| import OAI.Dynamics.SmoothObstruction.PairLaw |
| import OAI.Dynamics.SmoothObstruction.ProductCoordinates |
| import OAI.Dynamics.SmoothObstruction.GridCells |
|
|
| 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.PairLaw |
| open MeasureTheory Set Finset |
| open scoped ENNReal |
| variable {X α : Type*} [MeasurableSpace X] [Fintype α] |
| [MeasurableSpace α] [MeasurableSingletonClass α] |
|
|
| omit [MeasurableSpace α] [MeasurableSingletonClass α] in |
| / |
| is no renormalization after restricting to G or to endpoint patterns. -/ |
| theorem law_domination (μ : Measure X) (P : X → α) |
| (G : Set (X × X)) (Φ : X × X → X × X) (hΦ : Measurable Φ) |
| (C : ℝ≥0∞) |
| (hc : ∀ a,(((cell μ P a).prod (cell μ P a)).restrict G).map Φ≤ |
| C • (cell μ P a).prod (cell μ P a)) : |
| ((law μ P).restrict G).map Φ≤C • law μ P := by |
| have he : ((law μ P).restrict G).map Φ= |
| ∑ a,(μ (P ⁻¹' {a}))⁻¹ • (((cell μ P a).prod (cell μ P a)).restrict G).map Φ := by |
| rw [law,←Measure.restrictₗ_apply,map_sum, |
| ←Measure.mapₗ_apply_of_measurable hΦ,map_sum] |
| simp only [Measure.restrictₗ_apply,Measure.restrict_smul, |
| Measure.mapₗ_apply_of_measurable hΦ,Measure.map_smul _ hΦ.aemeasurable] |
| rw [he,law,Finset.smul_sum] |
| apply Finset.sum_le_sum |
| intro a _ |
| simpa only [smul_smul,mul_comm] using |
| smul_le_smul_left ((μ (P ⁻¹' {a}))⁻¹) (hc a) |
|
|
| / |
| a law normalized by the retained mass. -/ |
| theorem first_output_domination (μ : Measure X) [IsFiniteMeasure μ] |
| (P : X → α) (hP : Measurable P) (G : Set (X × X)) |
| (Φ : X × X → X × X) (hΦ : Measurable Φ) (C : ℝ≥0∞) |
| (hdom : ((law μ P).restrict G).map Φ≤C • law μ P) : |
| ((law μ P).restrict G).map (fun z => (Φ z).1)≤C • μ := by |
| have h := Measure.map_mono hdom measurable_fst |
| rw [Measure.map_map measurable_fst hΦ, |
| Measure.map_smul _ measurable_fst.aemeasurable,first_marginal μ P hP] at h |
| exact h |
| end SmoothObstruction.PairLaw |
|
|
| namespace SmoothObstruction.PairLaw |
| open MeasureTheory Set Finset |
| open scoped ENNReal |
| variable {X α : Type*} [MeasurableSpace X] [Fintype α] |
| [MeasurableSpace α] [MeasurableSingletonClass α] |
|
|
| / |
| cell, including zero-mass cells without any special convention. -/ |
| lemma same_cell_ae (μ : Measure X) [IsFiniteMeasure μ] (P : X → α) |
| (hP : Measurable P) : ∀ᵐ z ∂law μ P,P z.1=P z.2 := by |
| classical |
| rw [law,ae_finsetSum_measure_iff] |
| intro a _ |
| apply Measure.ae_smul_measure |
| have hQ : MeasurableSet (P ⁻¹' {a}) := hP (measurableSet_singleton a) |
| rw [cell,Measure.prod_restrict] |
| filter_upwards [ae_restrict_mem (hQ.prod hQ)] with z hz |
| exact (show P z.1=a from hz.1).trans (show P z.2=a from hz.2).symm |
| |
| /-- Retention is measured in the original pair probability, rather than in |
| any conditional law on the good set. -/ |
| theorem retained_bound (μ : Measure X) [IsProbabilityMeasure μ] |
| (P : X → α) (hP : Measurable P) (R : Set X) (hR : MeasurableSet R) : |
| let G := (R ×ˢ R) ∩ {z : X×X | P z.1=P z.2} |
| MeasurableSet G ∧ (law μ P) Gᶜ≤2*μ Rᶜ ∧ |
| 1-2*μ.real Rᶜ≤(law μ P).real G := by |
| intro G |
| have hsame : MeasurableSet {z : X×X | P z.1=P z.2} := |
| measurableSet_eq_fun (hP.comp measurable_fst) (hP.comp measurable_snd) |
| have hG : MeasurableSet G := (hR.prod hR).inter hsame |
| have he : G=ᵐ[law μ P] R ×ˢ R := by |
| filter_upwards [same_cell_ae μ P hP] with z hz |
| apply propext |
| change (z∈R ×ˢ R ∧ P z.1=P z.2) ↔ z∈R ×ˢ R |
| simp only [hz,and_true] |
| have hl : (law μ P) Gᶜ≤2*μ Rᶜ := by |
| rw [measure_congr he.compl] |
| exact bad_pair_bound μ P hP R hR |
| let := law_probability μ P hP |
| refine ⟨hG,hl,?_⟩ |
| have hreal : (law μ P).real Gᶜ≤2*μ.real Rᶜ := by |
| have h := ENNReal.toReal_mono (show 2*μ Rᶜ≠∞ by finiteness) hl |
| simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofNat,Measure.real] using h |
| rw [probReal_compl_eq_one_sub hG] at hreal |
| linarith |
| |
| /-- A measurable canonical pair map can be checked using any cellwise local |
| extension agreeing on the retained set. Different extensions off that set do |
| not affect the ORIGINAL unnormalized pushforward. -/ |
| theorem glue_cell_domination (μ : Measure X) [IsFiniteMeasure μ] |
| (P : X → α) (hP : Measurable P) (G : Set (X×X)) (hG : MeasurableSet G) |
| (Φ : X×X → X×X) (hΦ : Measurable Φ) (C : ℝ≥0∞) |
| (hc : ∀ a,∃ F : X×X → X×X,Measurable F ∧ |
| EqOn F Φ G ∧ |
| (((cell μ P a).prod (cell μ P a)).restrict G).map F≤ |
| C • (cell μ P a).prod (cell μ P a)) : |
| ((law μ P).restrict G).map Φ≤C • law μ P ∧ |
| ((law μ P).restrict G).map (fun z => (Φ z).1)≤C • μ := by |
| have hd : ((law μ P).restrict G).map Φ≤C • law μ P := by |
| apply law_domination μ P G Φ hΦ C |
| intro a |
| obtain ⟨F,_,he,hF⟩ := hc a |
| have he' : F=ᵐ[((cell μ P a).prod (cell μ P a)).restrict G] Φ := |
| (ae_restrict_of_forall_mem hG (fun _ hz => he hz)) |
| rwa [Measure.map_congr he'] at hF |
| exact ⟨hd,first_output_domination μ P hP G Φ hΦ C hd⟩ |
| end SmoothObstruction.PairLaw |
| |
| namespace SmoothObstruction.OrbitSafety |
| open Set MeasureTheory |
| open scoped ENNReal |
| variable {X : Type*} [MeasurableSpace X] |
| |
| /-- A finite two-sided safety exclusion is charged to the original invariant |
| measure at every time, without conditioning the retained law. -/ |
| theorem bilateral_bad_bound (μ : Measure X) (f g : X → X) |
| (hf : MeasurePreserving f μ μ) (hg : MeasurePreserving g μ μ) |
| (W : Set X) (hW : MeasurableSet W) (n : ℕ) : |
| let Z := (⋃ t : Fin (n+1),(f^[t.val]) ⁻¹' W) ∪ |
| (⋃ t : Fin (n+1),(g^[t.val]) ⁻¹' W) |
| MeasurableSet Z ∧ μ Z≤(2*(n+1) : ℝ≥0∞)*μ W ∧ |
| ∀ x,x∉Z → ∀ t≤n,(f^[t]) x∉W ∧ (g^[t]) x∉W := by |
| intro Z |
| have hm : MeasurableSet Z := |
| (MeasurableSet.iUnion (fun t : Fin (n+1) => hW.preimage (hf.iterate t.val).measurable)).union |
| (MeasurableSet.iUnion (fun t : Fin (n+1) => hW.preimage (hg.iterate t.val).measurable)) |
| have h1 : μ (⋃ t : Fin (n+1),(f^[t.val]) ⁻¹' W)≤(n+1 : ℝ≥0∞)*μ W := by |
| calc |
| _ ≤ ∑ t : Fin (n+1),μ ((f^[t.val]) ⁻¹' W) := measure_iUnion_fintype_le μ _ |
| _ = _ := by simp only [(hf.iterate _).measure_preimage hW.nullMeasurableSet, |
| Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_add,Nat.cast_one] |
| have h2 : μ (⋃ t : Fin (n+1),(g^[t.val]) ⁻¹' W)≤(n+1 : ℝ≥0∞)*μ W := by |
| calc |
| _ ≤ ∑ t : Fin (n+1),μ ((g^[t.val]) ⁻¹' W) := measure_iUnion_fintype_le μ _ |
| _ = _ := by simp only [(hg.iterate _).measure_preimage hW.nullMeasurableSet, |
| Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_add,Nat.cast_one] |
| refine ⟨hm,?_,?_⟩ |
| · exact (measure_union_le _ _).trans ((add_le_add h1 h2).trans_eq (by ring)) |
| · intro x hx t ht |
| constructor |
| · intro h |
| exact hx (Or.inl (mem_iUnion.mpr ⟨⟨t,Nat.lt_succ_of_le ht⟩,h⟩)) |
| · intro h |
| exact hx (Or.inr (mem_iUnion.mpr ⟨⟨t,Nat.lt_succ_of_le ht⟩,h⟩)) |
| end SmoothObstruction.OrbitSafety |
| |
| namespace SmoothObstruction.RetainedGlobal |
| open Set MeasureTheory |
| open scoped ENNReal |
| variable {X : Type*} [MeasurableSpace X] |
| |
| lemma real_union_le (μ : Measure X) [IsFiniteMeasure μ] (A B : Set X) : |
| μ.real (A∪B)≤μ.real A+μ.real B := by |
| have h := ENNReal.toReal_mono (show μ A+μ B≠∞ by finiteness) (measure_union_le A B) |
| simpa only [Measure.real,ENNReal.toReal_add (measure_ne_top _ _) (measure_ne_top _ _)] using h |
| |
| /-- The actual one-point retained carrier. The fixed atlas core is checked |
| only at 0 and the two endpoints. Coarse margins, product margins, and all |
| boundary-tube checks are charged in the original stationary measure. -/ |
| theorem exists_retained (μ : Measure X) [IsProbabilityMeasure μ] |
| (f g : X → X) (hf : MeasurePreserving f μ μ) (hg : MeasurePreserving g μ μ) |
| (G V Wc Wp Wb : Set X) (hG : MeasurableSet G) (hV : MeasurableSet V) |
| (hWc : MeasurableSet Wc) (hWp : MeasurableSet Wp) (hWb : MeasurableSet Wb) (n : ℕ) : |
| ∃ R : Set X,MeasurableSet R ∧ R⊆G ∧ |
| μ.real Rᶜ≤μ.real Gᶜ+3*(μ.real Vᶜ+μ.real Wc)+μ.real Wp+ |
| 2*((n:ℝ)+1)*μ.real Wb ∧ |
| ∀ x∈R,x∈V ∧ x∉Wc ∧ x∉Wp ∧ |
| f^[n] x∈V ∧ f^[n] x∉Wc ∧ g^[n] x∈V ∧ g^[n] x∉Wc ∧ |
| ∀ t≤n,f^[t] x∉Wb ∧ g^[t] x∉Wb := by |
| let B := Vᶜ∪Wc |
| have hB : MeasurableSet B := hV.compl.union hWc |
| let Z := (⋃ t : Fin (n+1),(f^[t.val]) ⁻¹' Wb) ∪ |
| (⋃ t : Fin (n+1),(g^[t.val]) ⁻¹' Wb) |
| obtain ⟨hZ,hZm,hZsafe⟩ := OrbitSafety.bilateral_bad_bound μ f g hf hg Wb hWb n |
| let R := G∩(B∪(f^[n]) ⁻¹' B∪(g^[n]) ⁻¹' B∪Wp∪Z)ᶜ |
| have hR : MeasurableSet R := hG.inter |
| (((((hB.union (hB.preimage (hf.iterate n).measurable)).union |
| (hB.preimage (hg.iterate n).measurable)).union hWp).union hZ).compl) |
| have hmB : μ.real B≤μ.real Vᶜ+μ.real Wc := real_union_le μ _ _ |
| have hmF : μ.real ((f^[n]) ⁻¹' B)=μ.real B := by |
| exact congrArg ENNReal.toReal ((hf.iterate n).measure_preimage hB.nullMeasurableSet) |
| have hmG : μ.real ((g^[n]) ⁻¹' B)=μ.real B := by |
| exact congrArg ENNReal.toReal ((hg.iterate n).measure_preimage hB.nullMeasurableSet) |
| have hmZ : μ.real Z≤2*((n:ℝ)+1)*μ.real Wb := by |
| have hh := ENNReal.toReal_mono (show (2*((n:ℝ≥0∞)+1))*μ Wb≠∞ by finiteness) hZm |
| simpa only [Measure.real,ENNReal.toReal_mul,ENNReal.toReal_ofNat, |
| ENNReal.toReal_add (by finiteness : (n:ℝ≥0∞)≠∞) (by norm_num : (1:ℝ≥0∞)≠∞), |
| ENNReal.toReal_natCast,ENNReal.toReal_one] using hh |
| refine ⟨R,hR,inter_subset_left,?_,?_⟩ |
| · have he : Rᶜ=Gᶜ∪(B∪(f^[n]) ⁻¹' B∪(g^[n]) ⁻¹' B∪Wp∪Z) := by |
| simp only [R,compl_inter,compl_compl] |
| rw [he] |
| have h0 := real_union_le μ Gᶜ (B∪(f^[n]) ⁻¹' B∪(g^[n]) ⁻¹' B∪Wp∪Z) |
| have h1 := real_union_le μ (B∪(f^[n]) ⁻¹' B∪(g^[n]) ⁻¹' B∪Wp) Z |
| have h2 := real_union_le μ (B∪(f^[n]) ⁻¹' B∪(g^[n]) ⁻¹' B) Wp |
| have h3 := real_union_le μ (B∪(f^[n]) ⁻¹' B) ((g^[n]) ⁻¹' B) |
| have h4 := real_union_le μ B ((f^[n]) ⁻¹' B) |
| rw [hmF] at h4 |
| rw [hmG] at h3 |
| linarith |
| · intro x hx |
| have hh : x∉B ∧ f^[n] x∉B ∧ g^[n] x∉B ∧ x∉Wp ∧ x∉Z := by |
| have h := hx.2 |
| simp only [mem_compl_iff,mem_union,Set.mem_preimage,not_or] at h |
| exact ⟨h.1.1.1.1,h.1.1.1.2,h.1.1.2,h.1.2,h.2⟩ |
| have hb0 : x∈V ∧ x∉Wc := by simpa only [B,mem_union,mem_compl_iff,not_or,not_not] using hh.1 |
| have hbp : f^[n] x∈V ∧ f^[n] x∉Wc := by |
| simpa only [B,mem_union,mem_compl_iff,not_or,not_not] using hh.2.1 |
| have hbm : g^[n] x∈V ∧ g^[n] x∉Wc := by |
| simpa only [B,mem_union,mem_compl_iff,not_or,not_not] using hh.2.2.1 |
| exact ⟨hb0.1,hb0.2,hh.2.2.2.1,hbp.1,hbp.2,hbm.1,hbm.2,hZsafe x hh.2.2.2.2⟩ |
|
|
| / |
| without replacing cell weights by good-set weights. -/ |
| theorem pair_mass {α : Type*} [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α] |
| (μ : Measure X) [IsProbabilityMeasure μ] (P : X → α) (hP : Measurable P) |
| (R : Set X) (hR : MeasurableSet R) {η : ℝ} (hm : μ.real Rᶜ≤η/2) : |
| let G := (R ×ˢ R)∩{z : X×X | P z.1=P z.2} |
| MeasurableSet G ∧ 1-η≤(PairLaw.law μ P).real G := by |
| intro G |
| obtain ⟨hG,_,hb⟩ := PairLaw.retained_bound μ P hP R hR |
| exact ⟨hG,by linarith⟩ |
| end SmoothObstruction.RetainedGlobal |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set MeasureTheory FiniteBins |
| 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] [ChartedSpace H M] |
| [MeasurableSpace M] [BorelSpace M] |
| {ι C : Type*} [Fintype ι] [Fintype C] |
| [MeasurableSpace C] [MeasurableSingletonClass C] |
| open ChartDensity |
|
|
| / |
| gauge reference per active starting bin. Wall strips are charged to the ORIGINAL |
| measure. No endpoint or orbit chart code is included in the cell alphabet. The |
| recovery assertion is precisely the ideal exchange needed by both joins. -/ |
| theorem finite_product_partition [IsManifold I ∞ M] (μ : Measure M) [IsProbabilityMeasure μ] |
| (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) |
| (c : M → ι) (hc : ∀ x,x∈K (c x)) : |
| ∃ D : ℝ,1≤D ∧ ∀ (P : M → C),Measurable P → ∀ (chart : C → ι), |
| (∀ x,chart (P x)=c x) → ∀ (G : Set M) (n : ℕ) (B R : ℝ),0≤B → |
| ∀ (A : ∀ x:G,BilateralGauge I S a c n B R x) (T : G → Finset (Fin d)) |
| (δ : ℝ),0<δ → ∀ N : ℕ,0<N → |
| let H₀ := Real.sqrt 2*Real.exp (2*B) |
| let m := ⌈H₀*D/(δ*N)+1⌉₊ |
| ∃ (L : M → C × (Fin d → Fin (2*m+1))) (W : Set M), |
| Measurable L ∧ MeasurableSet W ∧ μ.real W≤(d:ℝ)/N ∧ |
| (∀ x,(L x).1=P x) ∧ |
| (∀ x y,L x=L y → |
| ‖coordinates I (a (c x)) x-coordinates I (a (c x)) y‖≤ |
| Real.sqrt 2*((d:ℝ)+1)*(δ*N)) ∧ |
| (∀ (x y z : M) (hx : x∈G),L x=L y → x∉W → y∉W → P z=P x → |
| let ref : G := (reference P G (key P G ⟨x,hx⟩)).val |
| H₀*‖coordinates I (a (c x)) z-(coordinates I (a (c x)) x+ |
| (((A ref).forward I S a c).splitting (T ref)ᶜ).slow 0 |
| (coordinates I (a (c x)) y-coordinates I (a (c x)) x))‖≤δ/4 → |
| L z=L x) ∧ |
| Fintype.card (C × (Fin d → Fin (2*m+1)))=Fintype.card C*(2*m+1)^d := by |
| obtain ⟨D,hD,hbound⟩ := selected_coordinate_bound I a K hK hKU c hc |
| refine ⟨D,hD,?_⟩ |
| intro P hP chart hchart G n B R hB A T δ hδ N hN H₀ m |
| let ref (Q : Active P G) : G := (reference P G Q).val |
| obtain ⟨E,hE,hEi,hmix⟩ := product_coordinates P G hB |
| (fun Q => (A (ref Q)).forward I S a c) (fun Q => (T (ref Q))ᶜ) |
| let φ (Q : C) (x : M) := coordinates I (a (chart Q)) x |
| have hb (x : M) : ‖φ (P x) x‖≤D := by simpa only [φ,hchart] using hbound x |
| obtain ⟨L,W,hL,hW,hWm,hLP,hdiam,hstable,hcard⟩ := ProductCells.exists_cells μ P hP φ |
| (fun Q => measurable_coordinates I (a (chart Q))) E (show 0≤D by linarith) |
| (show 0≤H₀ by dsimp [H₀]; positivity) (Real.sqrt_nonneg _) hδ hb hEi hE hN |
| refine ⟨L,W,hL,hW,hWm,hLP,?_,?_,hcard⟩ |
| · intro x y hxy |
| simpa only [φ,hchart] using hdiam x y hxy |
| · intro x y z hx hxy hxW hyW hz ref₀ herror |
| apply hstable x y z hxy hxW hyW hz |
| let Q : Active P G := key P G ⟨x,hx⟩ |
| let ideal := φ (P x) x+(((A (ref Q)).forward I S a c).splitting (T (ref Q))ᶜ).slow 0 |
| (φ (P x) y-φ (P x) x) |
| have herr : H₀*‖φ (P x) z-ideal‖≤δ/4 := by |
| simpa only [ideal,φ,hchart,ref,Q] using herror |
| have he (i : Fin d) : |((E (P x)).symm (φ (P x) z)) i-((E (P x)).symm ideal) i|≤δ/4 := by |
| calc |
| _ = ‖(((E (P x)).symm (φ (P x) z-ideal)) i)‖ := by rw [map_sub]; rfl |
| _ ≤ ‖(E (P x)).symm (φ (P x) z-ideal)‖ := PiLp.norm_apply_le _ _ |
| _ ≤ ‖(E (P x)).symm.toContinuousLinearMap‖*‖φ (P x) z-ideal‖ := |
| (E (P x)).symm.toContinuousLinearMap.le_opNorm _ |
| _ ≤ H₀*‖φ (P x) z-ideal‖ := mul_le_mul_of_nonneg_right (hEi _) (norm_nonneg _) |
| _ ≤ δ/4 := herr |
| intro i |
| have hh := hmix Q (φ (P x) x) (φ (P x) y) i |
| change ((E (P x)).symm ideal) i=((E (P x)).symm (φ (P x) x)) i ∨ |
| ((E (P x)).symm ideal) i=((E (P x)).symm (φ (P x) y)) i at hh |
| rcases hh with hh|hh |
| · exact Or.inl (by simpa only [hh] using he i) |
| · exact Or.inr (by simpa only [hh] using he i) |
| 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*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| balls. This is a consequence of the identity diffeomorphism's genuine smooth |
| coordinate maps, not an assumed Riemannian/Lipschitz metric. -/ |
| theorem uniform_chart_transfer (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 ∧ ∀ (i j : ι) (x y : M) (δ : ℝ), |
| x∈K i → x∈K j → 0≤δ → δ≤r → |
| closedBall ((extChartAt I (a i)) x) δ⊆interior (range I) → |
| y∈(chartAt H (a i)).source → |
| ‖(extChartAt I (a i)) y-(extChartAt I (a i)) x‖≤δ → |
| y∈(chartAt H (a j)).source ∧ |
| ‖(extChartAt I (a j)) y-(extChartAt I (a j)) x‖≤ |
| C*‖(extChartAt I (a i)) y-(extChartAt I (a i)) x‖ := by |
| let Id : Diffeomorph I I M M ∞ := Diffeomorph.refl I M ∞ |
| obtain ⟨r,C,hr,hC,ht⟩ := actual_tube_data I hclosed Id a K hK hKU |
| refine ⟨r,C,hr,hC,fun i j x y δ hxi hxj hδ hδr hsafe hyi hxy => ?_⟩ |
| have hd := ht i j x δ hxi hxj hδ hδr hsafe |
| have hyb : (extChartAt I (a i)) y∈closedBall ((extChartAt I (a i)) x) δ := |
| mem_closedBall_iff_norm.mpr hxy |
| have hyj := (hd.2.1 hyb).2 |
| have hiy : (extChartAt I (a i)).symm ((extChartAt I (a i)) y)=y := |
| (extChartAt I (a i)).left_inv (by simpa only [extChartAt_source] using hyi) |
| have hix : (extChartAt I (a i)).symm ((extChartAt I (a i)) x)=x := |
| (extChartAt I (a i)).left_inv (by simpa only [extChartAt_source] using hKU i hxi) |
| change Id ((extChartAt I (a i)).symm ((extChartAt I (a i)) y))∈(chartAt H (a j)).source at hyj |
| rw [hiy] at hyj |
| refine ⟨hyj,?_⟩ |
| have hbound := (convex_closedBall ((extChartAt I (a i)) x) δ).norm_image_sub_le_of_norm_fderiv_le |
| (𝕜:=ℝ) (f:=localMap I Id (a i) (a j)) |
| (fun u hu => (hd.2.2 u hu).1.differentiableAt (by norm_num)) |
| (fun u hu => (hd.2.2 u hu).2.1) (mem_closedBall_self hδ) hyb |
| simpa only [localMap,Function.comp_apply,hix,hiy,Id,Diffeomorph.coe_refl,id_eq] using hbound |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| 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] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| |
| /-- A common output in two small selected tubes puts their centers in each |
| other's original chart, with a uniform linear coordinate bound. The chart |
| selector can be discontinuous: only its finite compact carrying patches are |
| used. This is the comparison needed to invoke old-tube rigidity when two |
| canonical switches have the same output. -/ |
| theorem uniform_common_output (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 ∧ ∀ (i j : ι) (x y z : M) (ρ : ℝ), |
| x∈K i → y∈K j → 0≤ρ → ρ≤r → |
| closedBall ((extChartAt I (a j)) y) ρ⊆interior (range I) → |
| z∈(chartAt H (a i)).source → z∈(chartAt H (a j)).source → |
| ‖(extChartAt I (a i)) z-(extChartAt I (a i)) x‖≤ρ → |
| ‖(extChartAt I (a j)) z-(extChartAt I (a j)) y‖≤ρ → |
| y∈(chartAt H (a i)).source ∧ |
| ‖(extChartAt I (a i)) y-(extChartAt I (a i)) x‖≤C*ρ := by |
| have hex (i : ι) : ∃ v : ℝ,0<v ∧ cthickening v (K i)⊆(chartAt H (a i)).source := |
| (hK i).exists_cthickening_subset_open (chartAt H (a i)).open_source (hKU i) |
| choose v hv hvU using hex |
| obtain ⟨v₀,hv₀,hv₀i⟩ := CarriedCharts.finite_positive_lower v hv |
| let L (i : ι) := cthickening v₀ (K i) |
| have hL (i : ι) : IsCompact (L i) := (hK i).cthickening |
| have hLU (i : ι) : L i⊆(chartAt H (a i)).source := |
| (cthickening_mono (hv₀i i) (K i)).trans (hvU i) |
| obtain ⟨r,C,hr,hC,ht⟩ := uniform_chart_transfer I hclosed a L hL hLU |
| obtain ⟨s,hs,hm⟩ := CoordinateTubes.uniform_inverse_modulus I hclosed a K hK hKU |
| (ε:=v₀/3) (by positivity) |
| refine ⟨min r s,C+1,lt_min hr hs,by linarith,?_⟩ |
| intro i j x y z ρ hxi hyj hρ hρr hsafe hzi hzj hzx hzy |
| have hρs : ρ ≤ s := hρr.trans (min_le_right _ _) |
| have hzx' := hm i x hxi z hzi (hzx.trans hρs) |
| have hzy' := hm j y hyj z hzj (hzy.trans hρs) |
| have hyx : dist y x≤v₀ := by |
| have hh := dist_triangle y z x |
| rw [dist_comm y z] at hh |
| linarith |
| have hyi : y∈L i := mem_cthickening_of_dist_le _ _ _ _ hxi hyx |
| have hyj' : y∈L j := self_subset_cthickening (K j) hyj |
| have ht' := ht j i y z ρ hyj' hyi hρ (hρr.trans (min_le_left _ _)) hsafe hzj hzy |
| refine ⟨hLU i hyi,?_⟩ |
| have htri := norm_sub_le_norm_sub_add_norm_sub ((extChartAt I (a i)) y) ((extChartAt I (a i)) z) |
| ((extChartAt I (a i)) x) |
| rw [norm_sub_rev ((extChartAt I (a i)) y) ((extChartAt I (a i)) z)] at htri |
| have hcz : C*‖(extChartAt I (a j)) z-(extChartAt I (a j)) y‖≤C*ρ := |
| mul_le_mul_of_nonneg_left hzy (by linarith) |
| nlinarith [ht'.2] |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.SwitchRecovery |
| open Set |
| variable {X α β E F : Type*} |
|
|
| / |
| maps even when the two inputs were certified by different frame witnesses. -/ |
| theorem global_injective_tubes |
| (c : X → α) (b : X → β) (future past : X → X) |
| (U : α → β → X → E) (V : α → β → X → F) |
| (TF TB RF RB : X → X → Prop) (R : Set X) (G : Set (X × X)) |
| (Φ : X × X → X × X) |
| (hcommonF : ∀ x∈R,∀ y∈R,∀ z,TF x z → TF y z → RF x y) |
| (hcommonB : ∀ x∈R,∀ y∈R,∀ z,TB x z → TB y z → RB x y) |
| (hgood : ∀ u∈G,u.1∈R ∧ u.2∈R) |
| (hc : ∀ u∈G,c u.1=c u.2 ∧ c (Φ u).1=c u.1) |
| (hb : ∀ u∈G, |
| b (future (Φ u).1)=b (future u.1) ∧ |
| b (past (Φ u).2)=b (past u.1) ∧ |
| b (future (Φ u).2)=b (future u.2) ∧ |
| b (past (Φ u).1)=b (past u.2)) |
| (heq : ∀ u∈G, |
| U (c u.1) (b (future u.1)) (Φ u).1=U (c u.1) (b (future u.1)) u.1 ∧ |
| V (c u.1) (b (past u.1)) (Φ u).2=V (c u.1) (b (past u.1)) u.1 ∧ |
| U (c u.2) (b (future u.2)) (Φ u).2=U (c u.2) (b (future u.2)) u.2 ∧ |
| V (c u.2) (b (past u.2)) (Φ u).1=V (c u.2) (b (past u.2)) u.2) |
| (hshadow : ∀ u∈G,TF u.1 (Φ u).1 ∧ TB u.1 (Φ u).2 ∧ |
| TF u.2 (Φ u).2 ∧ TB u.2 (Φ u).1) |
| (hrigid : ∀ x∈R,∀ y, |
| U (c x) (b (future x)) y=U (c x) (b (future x)) x → |
| V (c x) (b (past x)) y=V (c x) (b (past x)) x → |
| RF x y → RB x y → y=x) : |
| InjOn Φ G := by |
| intro u hu v hv huv |
| have hc₁ : c v.1=c u.1 := (hc v hv).2.symm.trans (by rw [←huv]; exact (hc u hu).2) |
| have hc₂ : c v.2=c u.2 := (hc v hv).1.symm.trans (hc₁.trans (hc u hu).1) |
| have hb₁ : b (future v.1)=b (future u.1) := (hb v hv).1.symm.trans (by rw [←huv]; exact (hb u hu).1) |
| have hb₂ : b (past v.1)=b (past u.1) := (hb v hv).2.1.symm.trans (by rw [←huv]; exact (hb u hu).2.1) |
| have hb₃ : b (future v.2)=b (future u.2) := (hb v hv).2.2.1.symm.trans (by rw [←huv]; exact (hb u hu).2.2.1) |
| have hb₄ : b (past v.2)=b (past u.2) := (hb v hv).2.2.2.symm.trans (by rw [←huv]; exact (hb u hu).2.2.2) |
| have he₁ := (heq v hv).1.symm |
| have he₂ := (heq v hv).2.1.symm |
| have he₃ := (heq v hv).2.2.1.symm |
| have he₄ := (heq v hv).2.2.2.symm |
| rw [hc₁,hb₁,←huv] at he₁ |
| rw [hc₁,hb₂,←huv] at he₂ |
| rw [hc₂,hb₃,←huv] at he₃ |
| rw [hc₂,hb₄,←huv] at he₄ |
| have hsu := hshadow u hu |
| have hsv := hshadow v hv |
| rw [←huv] at hsv |
| have hxu := (hgood u hu).1 |
| have hyu := (hgood u hu).2 |
| have hxv := (hgood v hv).1 |
| have hyv := (hgood v hv).2 |
| exact Prod.ext |
| (hrigid u.1 hxu v.1 (he₁.trans (heq u hu).1) (he₂.trans (heq u hu).2.1) |
| (hcommonF u.1 hxu v.1 hxv (Φ u).1 hsu.1 hsv.1) |
| (hcommonB u.1 hxu v.1 hxv (Φ u).2 hsu.2.1 hsv.2.1)).symm |
| (hrigid u.2 hyu v.2 (he₃.trans (heq u hu).2.2.1) (he₄.trans (heq u hu).2.2.2) |
| (hcommonF u.2 hyu v.2 hyv (Φ u).2 hsu.2.2.1 hsv.2.2.1) |
| (hcommonB u.2 hyu v.2 hyv (Φ u).1 hsu.2.2.2 hsv.2.2.2)).symm |
| end SmoothObstruction.SwitchRecovery |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| 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] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| intermediate selected charts being allowed to differ. -/ |
| theorem original_common_tubes (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 ∧ ∀ (S : Diffeomorph I I M M ∞) |
| (c : M → ι) (n : ℕ) (ρ δ : ℝ) (x y z : M), |
| (∀ v,v∈K (c v)) → 0≤ρ → ρ≤r → C*ρ≤δ → |
| (∀ t : Fin (n+1),closedBall ((extChartAt I (a (c (S^[t.val] y)))) (S^[t.val] y)) ρ |
| ⊆interior (range I)) → |
| originalTube I S a c (n:=n) ρ x z → originalTube I S a c (n:=n) ρ y z → |
| originalTube I S a c (n:=n) δ x y := by |
| obtain ⟨r,C,hr,hC,hh⟩ := uniform_common_output I hclosed a K hK hKU |
| refine ⟨r,C,hr,hC,?_⟩ |
| intro S c n ρ δ x y z hc hρ hρr hδ hsafe hx hy t |
| obtain ⟨hs,hb⟩ := hh (c (S^[t.val] x)) (c (S^[t.val] y)) |
| (S^[t.val] x) (S^[t.val] y) (S^[t.val] z) ρ (hc _) (hc _) |
| hρ hρr (hsafe t) (hx t).1 (hy t).1 (hx t).2 (hy t).2 |
| exact ⟨hs,hb.trans hδ⟩ |
|
|
| omit [ProperSpace M] [IsManifold I ∞ M] [Fintype ι] in |
| / |
| switch globally one-to-one. The four endpoint equations are the original |
| ones, and rigidity is invoked in OLD charts, not independently chosen ones. -/ |
| theorem original_switch_injective {C J : Type*} |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (c : M → ι) |
| (P : M → C) (L : M → J) (π : J → C) (chart : C → ι) |
| (hπ : ∀ x,π (L x)=P x) (hchart : ∀ x,chart (P x)=c x) |
| (n : ℕ) (δ ρ : ℝ) (R : Set M) (G : Set (M×M)) (Φ : M×M → M×M) |
| (Wf Wb : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))) |
| (hcommonF : ∀ x∈R,∀ y∈R,∀ z, |
| originalTube I S a c (n:=n) ρ x z → originalTube I S a c (n:=n) ρ y z → |
| originalTube I S a c (n:=n) δ x y) |
| (hcommonB : ∀ x∈R,∀ y∈R,∀ z, |
| originalTube I S.symm a c (n:=n) ρ x z → originalTube I S.symm a c (n:=n) ρ y z → |
| originalTube I S.symm a c (n:=n) δ x y) |
| (hgood : ∀ u∈G,u.1∈R ∧ u.2∈R) |
| (hcell : ∀ u∈G,L u.1=L u.2 ∧ L (Φ u).1=L u.1) |
| (hlabels : ∀ u∈G, |
| P (S^[n] (Φ u).1)=P (S^[n] u.1) ∧ |
| P (S.symm^[n] (Φ u).2)=P (S.symm^[n] u.1) ∧ |
| P (S^[n] (Φ u).2)=P (S^[n] u.2) ∧ |
| P (S.symm^[n] (Φ u).1)=P (S.symm^[n] u.2)) |
| (heq : ∀ u∈G, |
| originalEndpoint I S a c (n:=n) (Wf (P u.1,P (S^[n] u.1))) u.1 (Φ u).1 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u.1,P (S.symm^[n] u.1))) u.1 (Φ u).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P u.2,P (S^[n] u.2))) u.2 (Φ u).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u.2,P (S.symm^[n] u.2))) u.2 (Φ u).1) |
| (hshadow : ∀ u∈G, |
| originalTube I S a c (n:=n) ρ u.1 (Φ u).1 ∧ |
| originalTube I S.symm a c (n:=n) ρ u.1 (Φ u).2 ∧ |
| originalTube I S a c (n:=n) ρ u.2 (Φ u).2 ∧ |
| originalTube I S.symm a c (n:=n) ρ u.2 (Φ u).1) |
| (hrigid : ∀ x∈R,∀ z, |
| originalTube I S a c (n:=n) δ x z → originalTube I S.symm a c (n:=n) δ x z → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x z → z=x) : |
| InjOn Φ G := by |
| let U : J → C → M → EuclideanSpace ℝ (Fin d) := fun Q b z => |
| (Wf (π Q,b)).starProjection ((extChartAt I (a (chart b))) (S^[n] z)) |
| let V : J → C → M → EuclideanSpace ℝ (Fin d) := fun Q b z => |
| (Wb (π Q,b)).starProjection ((extChartAt I (a (chart b))) (S.symm^[n] z)) |
| have hf (x z : M) : U (L x) (P (S^[n] x)) z=U (L x) (P (S^[n] x)) x ↔ |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z := by |
| simp only [U,hπ,hchart,originalEndpoint] |
| have hb (x z : M) : V (L x) (P (S.symm^[n] x)) z=V (L x) (P (S.symm^[n] x)) x ↔ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x z := by |
| simp only [V,hπ,hchart,originalEndpoint] |
| refine SwitchRecovery.global_injective_tubes L P (S^[n]) (S.symm^[n]) U V |
| (originalTube I S a c (n:=n) ρ) (originalTube I S.symm a c (n:=n) ρ) |
| (originalTube I S a c (n:=n) δ) (originalTube I S.symm a c (n:=n) δ) |
| R G Φ hcommonF hcommonB hgood hcell hlabels ?_ hshadow ?_ |
| · intro u hu |
| obtain ⟨h1,h2,h3,h4⟩ := heq u hu |
| exact ⟨(hf _ _).mpr h1,(hb _ _).mpr h2,(hf _ _).mpr h3,(hb _ _).mpr h4⟩ |
| · intro x hx z h1 h2 h3 h4 |
| exact hrigid x hx z h3 h4 ((hf _ _).mp h1) ((hb _ _).mp h2) |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.AtlasCores |
| open Set Filter Topology Metric |
| variable {X E ι C : Type*} [MetricSpace X] [NormedAddCommGroup E] |
|
|
| / |
| selected chart makes the original coarse label locally constant. -/ |
| theorem coarse_label_germ (c : X → ι) (P : X → C) (φ : ι → X → E) |
| (U : ι → Set X) (hU : ∀ i,IsOpen (U i)) |
| (hφ : ∀ i,ContinuousOn (φ i) (U i)) |
| (x : X) (hxU : x∈U (c x)) (s δ : ℝ) (hs : 0<s) (hδ : 0<δ) |
| (hc : ∀ y,dist y x ≤ s → c y=c x) |
| (hrec : ∀ y,c y=c x → ‖φ (c x) y-φ (c x) x‖≤δ → P y=P x) : |
| ∀ᶠ y in 𝓝 x,P y=P x := by |
| have hcont : ContinuousAt (fun y => ‖φ (c x) y-φ (c x) x‖) x := |
| (((hφ _).continuousAt ((hU _).mem_nhds hxU)).sub continuousAt_const).norm |
| have hz : ‖φ (c x) x-φ (c x) x‖<δ := by simpa using hδ |
| filter_upwards [Metric.ball_mem_nhds x hs,hcont.eventually (gt_mem_nhds hz)] with y hy hy' |
| exact hrec y (hc y (Metric.mem_ball.mp hy).le) hy'.le |
|
|
| / |
| this finite local intersection is not charged to a partition alphabet. -/ |
| theorem finite_label_germ {T : Type*} [Finite T] |
| (P : X → C) (f : T → X → X) (x : X) |
| (hf : ∀ t,ContinuousAt (f t) x) |
| (hP : ∀ t,∀ᶠ y in 𝓝 (f t x),P y=P (f t x)) : |
| ∀ᶠ y in 𝓝 x,∀ t,P (f t y)=P (f t x) := by |
| exact Filter.eventually_all.mpr (fun t => (hf t).eventually (hP t)) |
| end SmoothObstruction.AtlasCores |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric MeasureTheory 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] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| before the time length, orbit, chart selector, or frame witness. -/ |
| theorem uniform_original_shadow (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) |
| {ε : ℝ} (hε : 0<ε) : |
| ∃ σ : ℝ,0<σ ∧ ∀ (S : Diffeomorph I I M M ∞) (c : M → ι) |
| (n : ℕ) (ρ : ℝ) (x z : M), (∀ u,u∈K (c u)) → ρ≤σ → |
| originalTube I S a c (n:=n) ρ x z → |
| ∀ t : Fin (n+1),dist (S^[t.val] z) (S^[t.val] x)<ε := by |
| obtain ⟨σ,hσ,hm⟩ := CoordinateTubes.uniform_inverse_modulus I hclosed a K hK hKU hε |
| refine ⟨σ,hσ,?_⟩ |
| intro S c n ρ x z hc hρ ht t |
| exact hm _ _ (hc _) _ (ht t).1 ((ht t).2.trans hρ) |
|
|
| / |
| itinerary in the partition. The only exclusions are the fixed chart core and |
| old coarse-grid wall strip at the particular time under consideration. -/ |
| theorem original_tube_labels |
| (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) |
| {s : ℝ} (hs : 0<s) : |
| ∃ σ : ℝ,0<σ ∧ ∀ {C : Type*} (S : Diffeomorph I I M M ∞) (c : M → ι) |
| (P : M → C) (V W : Set M) (δ ρ : ℝ) (n : ℕ), |
| (∀ u,u∈K (c u)) → |
| (∀ x∈V,∀ z,dist z x ≤ s → c z=c x) → |
| (∀ x z,x∉W → c z=c x → |
| ‖ChartDensity.coordinates I (a (c x)) z-ChartDensity.coordinates I (a (c x)) x‖≤δ/4 → |
| P z=P x) → ρ≤σ → ρ≤δ/4 → |
| ∀ x z,originalTube I S a c (n:=n) ρ x z → |
| ∀ t : Fin (n+1),S^[t.val] x∈V → S^[t.val] x∉W → P (S^[t.val] z)=P (S^[t.val] x) := by |
| obtain ⟨σ,hσ,hm⟩ := uniform_original_shadow I hclosed a K hK hKU hs |
| refine ⟨σ,hσ,?_⟩ |
| intro C S c P V W δ ρ n hc hcore hrec hρσ hρδ x z ht t hv hw |
| apply hrec _ _ hw (hcore _ hv _ ((hm S c n ρ x z hc hρσ ht t).le)) |
| rw [ChartDensity.coordinates_eq I _ _ (by simpa only [extChartAt_source] using (ht t).1), |
| ChartDensity.coordinates_eq I _ _ (by simpa only [extChartAt_source] using hKU _ (hc _))] |
| exact (ht t).2.trans hρδ |
|
|
| / |
| sharp tube radii tending to zero. This does not assume a rate for an arbitrary |
| compatible metric on the compact manifold. -/ |
| theorem original_shadow_tendsto (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) |
| (S : Diffeomorph I I M M ∞) (c : M → ι) (hc : ∀ u,u∈K (c u)) |
| (n : ℕ → ℕ) (ρ : ℕ → ℝ) (hρ : Tendsto ρ atTop (𝓝 0)) |
| (G : ℕ → Set (M×M)) (Φ : ℕ → M×M → M×M) |
| (ht : ∀ᶠ k in atTop,∀ u∈G k, |
| originalTube I S a c (n:=n k) (ρ k) u.1 (Φ k u).1 ∧ |
| originalTube I S.symm a c (n:=n k) (ρ k) u.2 (Φ k u).1) : |
| ∀ ε : ℝ,0<ε → ∀ᶠ k in atTop,∀ u∈G k,∀ t v : Fin (n k+1), |
| dist (S^[t.val] (Φ k u).1) (S^[t.val] u.1)+ |
| dist (S.symm^[v.val] (Φ k u).1) (S.symm^[v.val] u.2)<ε := by |
| intro ε hε |
| obtain ⟨σ,hσ,hm⟩ := uniform_original_shadow I hclosed a K hK hKU (half_pos hε) |
| filter_upwards [ht,hρ.eventually (gt_mem_nhds hσ)] with k hk hkρ u hu t v |
| have h₁ := hm S c (n k) (ρ k) u.1 (Φ k u).1 hc hkρ.le (hk u hu).1 t |
| have h₂ := hm S.symm c (n k) (ρ k) u.2 (Φ k u).1 hc hkρ.le (hk u hu).2 v |
| linarith |
|
|
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric MeasureTheory |
| open scoped Manifold ContDiff |
| 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] |
|
|
| / |
| full ORIGINAL cells and all four endpoint labels. Common-output old tubes |
| then prove global injectivity, even for unequal selected intermediate charts. |
| No partition is refined by an endpoint, frame witness, or orbit itinerary. -/ |
| theorem recovered_original_pair (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) |
| {s : ℝ} (hs : 0<s) : |
| ∃ σ r₁ C₁ : ℝ,0<σ ∧ 0<r₁ ∧ 1≤C₁ ∧ |
| ∀ {C J : Type*} (S : Diffeomorph I I M M ∞) (c : M → ι) |
| (P : M → C) (L : M → J) (π : J → C) (chart : C → ι) |
| (_ : ∀ x,π (L x)=P x) (_ : ∀ x,chart (P x)=c x) |
| (n : ℕ) (δ ρ δc δp H₀ E : ℝ) (R V Wc Wp : Set M) |
| (U : M → EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)) |
| (Φ : M×M → M×M) (Wf Wb : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))), |
| (∀ x,x∈K (c x)) → 0≤ρ → ρ≤σ → ρ≤δc/4 → ρ≤r₁ → C₁*ρ≤δ → |
| 0≤H₀ → H₀*E≤δp/4 → |
| (∀ x∈V,∀ z,dist z x ≤ s → c z=c x) → |
| (∀ x z,x∉Wc → c z=c x → |
| ‖ChartDensity.coordinates I (a (c x)) z-ChartDensity.coordinates I (a (c x)) x‖≤δc/4 → P z=P x) → |
| (∀ x y z,x∈R → L x=L y → x∉Wp → y∉Wp → P z=P x → |
| H₀*‖ChartDensity.coordinates I (a (c x)) z-(ChartDensity.coordinates I (a (c x)) x+ |
| U x (ChartDensity.coordinates I (a (c x)) y-ChartDensity.coordinates I (a (c x)) x))‖≤δp/4 → L z=L x) → |
| (∀ x∈R,x∈V ∧ x∉Wc ∧ x∉Wp ∧ |
| S^[n] x∈V ∧ S^[n] x∉Wc ∧ S.symm^[n] x∈V ∧ S.symm^[n] x∉Wc) → |
| (∀ x∈R,∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) δ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) δ⊆interior (range I)) → |
| (∀ u : M×M,u.1∈R → u.2∈R → L u.1=L u.2 → |
| originalTube I S a c (n:=n) ρ u.1 (Φ u).1 ∧ |
| originalTube I S.symm a c (n:=n) ρ u.2 (Φ u).1 ∧ |
| originalTube I S a c (n:=n) ρ u.2 (Φ u).2 ∧ |
| originalTube I S.symm a c (n:=n) ρ u.1 (Φ u).2 ∧ |
| ‖(extChartAt I (a (c u.1))) (Φ u).1-(extChartAt I (a (c u.1))) u.1- |
| U u.1 ((extChartAt I (a (c u.2))) u.2-(extChartAt I (a (c u.1))) u.1)‖≤E ∧ |
| ‖(extChartAt I (a (c u.2))) (Φ u).2-(extChartAt I (a (c u.2))) u.2- |
| U u.2 ((extChartAt I (a (c u.1))) u.1-(extChartAt I (a (c u.2))) u.2)‖≤E) → |
| (∀ u : M×M,u.1∈R → u.2∈R → L u.1=L u.2 → |
| originalEndpoint I S a c (n:=n) (Wf (P u.1,P (S^[n] u.1))) u.1 (Φ u).1 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u.1,P (S.symm^[n] u.1))) u.1 (Φ u).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P u.2,P (S^[n] u.2))) u.2 (Φ u).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u.2,P (S.symm^[n] u.2))) u.2 (Φ u).1) → |
| (∀ x∈R,∀ z, |
| originalTube I S a c (n:=n) δ x z → originalTube I S.symm a c (n:=n) δ x z → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x z → z=x) → |
| let G := {u : M×M | u.1∈R ∧ u.2∈R ∧ L u.1=L u.2} |
| (∀ u∈G,L (Φ u).1=L u.1 ∧ L (Φ u).2=L u.2) ∧ |
| (∀ u∈G,P (S^[n] (Φ u).1)=P (S^[n] u.1) ∧ |
| P (S.symm^[n] (Φ u).2)=P (S.symm^[n] u.1) ∧ |
| P (S^[n] (Φ u).2)=P (S^[n] u.2) ∧ |
| P (S.symm^[n] (Φ u).1)=P (S.symm^[n] u.2)) ∧ InjOn Φ G := by |
| obtain ⟨σ,hσ,hl⟩ := original_tube_labels I hclosed a K hK hKU hs |
| obtain ⟨r₁,C₁,hr₁,hC₁,hcommon⟩ := original_common_tubes I hclosed a K hK hKU |
| refine ⟨σ,r₁,C₁,hσ,hr₁,hC₁,?_⟩ |
| intro C J S c P L π chart hπ hchart n δ ρ δc δp H₀ E R V Wc Wp U Φ Wf Wb |
| hc hρ hρσ hρc hρr hρδ hH₀ hE hcore hcoarse hprod hret hsafe ht heq hrigid G |
| have hρδ' : ρ≤δ := (le_mul_of_one_le_left hρ hC₁).trans hρδ |
| have hPs {x y : M} (hh : L x=L y) : P x=P y := by rw [←hπ x,←hπ y,hh] |
| have hcs {x y : M} (hh : L x=L y) : c x=c y := by rw [←hchart x,←hchart y,hPs hh] |
| have hlabel (F : Diffeomorph I I M M ∞) (x z : M) |
| (hh : originalTube I F a c (n:=n) ρ x z) (t : Fin (n+1)) |
| (hv : F^[t.val] x∈V) (hw : F^[t.val] x∉Wc) : P (F^[t.val] z)=P (F^[t.val] x) := |
| hl F c P V Wc δc ρ n hc hcore hcoarse hρσ hρc x z hh t hv hw |
| have hret0 (x : M) (hx : x∈R) := hret x hx |
| have hsrc (x : M) : x∈(extChartAt I (a (c x))).source := by |
| simpa only [extChartAt_source] using hKU _ (hc x) |
| have hcell (u : M×M) (hu : u∈G) : L (Φ u).1=L u.1 ∧ L (Φ u).2=L u.2 := by |
| obtain ⟨htf,htb,hsf,hsb,hiz,hiw⟩ := ht u hu.1 hu.2.1 hu.2.2 |
| have hzP : P (Φ u).1=P u.1 := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply] using hlabel S u.1 (Φ u).1 htf 0 |
| (by simpa using (hret0 u.1 hu.1).1) (by simpa using (hret0 u.1 hu.1).2.1) |
| have hwP : P (Φ u).2=P u.2 := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply] using hlabel S u.2 (Φ u).2 hsf 0 |
| (by simpa using (hret0 u.2 hu.2.1).1) (by simpa using (hret0 u.2 hu.2.1).2.1) |
| have htz : (Φ u).1∈(extChartAt I (a (c u.1))).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (htf 0).1 |
| have htw : (Φ u).2∈(extChartAt I (a (c u.2))).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (hsf 0).1 |
| have hcross1 : u.2∈(extChartAt I (a (c u.1))).source := by rw [hcs hu.2.2]; exact hsrc u.2 |
| have hcross2 : u.1∈(extChartAt I (a (c u.2))).source := by rw [←hcs hu.2.2]; exact hsrc u.1 |
| constructor |
| · apply hprod u.1 u.2 (Φ u).1 hu.1 hu.2.2 (hret0 u.1 hu.1).2.2.1 (hret0 u.2 hu.2.1).2.2.1 hzP |
| rw [ChartDensity.coordinates_eq I _ _ htz,ChartDensity.coordinates_eq I _ _ (hsrc u.1), |
| ChartDensity.coordinates_eq I _ _ hcross1] |
| have hh := mul_le_mul_of_nonneg_left hiz hH₀ |
| rw [←hcs hu.2.2] at hh |
| simpa only [sub_add_eq_sub_sub] using hh.trans hE |
| · apply hprod u.2 u.1 (Φ u).2 hu.2.1 hu.2.2.symm (hret0 u.2 hu.2.1).2.2.1 (hret0 u.1 hu.1).2.2.1 hwP |
| rw [ChartDensity.coordinates_eq I _ _ htw,ChartDensity.coordinates_eq I _ _ (hsrc u.2), |
| ChartDensity.coordinates_eq I _ _ hcross2] |
| have hh := mul_le_mul_of_nonneg_left hiw hH₀ |
| rw [hcs hu.2.2] at hh |
| simpa only [sub_add_eq_sub_sub] using hh.trans hE |
| have hlabels (u : M×M) (hu : u∈G) : |
| P (S^[n] (Φ u).1)=P (S^[n] u.1) ∧ |
| P (S.symm^[n] (Φ u).2)=P (S.symm^[n] u.1) ∧ |
| P (S^[n] (Φ u).2)=P (S^[n] u.2) ∧ |
| P (S.symm^[n] (Φ u).1)=P (S.symm^[n] u.2) := by |
| obtain ⟨htf,htb,hsf,hsb,_,_⟩ := ht u hu.1 hu.2.1 hu.2.2 |
| have hfx := hlabel S u.1 (Φ u).1 htf (Fin.last n) (hret0 u.1 hu.1).2.2.2.1 (hret0 u.1 hu.1).2.2.2.2.1 |
| have hbx := hlabel S.symm u.1 (Φ u).2 hsb (Fin.last n) (hret0 u.1 hu.1).2.2.2.2.2.1 (hret0 u.1 hu.1).2.2.2.2.2.2 |
| have hfy := hlabel S u.2 (Φ u).2 hsf (Fin.last n) (hret0 u.2 hu.2.1).2.2.2.1 (hret0 u.2 hu.2.1).2.2.2.2.1 |
| have hby := hlabel S.symm u.2 (Φ u).1 htb (Fin.last n) (hret0 u.2 hu.2.1).2.2.2.2.2.1 (hret0 u.2 hu.2.1).2.2.2.2.2.2 |
| exact ⟨hfx,hbx,hfy,hby⟩ |
| refine ⟨hcell,hlabels,?_⟩ |
| apply original_switch_injective I S a c P L π chart hπ hchart n δ ρ R G Φ Wf Wb |
| · intro x hx y hy z hxt hyt |
| exact hcommon S c n ρ δ x y z hc hρ hρr hρδ |
| (fun t => (closedBall_subset_closedBall hρδ').trans (hsafe y hy t).1) hxt hyt |
| · intro x hx y hy z hxt hyt |
| exact hcommon S.symm c n ρ δ x y z hc hρ hρr hρδ |
| (fun t => (closedBall_subset_closedBall hρδ').trans (hsafe y hy t).2) hxt hyt |
| · exact fun u hu => ⟨hu.1,hu.2.1⟩ |
| · exact fun u hu => ⟨hu.2.2,(hcell u hu).1⟩ |
| · exact hlabels |
| · exact fun u hu => heq u hu.1 hu.2.1 hu.2.2 |
| · intro u hu |
| obtain ⟨h1,h2,h3,h4,_,_⟩ := ht u hu.1 hu.2.1 hu.2.2 |
| exact ⟨h1,h4,h3,h2⟩ |
| · exact hrigid |
| end SmoothObstruction.ManifoldCoordinates |
| |
| end |
| |
| end OAI |
| |
