| import OAI.Dynamics.SmoothObstruction.DensityLocalization |
| import OAI.Dynamics.SmoothObstruction.SwitchScales |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace SmoothObstruction.EndpointBudget |
| open Set Filter Topology ExteriorStability |
| /-- The literal endpoint hypotheses, simultaneously at every rank of the |
| unchanged ambient dimension. -/ |
| theorem eventual_admissible {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) (Dg R C K₀ Crel C₁ c β : ℝ) (hc : 0<c) (hβ : 0<β) |
| (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≤d, |
| admissible d p Dg R C K₀ Crel C₁ (b k) (n k) (b k^(1/2:ℝ)) (δ k) β := by |
| have he := all_ranks hbpos hb hn hfloor d C Crel C₁ (d*R) K₀ Dg c hc |
| (fun p : Fin (d+1) => frameWedgeConstant (E:=EuclideanSpace ℝ (Fin d)) p.val) |
| (fun p : Fin (d+1) => (Fintype.card (powersetCard (Fin d) p.val):ℝ)) |
| (fun p : Fin (d+1) => ‖continuousWedge (E:=EuclideanSpace ℝ (Fin d)) p.val‖) |
| (fun p : Fin (d+1) => unitWedgeConstant (E:=EuclideanSpace ℝ (Fin d)) p.val) |
| (fun _ => β) |
| (fun p => (by norm_num : (0:ℝ)≤1).trans (one_le_frameWedgeConstant p.val)) |
| (fun p => Nat.cast_nonneg _) (fun p => ContinuousMultilinearMap.opNorm_nonneg _) |
| (fun p => (by norm_num : (0:ℝ)≤1).trans (one_le_unitWedgeConstant p.val)) |
| (fun _ => hβ) hδ |
| filter_upwards [he] with k hk |
| intro p hp |
| have h := hk ⟨p,by omega⟩ |
| simpa only [admissible,leading,normCost,mixedCost,relative,graph,mul_assoc] using h |
| end SmoothObstruction.EndpointBudget |
|
|
| namespace SmoothObstruction.GridCount |
| open Filter Topology |
| lemma ceil_grid_bound {A t : ℝ} (hA : 0≤A) (ht : 0≤t) : |
| ((2*⌈A*Real.exp t+1⌉₊+1:ℕ):ℝ)≤(2*A+5)*Real.exp t := by |
| have hh := Nat.ceil_lt_add_one (by positivity : (0:ℝ)≤A*Real.exp t+1) |
| have he := Real.one_le_exp_iff.mpr ht |
| push_cast |
| nlinarith |
|
|
| /-- A uniform exponential count for the two original starting grids. |
| The strip-thinning denominator N cancels from the actual grid period. -/ |
| theorem original_grid_count (d j : ℕ) (Dc Dp Dg : ℝ) (hDc : 1≤Dc) (hDp : 1≤Dp) (hDg : 0≤Dg) : |
| ∃ C : ℝ,0≤C ∧ ∀ (b : ℝ) (n N : ℕ),0<b → b≤1 → 1≤b*n → 0<N → |
| let B := Dg*(b*n) |
| let t := b^(1/8:ℝ)*n |
| let dc := Real.exp (-t)/(((d:ℝ)+1)*N) |
| let dp := Real.exp (-2*t)/(Real.sqrt 2*((d:ℝ)+1)*N) |
| let mc := ⌈Dc/(dc*N)+1⌉₊ |
| let mp := ⌈(Real.sqrt 2*Real.exp (2*B))*Dp/(dp*N)+1⌉₊ |
| ((j*(2*mc+1)^d*(2*mp+1)^d:ℕ):ℝ)≤Real.exp (C*t) := by |
| let A₁ := Dc*((d:ℝ)+1) |
| let A₂ := 2*Dp*((d:ℝ)+1) |
| let K := (j:ℝ)*(2*A₁+5)^d*(2*A₂+5)^d |
| have hA₁ : 0≤A₁ := by dsimp only [A₁]; positivity |
| have hA₂ : 0≤A₂ := by dsimp only [A₂]; positivity |
| have hK : 0≤K := by dsimp only [K]; positivity |
| refine ⟨K+(3+2*Dg)*d,by positivity,?_⟩ |
| intro b n N hb hb1 hbn hN B t dc dp mc mp |
| have hbroot : b≤b^(1/8:ℝ) := by |
| simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_ge hb hb1 (by norm_num : (1/8:ℝ)≤1) |
| have htn : b*n≤t := mul_le_mul_of_nonneg_right hbroot (Nat.cast_nonneg _) |
| have ht : 1≤t := hbn.trans htn |
| have hB : B≤Dg*t := mul_le_mul_of_nonneg_left htn hDg |
| have hB0 : 0≤B := by dsimp only [B]; positivity |
| have hcs : Dc/(dc*N)=A₁*Real.exp t := by |
| dsimp only [dc,A₁] |
| rw [Real.exp_neg] |
| field_simp |
| have hps : (Real.sqrt 2*Real.exp (2*B))*Dp/(dp*N)=A₂*Real.exp (2*B+2*t) := by |
| dsimp only [dp,A₂] |
| rw [show -2*t= -(2*t) by ring,Real.exp_neg,Real.exp_add] |
| have hs : Real.sqrt 2*Real.sqrt 2=2 := Real.mul_self_sqrt (by norm_num) |
| field_simp |
| nlinarith only [hs] |
| have hc := ceil_grid_bound hA₁ (by linarith : 0≤t) |
| have hp := ceil_grid_bound hA₂ (by linarith : 0≤2*B+2*t) |
| rw [←hcs] at hc |
| rw [←hps] at hp |
| have hcp := pow_le_pow_left₀ (Nat.cast_nonneg _) hc d |
| have hpp := pow_le_pow_left₀ (Nat.cast_nonneg _) hp d |
| have hm := mul_le_mul (mul_le_mul_of_nonneg_left hcp (Nat.cast_nonneg j)) hpp |
| (by positivity) (by positivity) |
| have hmain : ((j*(2*mc+1)^d*(2*mp+1)^d:ℕ):ℝ)≤K*Real.exp ((d:ℝ)*t+(d:ℝ)*(2*B+2*t)) := by |
| calc |
| _ ≤ (j:ℝ)*((2*A₁+5)*Real.exp t)^d*((2*A₂+5)*Real.exp (2*B+2*t))^d := by |
| simpa only [Nat.cast_mul,Nat.cast_pow] using hm |
| _ = K*Real.exp ((d:ℝ)*t+(d:ℝ)*(2*B+2*t)) := by |
| dsimp only [K] |
| rw [mul_pow,mul_pow,←Real.exp_nat_mul,←Real.exp_nat_mul,Real.exp_add] |
| ring |
| have hKexp : K≤Real.exp K := by linarith [Real.add_one_le_exp K] |
| have hKt : K≤K*t := le_mul_of_one_le_right hK ht |
| have hBd := mul_le_mul_of_nonneg_left hB (show 0≤2*(d:ℝ) by positivity) |
| calc |
| _ ≤ K*Real.exp ((d:ℝ)*t+(d:ℝ)*(2*B+2*t)) := hmain |
| _ ≤ Real.exp K*Real.exp ((d:ℝ)*t+(d:ℝ)*(2*B+2*t)) := mul_le_mul_of_nonneg_right hKexp (Real.exp_pos _).le |
| _ = Real.exp (K+((d:ℝ)*t+(d:ℝ)*(2*B+2*t))) := (Real.exp_add _ _).symm |
| _ ≤ Real.exp ((K+(3+2*Dg)*d)*t) := by apply Real.exp_le_exp.mpr; nlinarith only [hKt,hBd] |
| end SmoothObstruction.GridCount |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology MeasureTheory |
| open scoped Manifold ContDiff ENNReal |
| 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] |
| [MeasurableSpace M] [BorelSpace M] [StandardBorelSpace M] [Nonempty M] |
| {ι : Type} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
|
|
| omit [ProperSpace M] [MeasurableSpace M] [BorelSpace M] [StandardBorelSpace M] [Nonempty M] in |
| lemma interior_of_selected_ball (a x : M) (hx : x∈(chartAt H a).source) {r : ℝ} (hr : 0≤r) |
| (hs : closedBall ((extChartAt I a) x) r⊆interior (range I)) : I.IsInteriorPoint x := by |
| apply (I.isInteriorPoint_iff_of_mem_atlas (n:=∞) (by simp) (chart_mem_atlas H a) hx).mpr |
| exact (chartAt H a).mem_interior_extend_target ((chartAt H a).map_source hx) |
| (hs (mem_closedBall_self hr)) |
|
|
| /-- Finite translates, canonical references, the original retained carrier, |
| and the genuine global switch are compatible at all sufficiently large scales. |
| The explicit cell cardinality counts only two starting grids, never orbit labels. -/ |
| theorem grid_switches (hclosed : IsClosed (range I)) |
| (μ : Measure M) [IsProbabilityMeasure μ] (hμ : SmoothPositiveDensity I μ) |
| (S : Diffeomorph I I M M ∞) (hS : MeasurePreserving S μ μ) |
| (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 : Measurable c) |
| (hcover : ∀ x,x∈K (c x)) (V : Set M) (hV : MeasurableSet V) |
| (s : ℝ) (hs : 0<s) (hcore : ∀ x∈V,∀ y,dist y x ≤ s → c y=c x) |
| (η : ℝ) (hη : 0<η) (hVm : μ.real Vᶜ≤η/100) |
| (b : ℕ → ℝ) (n : ℕ → ℕ) (hbpos : ∀ k,0<b k) (hb : Tendsto b atTop (𝓝 0)) |
| (hn : Tendsto (fun k => (n k:ℝ)) atTop atTop) |
| (hfloor : ∀ k,(n k:ℝ)^(-1/10:ℝ)≤b k) |
| (Dg R₀ : ℝ) (hDg : 0≤Dg) (hR₀ : 0≤R₀) |
| (G : ℕ → Set M) (hG : ∀ k,MeasurableSet (G k)) (hGm : ∀ k,μ.real (G k)ᶜ≤η/100) |
| (hA : ∀ᶠ k in atTop,∀ x∈G k,Nonempty (BilateralGauge I S a c (n k) (Dg*(b k*n k)) R₀ x)) |
| (Cb : ℝ) (_ : 0≤Cb) |
| (hcollar : ∀ r : ℝ,0≤r → ∃ W : Set M,MeasurableSet W ∧ μ.real W≤Cb*r ∧ |
| ∀ i x,x∈K i → x∉W → closedBall ((extChartAt I (a i)) x) r⊆interior (range I)) : |
| ∃ (Dc Dp : ℝ) (N : ℕ) (ρ : ℕ → ℝ),1≤Dc ∧ 1≤Dp ∧ 0<N ∧ |
| Tendsto ρ atTop (𝓝 0) ∧ ∀ᶠ k in atTop, |
| let B := Dg*(b k*n k) |
| let h₀ := Real.exp (-b k^(1/8:ℝ)*n k) |
| let r := Real.exp (-2*b k^(1/8:ℝ)*n k) |
| let δc := h₀/(((d:ℝ)+1)*N) |
| let δp := r/(Real.sqrt 2*((d:ℝ)+1)*N) |
| let mc := ⌈Dc/(δc*N)+1⌉₊ |
| let mp := ⌈(Real.sqrt 2*Real.exp (2*B))*Dp/(δp*N)+1⌉₊ |
| ∃ (J : Type) (_ : Fintype J) (_ : MeasurableSpace J) (_ : MeasurableSingletonClass J) |
| (L : M → J) (Gp : Set (M×M)) (Φ : M×M → M×M), |
| Fintype.card J=Fintype.card ι*(2*mc+1)^d*(2*mp+1)^d ∧ |
| Measurable L ∧ MeasurableSet Gp ∧ Measurable Φ ∧ |
| 1-η≤(PairLaw.law μ L).real Gp ∧ |
| ((PairLaw.law μ L).restrict Gp).map (fun u => (Φ u).1)≤(2:ℝ≥0∞) • μ ∧ |
| ∀ u∈Gp,originalTube I S a c (n:=n k) (ρ k) u.1 (Φ u).1 ∧ |
| originalTube I S.symm a c (n:=n k) (ρ k) u.2 (Φ u).1 := by |
| classical |
| obtain ⟨r₀,C₀,σ,r₁,C₁,rg,Cg,β,K₀F,CreF,C₁F,K₀B,CreB,C₁B,rμ, |
| hr₀,hC₀,hσ,hr₁,hC₁,hrg,hCg,hβ,hKF,hCrF,hCF,hKB,hCrB,hCB,hrμ,hswitch⟩ := |
| genuine_pair_switch I hclosed μ hμ S a K hK hKU s Dg hs hDg |
| obtain ⟨Dc,hDc,hcoarse⟩ := ChartDensity.exists_coarse_cells I μ a K hK hKU c hc hcover |
| obtain ⟨Dp,hDp,hproduct⟩ := finite_product_partition_uniform I μ S a K hK hKU c hcover |
| obtain ⟨N,hN⟩ := exists_nat_gt ((d:ℝ)/(η/100)+1) |
| have hNpos : 0<N := by |
| have : 0≤(d:ℝ)/(η/100) := by positivity |
| exact_mod_cast (by linarith : (0:ℝ)<N) |
| have hNm : (d:ℝ)/N≤η/100 := by |
| apply (div_le_iff₀ (Nat.cast_pos.mpr hNpos)).mpr |
| have h := (div_le_iff₀ (by positivity : (0:ℝ)<η/100)).mp (show (d:ℝ)/(η/100)≤N by linarith) |
| simpa only [mul_comm] using h |
| let Ac : ℝ := ((d:ℝ)+1)*N |
| let Ap : ℝ := Real.sqrt 2*((d:ℝ)+1)*N |
| have hAc : 0<Ac := by dsimp only [Ac]; positivity |
| have hAp : 0<Ap := by dsimp only [Ap]; positivity |
| let BB (k : ℕ) := Dg*(b k*n k) |
| let g (k : ℕ) := b k^(1/2:ℝ) |
| let α (k : ℕ) := (2+8*(d:ℝ))*g k |
| let ε (k : ℕ) := Real.exp (-b k^(3/4:ℝ)*n k) |
| let h₀ (k : ℕ) := Real.exp (-b k^(1/8:ℝ)*n k) |
| let r (k : ℕ) := Real.exp (-2*b k^(1/8:ℝ)*n k) |
| let δ (k : ℕ) := Real.exp (-(3/2:ℝ)*b k^(1/8:ℝ)*n k) |
| let ρ (k : ℕ) := 2*C₀*ShortBin.bvpInverseBound (BB k) (α k) (n k)*r k |
| have hbfe : ∀ᶠ k in atTop,0<b k := Filter.Eventually.of_forall hbpos |
| have hffe : ∀ᶠ k in atTop,(n k:ℝ)^(-1/10:ℝ)≤b k := Filter.Eventually.of_forall hfloor |
| have hεenv : ∀ᶠ k in atTop,0≤ε k ∧ ε k≤Real.exp (-b k^(3/4:ℝ)*n k) := |
| Filter.Eventually.of_forall (fun k => ⟨(Real.exp_pos _).le,le_rfl⟩) |
| have hρ0 := (ShortBin.spatial_switch_limits hbfe hb hn hffe hεenv Dg (2+8*(d:ℝ)) C₀ hDg (by positivity)).2.2.1 |
| have hspatial := ShortBin.spatial_ready hbfe hb hn hffe Dg (2+8*(d:ℝ)) C₀ C₁ Cg K₀F K₀B |
| r₀ rg σ r₁ rμ Ac Ap hDg (by positivity) hC₀ hr₀ hrg hσ hr₁ hrμ hAc hAp |
| have hrefs := eventual_finite_references I hclosed S a K hK hKU hbfe hb hn hffe Dg R₀ hDg |
| choose Cp hCp hCpbound using fun p : Fin (d+1) => |
| ExteriorStability.exists_power_bound (E:=EuclideanSpace ℝ (Fin d)) p.val |
| let Cext := 1+∑ p,Cp p |
| have hCext : 0≤Cext := by |
| have := Finset.sum_nonneg (s:=Finset.univ) (fun p _ => (show (0:ℝ)≤1 by norm_num).trans (hCp p)) |
| dsimp only [Cext] |
| linarith |
| have hpoly (p : ℕ) (hp : p≤d) (F : EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)) : |
| ‖ExteriorStability.exterior p F‖≤Cext*(max 1 ‖F‖)^p := by |
| have hh := hCpbound ⟨p,by omega⟩ F |
| apply hh.trans |
| apply mul_le_mul_of_nonneg_right _ (by positivity) |
| have hc := Finset.single_le_sum (fun q _ => (show (0:ℝ)≤1 by norm_num).trans (hCp q)) (Finset.mem_univ (⟨p,by omega⟩ : Fin (d+1))) |
| dsimp only [Cext]; linarith |
| have hbudgetF := EndpointBudget.eventual_admissible hbfe hb hn hffe d Dg R₀ Cext K₀F CreF C₁F 1 β (by norm_num) hβ |
| (by simpa only [neg_one_mul] using hεenv) |
| have hbudgetB := EndpointBudget.eventual_admissible hbfe hb hn hffe d Dg R₀ Cext K₀B CreB C₁B 1 β (by norm_num) hβ |
| (by simpa only [neg_one_mul] using hεenv) |
| have hdec (m : ℕ) : Tendsto (fun k => (n k:ℝ)^m*h₀ k) atTop (𝓝 0) := by |
| have hh := ScaleHierarchy.polynomial_decay (p:=1/8) (q:=1) (c:=1) (C:=0) |
| (by norm_num) (by norm_num) (by norm_num) (by norm_num) hbfe hb hn hffe m |
| simpa only [zero_mul,zero_sub,one_mul,neg_mul,h₀] using hh |
| have hcollar0 : Tendsto (fun k => (2*(n k:ℝ)+3)*(Cb*h₀ k)) atTop (𝓝 0) := by |
| have hh := ((hdec 1).const_mul (2*Cb)).add ((hdec 0).const_mul (3*Cb)) |
| simp only [mul_zero,add_zero] at hh |
| apply hh.congr'; apply Filter.Eventually.of_forall; intro k; simp only [pow_one,pow_zero,one_mul]; ring |
| refine ⟨Dc,Dp,N,ρ,hDc,hDp,hNpos,hρ0,?_⟩ |
| filter_upwards [hspatial,hrefs,hbudgetF,hbudgetB,hA, |
| hcollar0.eventually_le_const (by positivity : (0:ℝ)<η/100)] with k hsp href hbudF hbudB hAk hcolmass |
| rcases hsp with ⟨hbn,heps,hsmall,hgap,hδr,hδrg,hρδ,hρσ,hρc,hρr₁,hρC,hρCg,hδF,hδB,hE,hcontract,hdata,hrrμ,hδh⟩ |
| dsimp only |
| let dc := h₀ k/Ac |
| let dp := r k/Ap |
| have hdc : 0<dc := div_pos (Real.exp_pos _) hAc |
| have hdp : 0<dp := div_pos (Real.exp_pos _) hAp |
| obtain ⟨P,Wc,hP,hWc,hWcm,hPc,hPdiam,hPrec,hPcard⟩ := hcoarse hdc hNpos |
| let C := ι × (Fin d → Fin (2*⌈Dc/(dc*N)+1⌉₊+1)) |
| have hPc' (x y : M) (he : P x=P y) : c x=c y := by rw [←hPc x,←hPc y,he] |
| have hsrc (x : M) : x∈(extChartAt I (a (c x))).source := by rw [extChartAt_source]; exact hKU _ (hcover x) |
| have hceq : ((d:ℝ)+1)*(dc*N)=h₀ k := by dsimp only [dc,Ac]; field_simp |
| have hpeq : Real.sqrt 2*((d:ℝ)+1)*(dp*N)=r k := by dsimp only [dp,Ap]; field_simp |
| obtain ⟨Wb,hWb,hWbm,hWs⟩ := hcollar (h₀ k) (Real.exp_pos _).le |
| let G' := G k\Wb |
| have hiG (x : M) (hx : x∈G') : I.IsInteriorPoint x := |
| interior_of_selected_ball I _ _ (hKU _ (hcover x)) (Real.exp_pos _).le (hWs _ x (hcover x) hx.2) |
| let A (x : G') : BilateralGauge I S a c (n k) (BB k) R₀ x := (hAk x x.property.1).some |
| have hco (x y : M) (he : P x=P y) : |
| ‖(extChartAt I (a (c y))) x-(extChartAt I (a (c y))) y‖≤h₀ k := by |
| have hh := hPdiam x y he |
| rw [hceq] at hh |
| rw [ChartDensity.coordinates_eq I _ _ (hsrc x),hPc' x y he] at hh |
| rw [ChartDensity.coordinates_eq I _ _ (hsrc y)] at hh |
| exact hh |
| obtain ⟨γ,T,ref,Wf,Wback,hrefeq,hα,hrefP,hcard,hreff,hslow,hfast,hstart,hend⟩ := |
| href C c P G' A hcover hPc' hiG hco |
| subst ref |
| have hBB : 0≤BB k := mul_nonneg hDg (mul_nonneg (hbpos k).le (Nat.cast_nonneg _)) |
| obtain ⟨L,Wp,hL,hWp,hWpm,hLP,hLdiam,hLrec,hLcard⟩ := |
| hproduct C P hP Prod.fst hPc G' (n k) (BB k) R₀ hBB A T dp hdp N hNpos |
| have hSsymm : MeasurePreserving S.symm μ μ := MeasurePreserving.symm S.toHomeomorph.toMeasurableEquiv hS |
| obtain ⟨R,hR,hRG,hRm,hret⟩ := RetainedGlobal.exists_retained μ S S.symm hS hSsymm |
| G' V Wc Wp Wb ((hG k).diff hWb) hV hWc hWp hWb (n k) |
| have hmassR : μ.real Rᶜ≤η/2 := by |
| have hG'm : μ.real G'ᶜ≤μ.real (G k)ᶜ+μ.real Wb := by |
| simpa only [G',compl_sdiff,union_comm] using RetainedGlobal.real_union_le μ (G k)ᶜ Wb |
| have hcoef : 0≤2*(n k:ℝ)+3 := by positivity |
| have hbc := mul_le_mul_of_nonneg_left hWbm hcoef |
| have hWc' := hWcm.trans hNm |
| have hWp' := hWpm.trans hNm |
| have hg' := hGm k |
| nlinarith only [hRm,hG'm,hbc,hWc',hWp',hVm,hg',hcolmass,hη] |
| have hLp (x y : M) (he : L x=L y) : P x=P y := by rw [←hLP x,←hLP y,he] |
| have hLc (x y : M) (he : L x=L y) : c x=c y := hPc' x y (hLp x y he) |
| have hLdiam' (x y : M) (he : L x=L y) : |
| ‖ChartDensity.coordinates I (a (c x)) x-ChartDensity.coordinates I (a (c x)) y‖≤r k := by |
| simpa only [hpeq] using hLdiam x y he |
| obtain ⟨Φ,hΦ,hGp,hinj,hlaw,hmarg,htube⟩ := hswitch (C:=C) c P L Prod.fst Prod.fst G' R V Wc Wp |
| (n k) (b k) R₀ Cext (ε k) (g k) (α k) (r k) dc dp |
| (Real.sqrt 2*Real.exp (2*BB k)) γ A T (fun x => (FiniteBins.reference P G' (FiniteBins.key P G' x)).val) Wf Wback hc hP hL hR hRG hcover hLP hPc |
| (fun x hx => hiG x (hRG hx)) hrefP hcard hreff (by simpa only [ε,SwitchScales.shortSignal,neg_mul] using hstart) hend |
| (hbpos k).le hbn hR₀ hCext (Real.exp_pos _).le heps hsmall (Real.rpow_nonneg (hbpos k).le _) |
| hslow hfast hgap (fun x => hα (P x)) (Real.exp_pos _).le hδr hδrg |
| (by exact mul_nonneg (mul_nonneg (by linarith) (ShortBin.bvp_nonneg _ _ _)) (Real.exp_pos _).le) |
| hρδ hρσ hρc hρr₁ hρC hρCg hδF hδB hdc (by positivity) hE hpoly hbudF hbudB |
| (fun x hx t => ⟨(closedBall_subset_closedBall hδh).trans (hWs _ _ (hcover _) ((hret x hx).2.2.2.2.2.2.2 _ (by omega)).1), |
| (closedBall_subset_closedBall hδh).trans (hWs _ _ (hcover _) ((hret x hx).2.2.2.2.2.2.2 _ (by omega)).2)⟩) |
| (fun x hx y hy he => by |
| have hh := hLdiam' x y he |
| rw [ChartDensity.coordinates_eq I _ _ (hsrc x),hLc x y he, |
| ChartDensity.coordinates_eq I _ _ (hsrc y),norm_sub_rev] at hh |
| rw [hLc x y he] |
| exact hh.trans (le_mul_of_one_le_left (Real.exp_pos _).le hC₀)) |
| hcontract hdata (fun x y he => (hLdiam' x y he).trans_lt hrrμ) hcore hPrec |
| (fun x y z hx he hxW hyW hz herr => by |
| apply hLrec x y z (hRG hx) he hxW hyW hz |
| exact herr) |
| (fun x hx => ⟨(hret x hx).1,(hret x hx).2.1,(hret x hx).2.2.1, |
| (hret x hx).2.2.2.1,(hret x hx).2.2.2.2.1,(hret x hx).2.2.2.2.2.1,(hret x hx).2.2.2.2.2.2.1⟩) |
| let Gp := {u : M×M | u.1∈R ∧ u.2∈R ∧ L u.1=L u.2} |
| have hmass := (RetainedGlobal.pair_mass μ L hL R hR hmassR).2 |
| refine ⟨_,inferInstance,inferInstance,inferInstance,L,Gp,Φ,?_,hL,hGp,hΦ,?_,hmarg,htube⟩ |
| · exact hLcard.trans (by rw [hPcard]) |
| · have he : Gp=R.prod R∩{u : M×M | L u.1=L u.2} := by |
| ext u; exact ⟨fun h => ⟨⟨h.1,h.2.1⟩,h.2.2⟩,fun h => ⟨h.1.1,h.1.2,h.2⟩⟩ |
| rw [he] |
| exact hmass |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology MeasureTheory |
| open scoped Manifold ContDiff ENNReal NNReal |
| variable {d : ℕ} {H : Type} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type} [MetricSpace M] [CompactSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [MeasurableSpace M] [BorelSpace M] [StandardBorelSpace M] [Nonempty M] |
| |
| /-- Smooth switching from a good gauge and a quantitative geometric |
| boundary collar, for Euclidean and half-space manifold models. -/ |
| theorem switching_from_gauges (hclosed : IsClosed (range I)) |
| (μ : ProbabilityMeasure M) (hμ : SmoothPositiveDensity I (μ : Measure M)) |
| (S : Diffeomorph I I M M ∞) (hS : MeasurePreserving S (μ : Measure M) (μ : Measure M)) |
| (hcollar : ∀ (ι : Type) [Fintype ι] (a : ι → M) (K : ι → Set M), |
| (∀ i,IsCompact (K i)) → (∀ i,K i⊆(chartAt H (a i)).source) → |
| ∃ Cb : ℝ,0≤Cb ∧ ∀ r : ℝ,0≤r → ∃ W : Set M,MeasurableSet W ∧ |
| (μ : Measure M).real W≤Cb*r ∧ |
| ∀ i x,x∈K i → x∉W → closedBall ((extChartAt I (a i)) x) r⊆interior (range I)) : |
| SmoothSwitching (μ : Measure M) S S.symm := by |
| classical |
| intro η hη |
| let ε : ℝ≥0 := ⟨η/100,by positivity⟩ |
| obtain ⟨ι,hfin,hmeas,hsc,a,K,L,c,s,hK,hKU,hc,hcover,hL,hLm,hs,hLK,hLc,hcore, |
| b,R₀,Dg,G,hR₀,hDg,hbpos,hfloor,hbs,hG,hA⟩ := good_stable_gauges I S μ hS hη ε (by change (0:ℝ)<η/100; positivity) |
| let : Fintype ι := hfin |
| let : MeasurableSpace ι := hmeas |
| let : MeasurableSingletonClass ι := hsc |
| let V := ⋃ i,L i |
| have hV : MeasurableSet V := (isCompact_iUnion hL).measurableSet |
| have hVm : (μ : Measure M).real Vᶜ≤η/100 := by |
| exact (ENNReal.toReal_mono (by simp) hLm).trans_eq (by rfl) |
| have hGm : ∀ k,(μ : Measure M).real (G k)ᶜ≤η/100 := by |
| intro k |
| rw [measureReal_compl (hG k).1.measurableSet,probReal_univ] |
| have hh := (hG k).2 |
| linarith |
| have hb : Tendsto b atTop (𝓝 0) := by |
| have hh := hbs.tendsto_atTop_zero.sqrt |
| simpa only [Real.sqrt_sq (hbpos _).le,Real.sqrt_zero] using hh |
| have hn : Tendsto (fun k => ((2^k:ℕ):ℝ)) atTop atTop := by |
| simpa only [Nat.cast_pow,Nat.cast_ofNat] using |
| tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ)<2) |
| have hfloor' : ∀ k,((2^k:ℕ):ℝ)^(-1/10:ℝ)≤b k := by |
| intro k |
| have he : ((2^k:ℕ):ℝ)^(-1/10:ℝ)=GoodStationary.floorAt k := by |
| rw [Nat.cast_pow,Nat.cast_ofNat,←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] |
| dsimp only [GoodStationary.floorAt] |
| congr 1 |
| ring |
| rw [he] |
| exact hfloor k |
| obtain ⟨Cb,hCb,hcol⟩ := hcollar ι a K hK hKU |
| obtain ⟨Dc,Dp,N,ρ,hDc,hDp,hN,hρ,hsw⟩ := grid_switches I hclosed (μ : Measure M) hμ S hS |
| a K hK hKU c hc hcover V hV s hs hcore η hη hVm b (fun k => 2^k) hbpos hb hn hfloor' |
| Dg R₀ hDg hR₀ G (fun k => (hG k).1.measurableSet) hGm (by simpa only [Nat.cast_pow,Nat.cast_ofNat] using hA) |
| Cb hCb hcol |
| obtain ⟨C,hC,hcount⟩ := GridCount.original_grid_count d (Fintype.card ι) Dc Dp Dg hDc hDp hDg |
| let B := fun k => SmoothScales.B (b k) |
| have hBs : Summable (fun k => (B k)^16) := by |
| simpa only [B,SmoothScales.root_sixteen (hbpos _).le] using hbs |
| have hex : ∀ᶠ k in atTop,∃ D : SwitchData (μ : Measure M),D.Good η C (B k) (2^k) ∧ |
| ∀ u∈D.carrier,originalTube I S a c (n:=2^k) (ρ k) u.1 (D.point u) ∧ |
| originalTube I S.symm a c (n:=2^k) (ρ k) u.2 (D.point u) := by |
| filter_upwards [hsw,hb.eventually_le_const (by norm_num : (0:ℝ)<1)] with k hk hb1 |
| obtain ⟨J,hjfin,hjmeas,hjsc,P,Gp,Φ,hcard,hP,hGp,hΦ,hm,hmarg,ht⟩ := hk |
| let : Fintype J := hjfin |
| let : MeasurableSpace J := hjmeas |
| let : MeasurableSingletonClass J := hjsc |
| let D : SwitchData (μ : Measure M) := ⟨J,hjfin,hjmeas,hjsc,P,Gp,(fun u => (Φ u).1),hP,hGp,measurable_fst.comp hΦ⟩ |
| have hbn := GoodStationary.burden_ge_one b hfloor k |
| have hcnt := hcount (b k) (2^k) N (hbpos k) hb1 (by simpa only [Nat.cast_pow,Nat.cast_ofNat] using hbn) hN |
| refine ⟨D,⟨?_,hm,hmarg⟩,ht⟩ |
| change (Fintype.card J:ℝ)≤_ |
| rw [hcard] |
| simpa only [B,SmoothScales.B,mul_assoc,neg_mul] using hcnt |
| let D₀ : SwitchData (μ : Measure M) := ⟨Unit,inferInstance,inferInstance,inferInstance, |
| (fun _ => ()),∅,Prod.fst,measurable_const,MeasurableSet.empty,measurable_fst⟩ |
| let D (k : ℕ) : SwitchData (μ : Measure M) := |
| if hk : ∃ D : SwitchData (μ : Measure M),D.Good η C (B k) (2^k) ∧ |
| ∀ u∈D.carrier,originalTube I S a c (n:=2^k) (ρ k) u.1 (D.point u) ∧ |
| originalTube I S.symm a c (n:=2^k) (ρ k) u.2 (D.point u) |
| then hk.choose else D₀ |
| have hD : ∀ᶠ k in atTop,(D k).Good η C (B k) (2^k) ∧ |
| ∀ u∈(D k).carrier,originalTube I S a c (n:=2^k) (ρ k) u.1 ((D k).point u) ∧ |
| originalTube I S.symm a c (n:=2^k) (ρ k) u.2 ((D k).point u) := by |
| filter_upwards [hex] with k hk |
| dsimp only [D]; rw [dite_eq_left hk]; exact hk.choose_spec |
| refine ⟨B,C,D,fun k => SmoothScales.positive (hbpos k),hBs,hD.mono (fun _ hk => hk.1),?_⟩ |
| have hout := original_shadow_tendsto I hclosed a K hK hKU S c hcover (fun k => 2^k) ρ hρ |
| (fun k => (D k).carrier) (fun k u => ((D k).point u,u.2)) (hD.mono (fun _ hk => hk.2)) |
| exact hout |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
