁ ((extChartAt I (a (c (S.symm^[n] y))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm) (Ψ t).1)= |
| EndpointFrame.coordinates c₁ s₁ ((extChartAt I (a (c (S.symm^[n] y))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm) t.2)) : |
| let e := Prod.map (extChartAt I a₀).symm (extChartAt I a₀).symm |
| ∀ᶠ t in 𝓝 u, |
| originalEndpoint I S a c (n:=n) (Wf (P (e t).1,P (S^[n] (e t).1))) (e t).1 (e (Ψ t)).1 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P (e t).2,P (S.symm^[n] (e t).2))) (e t).2 (e (Ψ t)).1 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P (e t).2,P (S^[n] (e t).2))) (e t).2 (e (Ψ t)).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P (e t).1,P (S.symm^[n] (e t).1))) (e t).1 (e (Ψ t)).2 := by |
| intro e |
| filter_upwards [hlabels,heq] with t ht he |
| exact ⟨(original_endpoint_iff_coordinates I S a c P hPc Wf a₀ x n t.1 (Ψ t).1 b₀ r₀ hb₀ hz₀ ht.1 ht.2.2.1).mpr he.1, |
| (original_endpoint_iff_coordinates I S.symm a c P hPc Wb a₀ y n t.2 (Ψ t).1 c₁ s₁ hc₁ hv₁ ht.2.1 ht.2.2.2.2.2).mpr he.2.2.2, |
| (original_endpoint_iff_coordinates I S a c P hPc Wf a₀ y n t.2 (Ψ t).2 b₁ r₁ hb₁ hz₁ ht.2.1 ht.2.2.2.1).mpr he.2.1, |
| (original_endpoint_iff_coordinates I S.symm a c P hPc Wb a₀ x n t.1 (Ψ t).2 c₀ s₀ hc₀ hv₀ ht.1 ht.2.2.2.2.1).mpr he.2.2.1⟩ |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology EndpointFrame |
| open scoped Manifold ContDiff |
| variable {d p q : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
|
|
| /-- The authentic eight endpoint minors give the local Jacobian germ of the |
| ORIGINAL measurable switch. Rank-zero blocks are included. The agreement is |
| relative to retained original inputs, not an extension or a different law. -/ |
| theorem controlled_original_germ (hclosed : IsClosed (range I)) |
| (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C β : ℝ,0<r ∧ 1≤C ∧ 0<β ∧ |
| ∀ {A J : Type*} (S : Diffeomorph I I M M ∞) (c : M → ι) |
| (P : M → A) (L : M → J) (R : Set M) |
| (Wf Wb : A×A → Submodule ℝ E₀) (n : ℕ) (ρ δ : ℝ) |
| (Φ : M×M → M×M) (G : Set (M×M)) (x y : M), |
| (∀ z,z∈K (c z)) → (∀ u v,P u=P v → c u=c v) → c x=c y → |
| 0≤ρ → ρ≤r → C*ρ<δ → |
| (∀ u∈G,u.1∈R ∧ u.2∈R ∧ L u.1=L u.2) → |
| (∀ u∈R,∀ v∈R,L u=L v → ∀ z, |
| originalTube I S a c (n:=n) δ u z → originalTube I S.symm a c (n:=n) δ v z → |
| originalEndpoint I S a c (n:=n) (Wf (P u,P (S^[n] u))) u z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P v,P (S.symm^[n] v))) v z → z=(Φ (u,v)).1) → |
| (∀ u∈R,∀ v∈R,L u=L v → |
| originalTube I S a c (n:=n) δ v (Φ (u,v)).2 ∧ |
| originalTube I S.symm a c (n:=n) δ u (Φ (u,v)).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P v,P (S^[n] v))) v (Φ (u,v)).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u,P (S.symm^[n] u))) u (Φ (u,v)).2) → |
| (∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S^[t.val] y)))) (S^[t.val] y)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] y)))) (S.symm^[t.val] y)) ρ⊆interior (range I)) → |
| let z := (Φ (x,y)).1 |
| let w := (Φ (x,y)).2 |
| originalTube I S a c (n:=n) ρ x z → originalTube I S.symm a c (n:=n) ρ y z → |
| originalTube I S a c (n:=n) ρ y w → originalTube I S.symm a c (n:=n) ρ x w → |
| let a₀ := a (c x) |
| let X := (extChartAt I a₀) x |
| let Y := (extChartAt I a₀) y |
| let Z := (extChartAt I a₀) z |
| let W := (extChartAt I a₀) w |
| let f₀ := extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm |
| let f₁ := extChartAt I (a (c (S^[n] y))) ∘ S^[n] ∘ (extChartAt I a₀).symm |
| let g₀ := extChartAt I (a (c (S.symm^[n] x))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm |
| let g₁ := extChartAt I (a (c (S.symm^[n] y))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm |
| ∀ (U V : Submodule ℝ E₀) (_ : IsCompl U V) |
| (u : OrthonormalBasis (Fin p) ℝ U) (v : OrthonormalBasis (Fin q) ℝ V) |
| (b₀ b₁ c₀ c₁ : OrthonormalBasis (Fin d) ℝ E₀) |
| (r₀ r₁ : powersetCard (Fin d) p) (s₀ s₁ : powersetCard (Fin d) q) |
| (A₀ A₁ : Matrix (Fin p) (Fin p) ℝ) (D₀ D₁ : Matrix (Fin q) (Fin q) ℝ), |
| (∀ j∈r₀,b₀ j∈Wf (P x,P (S^[n] x))) → |
| (∀ v∈Wf (P x,P (S^[n] x)),∀ j∉r₀,b₀.repr v j=0) → |
| (∀ j∈r₁,b₁ j∈Wf (P y,P (S^[n] y))) → |
| (∀ v∈Wf (P y,P (S^[n] y)),∀ j∉r₁,b₁.repr v j=0) → |
| (∀ j∈s₀,c₀ j∈Wb (P x,P (S.symm^[n] x))) → |
| (∀ v∈Wb (P x,P (S.symm^[n] x)),∀ j∉s₀,c₀.repr v j=0) → |
| (∀ j∈s₁,c₁ j∈Wb (P y,P (S.symm^[n] y))) → |
| (∀ v∈Wb (P y,P (S.symm^[n] y)),∀ j∉s₁,c₁.repr v j=0) → |
| GoodEndpoint β b₀ r₀ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₀ X A₀ → |
| GoodEndpoint β b₀ r₀ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₀ Z A₀ → |
| GoodEndpoint β b₁ r₁ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₁ Y A₁ → |
| GoodEndpoint β b₁ r₁ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₁ W A₁ → |
| GoodEndpoint β c₀ s₀ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₀ X D₀ → |
| GoodEndpoint β c₀ s₀ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₀ W D₀ → |
| GoodEndpoint β c₁ s₁ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₁ Y D₁ → |
| GoodEndpoint β c₁ s₁ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₁ Z D₁ → |
| coordinates b₀ r₀ (f₀ Z)=coordinates b₀ r₀ (f₀ X) → |
| coordinates b₁ r₁ (f₁ W)=coordinates b₁ r₁ (f₁ Y) → |
| coordinates c₀ s₀ (g₀ W)=coordinates c₀ s₀ (g₀ X) → |
| coordinates c₁ s₁ (g₁ Z)=coordinates c₁ s₁ (g₁ Y) → |
| (∀ᶠ t in 𝓝 (X,Y), |
| P ((extChartAt I a₀).symm t.1)=P x ∧ P ((extChartAt I a₀).symm t.2)=P y ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.1))=P (S^[n] x) ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.2))=P (S^[n] y) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.1))=P (S.symm^[n] x) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.2))=P (S.symm^[n] y)) → |
| let e := Prod.map (ChartDensity.coordinates I a₀) (ChartDensity.coordinates I a₀) |
| let s := Prod.map (ChartDensity.inverse I a₀) (ChartDensity.inverse I a₀) |
| ∃ Ψ : E₀×E₀ → E₀×E₀,HasFDerivAt Ψ (fderiv ℝ Ψ (X,Y)) (X,Y) ∧ |
| (81/121:ℝ)≤|(fderiv ℝ Ψ (X,Y)).det| ∧ |
| Ψ =ᶠ[𝓝[ChartDomination.coordinateSet e s G] (X,Y)] (e ∘ Φ ∘ s) := by |
| obtain ⟨r,C,hr,hC,hgerm⟩ := original_pair_germ I hclosed a K hK hKU |
| obtain ⟨β,hβ,hbranch⟩ := four_endpoint_branch (d:=d) (p:=p) (q:=q) |
| refine ⟨r,C,β,hr,hC,hβ,?_⟩ |
| intro A J S c P L R Wf Wb n ρ δ Φ G x y hc hPc hcxy hρ hρr hρδ hG huniq hsecond hsafe |
| z w ht1 ht2 ht3 ht4 a₀ X Y Z W f₀ f₁ g₀ g₁ U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ A₀ A₁ D₀ D₁ |
| hb₀ hz₀ hb₁ hz₁ hc₀ hv₀ hc₁ hv₁ hxF hzF hyF hwF hxB hwB hyB hzB he₀ he₁ he₂ he₃ hlabels e s |
| obtain ⟨Ψ,hΨ,hcd,hlo,_,heq⟩ := hbranch U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ |
| f₀ f₁ g₀ g₁ X Y Z W A₀ A₁ D₀ D₁ hxF hzF hyF hwF hxB hwB hyB hzB he₀ he₁ he₂ he₃ |
| let r₀' := Prod.map (extChartAt I a₀).symm (extChartAt I a₀).symm |
| have hsrcx : x∈(extChartAt I a₀).source := by simpa only [extChartAt_source] using hKU _ (hc x) |
| have hsrcy : y∈(extChartAt I a₀).source := by |
| dsimp only [a₀]; rw [hcxy]; simpa only [extChartAt_source] using hKU _ (hc y) |
| have hsrcz : z∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht1 0).1 |
| have hsrcw : w∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht4 0).1 |
| have hx : r₀' (X,Y)=(x,y) := Prod.ext ((extChartAt I a₀).left_inv hsrcx) ((extChartAt I a₀).left_inv hsrcy) |
| have hz : r₀' (Ψ (X,Y))=(z,w) := by |
| rw [hΨ]; exact Prod.ext ((extChartAt I a₀).left_inv hsrcz) ((extChartAt I a₀).left_inv hsrcw) |
| have hXi : X∈interior (range I) := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply] using (hsafe 0).1 (mem_closedBall_self hρ) |
| have hYi : Y∈interior (range I) := by |
| dsimp only [Y,a₀]; rw [hcxy] |
| simpa only [Fin.val_zero,Function.iterate_zero_apply] using (hsafe 0).2.2.1 (mem_closedBall_self hρ) |
| have hZi : Z∈interior (range I) := by |
| apply (hsafe 0).1 |
| exact mem_closedBall.mpr (by simpa only [Fin.val_zero,Function.iterate_zero_apply,dist_eq_norm] using (ht1 0).2) |
| have hWi : W∈interior (range I) := by |
| apply (hsafe 0).2.1 |
| exact mem_closedBall.mpr (by simpa only [Fin.val_zero,Function.iterate_zero_apply,dist_eq_norm] using (ht4 0).2) |
| have hct := (ChartDensity.pair_inverse_germ I a₀ (X,Y) |
| ((extChartAt I a₀).map_source hsrcx) ((extChartAt I a₀).map_source hsrcy) hXi hYi).1 |
| have hco : ContinuousAt r₀' (Z,W) := (ChartDensity.pair_inverse_germ I a₀ (Z,W) |
| ((extChartAt I a₀).map_source hsrcz) ((extChartAt I a₀).map_source hsrcw) hZi hWi).1 |
| have hctΨ : ContinuousAt (r₀' ∘ Ψ) (X,Y) := (hΨ ▸ hco).comp hcd.continuousAt |
| have heq' := original_equations_germ I S a c P hPc Wf Wb a₀ x y n b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ |
| hb₀ hz₀ hb₁ hz₁ hc₀ hv₀ hc₁ hv₁ Ψ (X,Y) hlabels heq |
| have hag := hgerm S c hc P L R Wf Wb n ρ δ Φ r₀' (r₀' ∘ Ψ) (X,Y) |
| hct hctΨ hρ hρr hρδ (by simpa only [hx] using hsafe) |
| (by simpa only [Function.comp_apply,hx,hz] using ht1) |
| (by simpa only [Function.comp_apply,hx,hz] using ht2) |
| (by simpa only [Function.comp_apply,hx,hz] using ht3) |
| (by simpa only [Function.comp_apply,hx,hz] using ht4) huniq hsecond heq' |
| refine ⟨Ψ,(hcd.differentiableAt (by simp)).hasFDerivAt,hlo,?_⟩ |
| apply ChartDensity.retained_pair_germ I a₀ G Φ Ψ (X,Y) hcd.continuousAt |
| ((extChartAt I a₀).map_source hsrcx) ((extChartAt I a₀).map_source hsrcy) hXi hYi |
| · simpa only [hΨ] using (extChartAt I a₀).map_source hsrcz |
| · simpa only [hΨ] using (extChartAt I a₀).map_source hsrcw |
| · simpa only [hΨ] using hZi |
| · simpa only [hΨ] using hWi |
| · filter_upwards [hag] with t ht |
| intro htG |
| exact ht (hG _ htG).1 (hG _ htG).2.1 (hG _ htG).2.2 |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric EuclideanSplitting ExteriorStability EndpointFrame |
| open scoped ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
| |
| /-- The original orbit-tube endpoint minor, now with the actual bilateral |
| finite-time gauge. This is uniform in every starting bin and every rank. -/ |
| theorem gauge_endpoint_minor (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (L : ι → Set M) |
| (hL : ∀ i,IsCompact (L i)) (hLU : ∀ i,L i⊆(chartAt H (a i)).source) |
| (Dg : ℝ) (hDg : 0≤Dg) : |
| ∃ r K₀ Crel C₁ : ℝ,0<r ∧ 1≤K₀ ∧ 0≤Crel ∧ 0≤C₁ ∧ |
| ∀ {p q : ℕ} (c : M → ι) (x y : M) (n : ℕ) (b R C δ β γ g : ℝ) |
| (A : BilateralGauge I S a c n (Dg*(b*n)) R x) |
| (T : Finset (Fin d)) (_ : T.card=p), |
| (∀ z,z∈L (c z)) → 0≤b → 1≤b*n → 0≤R → 0≤g → 0≤C → 0≤δ → |
| (∀ j∈T,γ+g≤A.lam j) → (∀ j∉T,A.lam j≤γ-g) → |
| let ρ := Real.exp (-(3/2:ℝ)*SmoothScales.B b*n) |
| K₀*ρ≤1 → ρ≤r → originalTube I S a c (n:=n) ρ x y → |
| (∀ t : Fin (n+1),closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I)) → |
| (∀ B : E₀ →L[ℝ] E₀,‖exterior p B‖≤C*(max 1 ‖B‖)^p) → |
| ∀ (W G Gslow : Submodule ℝ E₀),Module.finrank ℝ W=p → |
| (‖(A.forwardSpace I S a c n T).starProjection-W.starProjection‖≤1/2) → |
| (‖(A.forwardSpace I S a c 0 T).starProjection-G.starProjection‖≤δ) → |
| (‖(A.forwardSpace I S a c 0 Tᶜ).starProjection-Gslow.starProjection‖≤δ) → |
| ∀ (bE : OrthonormalBasis (Fin d) ℝ E₀) (rE : powersetCard (Fin d) p), |
| (∀ j∈rE,bE j∈W) → (∀ v∈W,∀ j∉rE,bE.repr v j=0) → |
| ∀ (v : Fin p → E₀) (z : Fin q → E₀),Orthonormal ℝ v → |
| (∀ i,v i∈G) → (∀ j,z j∈Gslow) → (∀ j,‖z j‖≤1) → |
| let K := Dg*(b*n) |
| let L₀ := (C*(Real.exp K)^p)^2*(C*2^p) |
| let N₀ := C*(C^2*(Real.exp K)^(2*p)*(Fintype.card (powersetCard (Fin d) p) : ℝ)) |
| let N₁ := C*(C*(Real.exp K)^p)^2*(Fintype.card (powersetCard (Fin d) p) : ℝ)*‖continuousWedge (E := E₀) p‖ |
| let ζ := RelativeAsymptotics.envelope Crel 0 C₁ (d*R) K₀ b n * |
| Real.exp (RelativeAsymptotics.envelope Crel 0 C₁ (d*R) K₀ b n) |
| let ε := 2*L₀*C*ζ*‖continuousWedge (E := E₀) p‖ |
| let θ := 2*L₀*(N₁*Real.exp (-2*g*n)+N₀*unitWedgeConstant (E := E₀) p*δ) |
| (frameWedgeConstant (E := E₀) p*δ)*(1+L₀*N₀)≤1/2 → |
| ε≤1/100 → (θ+ε)/(1-ε)≤β → |
| let F := extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I (a (c x))).symm |
| GoodEndpoint β bE rE v z F ((extChartAt I (a (c x))) y) |
| (endpointMatrix bE p rE (fderiv ℝ F ((extChartAt I (a (c x))) x)) v) := by |
| obtain ⟨r,K₀,Crel,C₁,hr,hK₀,hCrel,hC₁,hh⟩ := actual_endpoint_minor I hclosed S a L hL hLU Dg hDg |
| refine ⟨r,K₀,Crel,C₁,hr,hK₀,hCrel,hC₁,?_⟩ |
| intro p q c x y n b R C δ β γ g A T hTp hc hb hbn hR hg hC hδ hfast hslow ρ hρ hρr ht hs |
| apply hh c x y n A.UF A.VF A.lam T hTp γ g (Dg*(b*n)) b n R C δ β hc hb |
| (Nat.cast_nonneg n) hbn hR hg (by positivity) le_rfl (by nlinarith [show (0:ℝ)≤(n:ℝ) from Nat.cast_nonneg n]) hC hδ |
| (fun t => (A.invF t).1) (fun t => (A.invF t).2) A.stepF A.rate_bound hfast hslow |
| (fun t ht => (A.normF t ht).1) (fun t ht => (A.normF t ht).2) hρ hρr |
| (fun t htn => by simpa only [extChartAt_source] using (ht ⟨t,by omega⟩).1) (fun t htn => (ht ⟨t,by omega⟩).2) |
| (fun t htn => hs ⟨t,by omega⟩) |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.EndpointBudget |
| open Set EndpointFrame ExteriorStability |
| /-- Literal three errors entering the actual endpoint-minor theorem. -/ |
| def admissible (d p : ℕ) (Dg R C K₀ Crel C₁ b n g δ β : ℝ) : Prop := |
| let K := Dg*(b*n) |
| let L₀ := (C*(Real.exp K)^p)^2*(C*2^p) |
| let N₀ := C*(C^2*(Real.exp K)^(2*p)*(Fintype.card (powersetCard (Fin d) p) : ℝ)) |
| let N₁ := C*(C*(Real.exp K)^p)^2*(Fintype.card (powersetCard (Fin d) p) : ℝ)*‖continuousWedge (E := EuclideanSpace ℝ (Fin d)) p‖ |
| let ζ := RelativeAsymptotics.envelope Crel 0 C₁ (d*R) K₀ b n * |
| Real.exp (RelativeAsymptotics.envelope Crel 0 C₁ (d*R) K₀ b n) |
| let ε := 2*L₀*C*ζ*‖continuousWedge (E := EuclideanSpace ℝ (Fin d)) p‖ |
| let θ := 2*L₀*(N₁*Real.exp (-2*g*n)+N₀*unitWedgeConstant (E := EuclideanSpace ℝ (Fin d)) p*δ) |
| (frameWedgeConstant (E := EuclideanSpace ℝ (Fin d)) p*δ)*(1+L₀*N₀)≤1/2 ∧ |
| ε≤1/100 ∧ (θ+ε)/(1-ε)≤β |
| end SmoothObstruction.EndpointBudget |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric EuclideanSplitting ExteriorStability EndpointFrame |
| open scoped ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
| |
| /-- One authentic frame and one original leading matrix control the whole |
| old-chart orbit tube. Neither the frame nor the normalization is reselected |
| at the perturbed point. -/ |
| theorem gauge_endpoint_family (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (L : ι → Set M) |
| (hL : ∀ i,IsCompact (L i)) (hLU : ∀ i,L i⊆(chartAt H (a i)).source) |
| (Dg : ℝ) (hDg : 0≤Dg) : |
| ∃ r K₀ Crel C₁ : ℝ,0<r ∧ 1≤K₀ ∧ 0≤Crel ∧ 0≤C₁ ∧ |
| ∀ {p q : ℕ} (c : M → ι) (x : M) (n : ℕ) (b R C δ β γ g : ℝ) |
| (A : BilateralGauge I S a c n (Dg*(b*n)) R x) |
| (T : Finset (Fin d)) (_ : T.card=p), |
| (∀ z,z∈L (c z)) → 0≤b → 1≤b*n → 0≤R → 0≤g → 0≤C → 0≤δ → |
| (∀ j∈T,γ+g≤A.lam j) → (∀ j∉T,A.lam j≤γ-g) → |
| let ρ := Real.exp (-(3/2:ℝ)*SmoothScales.B b*n) |
| K₀*ρ≤1 → ρ≤r → |
| (∀ t : Fin (n+1),closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I)) → |
| (∀ B : E₀ →L[ℝ] E₀,‖exterior p B‖≤C*(max 1 ‖B‖)^p) → |
| ∀ (W G Gslow : Submodule ℝ E₀),Module.finrank ℝ W=p → |
| (‖(A.forwardSpace I S a c n T).starProjection-W.starProjection‖≤1/2) → |
| (‖(A.forwardSpace I S a c 0 T).starProjection-G.starProjection‖≤δ) → |
| (‖(A.forwardSpace I S a c 0 Tᶜ).starProjection-Gslow.starProjection‖≤δ) → |
| ∀ (u : OrthonormalBasis (Fin p) ℝ G) (v : OrthonormalBasis (Fin q) ℝ Gslow), |
| EndpointBudget.admissible d p Dg R C K₀ Crel C₁ b n g δ β → |
| ∃ (bE : OrthonormalBasis (Fin d) ℝ E₀) (rE : powersetCard (Fin d) p) |
| (D : Matrix (Fin p) (Fin p) ℝ), |
| (∀ j∈rE,bE j∈W) ∧ (∀ v∈W,∀ j∉rE,bE.repr v j=0) ∧ |
| ∀ y,originalTube I S a c (n:=n) ρ x y → |
| let F := extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I (a (c x))).symm |
| GoodEndpoint β bE rE (fun i => (u i:E₀)) (fun i => (v i:E₀)) F |
| ((extChartAt I (a (c x))) y) D := by |
| obtain ⟨r,K₀,Crel,C₁,hr,hK₀,hCrel,hC₁,hm⟩ := gauge_endpoint_minor I hclosed S a L hL hLU Dg hDg |
| refine ⟨r,K₀,Crel,C₁,hr,hK₀,hCrel,hC₁,?_⟩ |
| intro p q c x n b R C δ β γ g A T hTp hc hb hbn hR hg hC hδ hf hs ρ hρ hρr hsafe hpoly |
| W G Gslow hdim hend hstart hstartSlow u v hbud |
| obtain ⟨bE,rE,hrE,_,hzero⟩ := EndpointFrame.exists_frame W hdim |
| let F := extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I (a (c x))).symm |
| let D := endpointMatrix bE p rE (fderiv ℝ F ((extChartAt I (a (c x))) x)) (fun i => (u i:E₀)) |
| refine ⟨bE,rE,D,hrE,hzero,?_⟩ |
| intro y hy |
| have hu : Orthonormal ℝ (fun i => (u i:E₀)) := u.orthonormal.comp_linearIsometry G.subtypeₗᵢ |
| have hvnorm (i : Fin q) : ‖(v i:E₀)‖≤1 := by |
| change ‖v i‖≤1 |
| exact le_of_eq (v.norm_eq_one i) |
| exact hm c x y n b R C δ β γ g A T hTp hc hb hbn hR hg hC hδ hf hs hρ hρr hy hsafe |
| hpoly W G Gslow hdim hend hstart hstartSlow bE rE hrE hzero |
| (fun i => (u i:E₀)) (fun i => (v i:E₀)) hu (fun i => (u i).property) |
| (fun i => (v i).property) hvnorm hbud.1 hbud.2.1 hbud.2.2 |
| |
| omit [IsManifold I ∞ M] [Fintype ι] in |
| lemma originalTube_self (S : Diffeomorph I I M M ∞) (a : ι → M) (c : M → ι) |
| {n : ℕ} {ρ : ℝ} (hρ : 0≤ρ) (x : M) |
| (hsrc : ∀ z,z∈(chartAt H (a (c z))).source) : originalTube I S a c (n:=n) ρ x x := by |
| intro t |
| exact ⟨hsrc _,by simpa only [sub_self,norm_zero] using hρ⟩ |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology EndpointFrame |
| open scoped Manifold ContDiff |
| variable {d p q : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
| |
| def OriginalEndpointControl (S : Diffeomorph I I M M ∞) (a : ι → M) (c : M → ι) |
| (n : ℕ) (ρ β : ℝ) (W : Submodule ℝ E₀) (u : Fin p → E₀) (v : Fin q → E₀) (x : M) : Prop := |
| ∃ (b : OrthonormalBasis (Fin d) ℝ E₀) (r : powersetCard (Fin d) p) |
| (D : Matrix (Fin p) (Fin p) ℝ), |
| (∀ j∈r,b j∈W) ∧ (∀ z∈W,∀ j∉r,b.repr z j=0) ∧ |
| ∀ y,originalTube I S a c (n:=n) ρ x y → |
| GoodEndpoint β b r u v |
| (extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I (a (c x))).symm) |
| ((extChartAt I (a (c x))) y) D |
| |
| omit [IsManifold I ∞ M] [Fintype ι] in |
| lemma originalEndpoint_coordinates (S : Diffeomorph I I M M ∞) |
| (a : ι → M) (c : M → ι) (n : ℕ) (a₀ x z : M) (W : Submodule ℝ E₀) |
| (b : OrthonormalBasis (Fin d) ℝ E₀) (r : powersetCard (Fin d) p) |
| (hr : ∀ j∈r,b j∈W) (hz : ∀ z∈W,∀ j∉r,b.repr z j=0) |
| (hx : x∈(extChartAt I a₀).source) (hzt : z∈(extChartAt I a₀).source) |
| (he : originalEndpoint I S a c (n:=n) W x z) : |
| coordinates b r ((extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm) |
| ((extChartAt I a₀) z))= |
| coordinates b r ((extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm) |
| ((extChartAt I a₀) x)) := by |
| simp only [Function.comp_apply,(extChartAt I a₀).left_inv hx,(extChartAt I a₀).left_inv hzt] |
| exact (EndpointFrame.coordinates_eq_iff W b r hr hz _ _).mpr he |
| |
| /-- Fixed old-chart endpoint families give the actual retained-set relative germ. -/ |
| theorem family_original_germ (hclosed : IsClosed (range I)) |
| (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C β : ℝ,0<r ∧ 1≤C ∧ 0<β ∧ |
| ∀ {A J : Type*} (S : Diffeomorph I I M M ∞) (c : M → ι) |
| (P : M → A) (L : M → J) (R : Set M) |
| (Wf Wb : A×A → Submodule ℝ E₀) (n : ℕ) (ρ δ : ℝ) |
| (Φ : M×M → M×M) (G : Set (M×M)) (x y : M), |
| (∀ z,z∈K (c z)) → (∀ u v,P u=P v → c u=c v) → c x=c y → |
| 0≤ρ → ρ≤r → C*ρ<δ → |
| (∀ u∈G,u.1∈R ∧ u.2∈R ∧ L u.1=L u.2) → |
| (∀ u∈R,∀ v∈R,L u=L v → ∀ z, |
| originalTube I S a c (n:=n) δ u z → originalTube I S.symm a c (n:=n) δ v z → |
| originalEndpoint I S a c (n:=n) (Wf (P u,P (S^[n] u))) u z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P v,P (S.symm^[n] v))) v z → z=(Φ (u,v)).1) → |
| (∀ u∈R,∀ v∈R,L u=L v → |
| originalTube I S a c (n:=n) δ v (Φ (u,v)).2 ∧ |
| originalTube I S.symm a c (n:=n) δ u (Φ (u,v)).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P v,P (S^[n] v))) v (Φ (u,v)).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u,P (S.symm^[n] u))) u (Φ (u,v)).2) → |
| (∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S^[t.val] y)))) (S^[t.val] y)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] y)))) (S.symm^[t.val] y)) ρ⊆interior (range I)) → |
| let z := (Φ (x,y)).1 |
| let w := (Φ (x,y)).2 |
| originalTube I S a c (n:=n) ρ x z → originalTube I S.symm a c (n:=n) ρ y z → |
| originalTube I S a c (n:=n) ρ y w → originalTube I S.symm a c (n:=n) ρ x w → |
| let a₀ := a (c x) |
| let X := (extChartAt I a₀) x |
| let Y := (extChartAt I a₀) y |
| ∀ (U V : Submodule ℝ E₀) (_ : IsCompl U V) |
| (u : OrthonormalBasis (Fin p) ℝ U) (v : OrthonormalBasis (Fin q) ℝ V), |
| OriginalEndpointControl I S a c n ρ β (Wf (P x,P (S^[n] x))) (fun i => (u i:E₀)) (fun i => (v i:E₀)) x → |
| OriginalEndpointControl I S a c n ρ β (Wf (P y,P (S^[n] y))) (fun i => (u i:E₀)) (fun i => (v i:E₀)) y → |
| OriginalEndpointControl I S.symm a c n ρ β (Wb (P x,P (S.symm^[n] x))) (fun i => (v i:E₀)) (fun i => (u i:E₀)) x → |
| OriginalEndpointControl I S.symm a c n ρ β (Wb (P y,P (S.symm^[n] y))) (fun i => (v i:E₀)) (fun i => (u i:E₀)) y → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S a c (n:=n) (Wf (P y,P (S^[n] y))) y w → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x w → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z → |
| (∀ᶠ t in 𝓝 (X,Y), |
| P ((extChartAt I a₀).symm t.1)=P x ∧ P ((extChartAt I a₀).symm t.2)=P y ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.1))=P (S^[n] x) ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.2))=P (S^[n] y) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.1))=P (S.symm^[n] x) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.2))=P (S.symm^[n] y)) → |
| let e := Prod.map (ChartDensity.coordinates I a₀) (ChartDensity.coordinates I a₀) |
| let s := Prod.map (ChartDensity.inverse I a₀) (ChartDensity.inverse I a₀) |
| ∃ Ψ : E₀×E₀ → E₀×E₀,HasFDerivAt Ψ (fderiv ℝ Ψ (X,Y)) (X,Y) ∧ |
| (81/121:ℝ)≤|(fderiv ℝ Ψ (X,Y)).det| ∧ |
| Ψ =ᶠ[𝓝[ChartDomination.coordinateSet e s G] (X,Y)] (e ∘ Φ ∘ s) := by |
| obtain ⟨r,C,β,hr,hC,hβ,hh⟩ := controlled_original_germ (p:=p) (q:=q) I hclosed a K hK hKU |
| refine ⟨r,C,β,hr,hC,hβ,?_⟩ |
| intro A J S c P L R Wf Wb n ρ δ Φ G x y hc hPc hcxy hρ hρr hρδ hG huniq hsecond hsafe |
| z w ht1 ht2 ht3 ht4 a₀ X Y U V h u v hf₀ hf₁ hb₀ hb₁ he₀ he₁ he₂ he₃ hlabels e s |
| obtain ⟨b₀,r₀,A₀,hbF₀,hzF₀,hmF₀⟩ := hf₀ |
| obtain ⟨b₁,r₁,A₁,hbF₁,hzF₁,hmF₁⟩ := hf₁ |
| obtain ⟨c₀,s₀,D₀,hbB₀,hzB₀,hmB₀⟩ := hb₀ |
| obtain ⟨c₁,s₁,D₁,hbB₁,hzB₁,hmB₁⟩ := hb₁ |
| have hsrc (x : M) : x∈(chartAt H (a (c x))).source := hKU _ (hc x) |
| have hx : x∈(extChartAt I a₀).source := by simpa only [extChartAt_source] using hsrc x |
| have hy : y∈(extChartAt I a₀).source := by dsimp only [a₀]; rw [hcxy]; simpa only [extChartAt_source] using hsrc y |
| have hz : z∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht1 0).1 |
| have hw : w∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht4 0).1 |
| have hfx := hmF₀ x (originalTube_self I S a c hρ x hsrc) |
| have hfy := hmF₁ y (originalTube_self I S a c hρ y hsrc) |
| have hbx := hmB₀ x (originalTube_self I S.symm a c hρ x hsrc) |
| have hby := hmB₁ y (originalTube_self I S.symm a c hρ y hsrc) |
| have hfw := hmF₁ w ht3 |
| have hbz := hmB₁ z ht2 |
| rw [←hcxy] at hfy hby hfw hbz |
| exact hh S c P L R Wf Wb n ρ δ Φ G x y hc hPc hcxy hρ hρr hρδ hG huniq hsecond hsafe |
| ht1 ht2 ht3 ht4 U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ A₀ A₁ D₀ D₁ |
| hbF₀ hzF₀ hbF₁ hzF₁ hbB₀ hzB₀ hbB₁ hzB₁ |
| hfx (hmF₀ z ht1) hfy hfw hbx (hmB₀ w ht4) hby hbz |
| (originalEndpoint_coordinates I S a c n a₀ x z _ b₀ r₀ hbF₀ hzF₀ hx hz he₀) |
| (originalEndpoint_coordinates I S a c n a₀ y w _ b₁ r₁ hbF₁ hzF₁ hy hw he₁) |
| (originalEndpoint_coordinates I S.symm a c n a₀ x w _ c₀ s₀ hbB₀ hzB₀ hx hw he₂) |
| (originalEndpoint_coordinates I S.symm a c n a₀ y z _ c₁ s₁ hbB₁ hzB₁ hy hz he₃) hlabels |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction |
| lemma finite_positive_bound {ι : Type*} [Fintype ι] (f : ι → ℝ) (hf : ∀ i,0<f i) : |
| ∃ b : ℝ,0<b ∧ ∀ i,b≤f i := by |
| classical |
| by_cases h : Nonempty ι |
| · let := h |
| refine ⟨Finset.univ.inf' Finset.univ_nonempty f,?_,?_⟩ |
| · exact (Finset.lt_inf'_iff Finset.univ_nonempty).mpr (fun i _ => hf i) |
| · intro i; exact Finset.inf'_le _ (Finset.mem_univ i) |
| · exact ⟨1,by norm_num,fun i => (h ⟨i⟩).elim⟩ |
| end SmoothObstruction |
|
|
| end |
|
|
| end OAI |
|
|
