openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /TwoProfileComponentShell.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Estimates.ActualOuterMasks | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.AssemblyEndpointNormLe | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.TwoProfileOuterAngularBound | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.AssemblyPairMass | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.AssemblyListsCard | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND | |
| open Set MeasureTheory Filter PlanarFalconer | |
| open scoped ContDiff Classical Topology ComplexConjugate | |
| variable {n : ℕ} | |
| local notation "V" => EuclideanSpace ℝ (Fin (n+1)) | |
| theorem actual_two_profile_component_shell (hn : 1 ≤ n) (Ctest : ℝ) : | |
| ∃ Cd : ℝ, 0 ≤ Cd ∧ ∀ (chi beta sig kap eta : ℝ), | |
| 0 < chi → 0 < beta → beta < min (chi/10) (1/10) → | |
| 0 < kap → kap ≤ 1/1000 → 0 < sig → sig < kap/1000 → | |
| beta+(n+4:ℝ)*sig < chi/4 → 0 < eta → | |
| ∀ (Ψ : ℝ → ℝ), ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Ioo (1/4:ℝ) 4 → | |
| (∀ r, 0 ≤ Ψ r) → (∀ r ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ r) → | |
| ∀ (ψ : EuclideanSpace ℝ (Fin (n+1)) → ℂ), ContDiff ℝ ∞ ψ → | |
| (∀ w, ‖w‖=1 → ‖ψ w‖ ≤ 1) → | |
| ∀ (δ K : ℝ), 0 < δ → 0 < K → ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (ρ ν : Measure (EuclideanSpace ℝ (Fin (n+1)))) (B₀ : ℕ) (m₁ m₂ : ℝ → ℝ), | |
| RegularizedProfileData ρ N B₀ ((n+1:ℝ)/2+chi) sig m₁ → | |
| RegularizedProfileData ν N B₀ ((n+1:ℝ)/2+chi) sig m₂ → | |
| ρ (treeCell [])=1 → ν (treeCell [])=1 → | |
| let e₁ : RecursiveEnvironmentND n := ⟨ρ,m₁,N,sourcePolicyBlock N kap,sourceSmoothingLevels sig kap,sig,Ctest⟩ | |
| let e₂ : RecursiveEnvironmentND n := ⟨ν,m₂,N,sourcePolicyBlock N kap,sourceSmoothingLevels sig kap,sig,Ctest⟩ | |
| let c := GraphLiteralSupport.CurrentFalconer.entryDepthND (fun j : Fin 2 => if j=0 then e₁.profile else e₂.profile) N beta | |
| ∀ (X Y : Set (EuclideanSpace ℝ (Fin (n+1)))) | |
| (f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ), | |
| MeasurableSet X → MeasurableSet Y → X ⊆ treeCell [] → Y ⊆ treeCell [] → | |
| Measurable f → Measurable g → (∀ x, ‖f x‖ ≤ 1) → (∀ x, ‖g x‖ ≤ 1) → | |
| (∀ x ∈ X, ∀ y ∈ Y, δ ≤ dist x y ∧ dist x y ≤ K) → | |
| (∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), | |
| ψ =ᶠ[𝓝 ((dist z.1 z.2)⁻¹ • (z.1-z.2))] (fun _ => 1) ∧ | |
| ψ =ᶠ[𝓝 (-((dist z.1 z.2)⁻¹ • (z.1-z.2)))] (fun _ => 0)) → | |
| Integrable (fun r => ‖assemblyDistanceTransform e₁ e₂ c X Y f g r‖^2) | |
| (volume.restrict (Icc ((2:ℝ)^N/2) (2*(2:ℝ)^N))) ∧ | |
| (∫ r in Icc ((2:ℝ)^N/2) (2*(2:ℝ)^N), | |
| ‖assemblyDistanceTransform e₁ e₂ c X Y f g r‖^2) ≤ | |
| ((2:ℝ)^N)^(-beta+Cd*(kap+sig/kap)+eta) | |
| := by | |
| obtain ⟨Cd,hCd,hroot⟩ := actual_two_profile_outer_angular_bound hn Ctest | |
| refine ⟨Cd,hCd,?_⟩ | |
| intro chi beta sig kap eta hchi hbeta hbetasm hkap hkap1 hsig hsigkap hsmall heta | |
| Ψ hΨ hsupp hΨ0 hΨ1 ψ hψ hψ1 δ K hδ hK | |
| obtain ⟨φ,hφ,hφeq,P,hP⟩ := assembly_bounded_cutoff ψ hψ | |
| have hφ1 : ∀ w, ‖w‖=1 → ‖φ w‖ ≤ 1 := by | |
| intro w hw | |
| rw [(hφeq w hw).eq_of_nhds] | |
| exact hψ1 w hw | |
| have hsig1 : sig ≤ 1 := by linarith only [hsigkap,hkap1] | |
| choose D hD hderiv using (fun m : ℕ => uniform_assemblyEndpoint_derivatives (n:=n) m) | |
| obtain ⟨J,hsh⟩ := Attempt4PrefixShell.source_weighted_shellND (ι:=PolicyDescriptor) (κ:=PolicyDescriptor) | |
| φ hφ 2 D sig (eta/4) 2 0 16 δ K hsig hsig1 hD (by positivity) hδ hK | |
| obtain ⟨N₁,hsh⟩ := Filter.eventually_atTop.mp hsh | |
| obtain ⟨M,hroot⟩ := hroot chi beta sig kap (eta/4) hchi hbeta hbetasm hkap hkap1 hsig hsigkap | |
| hsmall (by positivity) (prefixInverseVocabulary n J) (2*J) (2*J) | |
| (fun _ hL => prefixInverseVocabulary_budget J hL) Ψ hΨ hsupp hΨ0 hΨ1 φ hφ hφ1 | |
| obtain ⟨N₂,hroot⟩ := hroot 1 zero_le_one | |
| obtain ⟨N₃,hconst⟩ := fixed_constant_absorbed (show 0<eta/2 by positivity) 2 | |
| refine ⟨max 1000 (max N₁ (max N₂ N₃)),?_⟩ | |
| intro N hN ρ ν B₀ m₁ m₂ hd₁ hd₂ hroot₁ hroot₂ e₁ e₂ c X Y f g hX hY hXroot hYroot hf hg hfn hgn hsep hcut | |
| let := hd₁.probability | |
| let := hd₂.probability | |
| have hNp : 0<N := by omega | |
| have hNR : 0<(N:ℝ) := by exact_mod_cast hNp | |
| have hq : 0<sourcePolicyBlock N kap := sourcePolicyBlock_pos hNp hkap | |
| have hentry := assembly_entry_estimates hn e₁ e₂ rfl (by change 1000≤N;omega) | |
| hd₁ hd₂ hchi hbeta (hbetasm.trans_le (min_le_right _ _)) hsmall hkap hkap1 rfl rfl | |
| have hcR : (c:ℝ)<(N:ℝ)/4 := hentry.2.1 | |
| have hc : 2*c≤N := by | |
| have hh : (2*c:ℝ)≤N := by linarith only [hcR,hNR] | |
| exact_mod_cast hh | |
| have hcN : c≤N := by omega | |
| let S := assemblyDescriptorList e₁.profile e₁.N e₁.q c | |
| let T := assemblyDescriptorList e₂.profile e₂.N e₂.q c | |
| let R : ℝ := (2:ℝ)^N | |
| let G : ℝ := -beta+Cd*(kap+sig/kap) | |
| let BHF : ℝ := R^(G+eta/4) | |
| have hR : 1≤R := one_le_pow₀ (by norm_num) | |
| have hRp : 0<R := by dsimp [R];positivity | |
| have hBHF : 0≤BHF := Real.rpow_nonneg hRp.le _ | |
| have hM₁ : (((2:ℝ)^e₁.N)^(-e₁.mass 0))=1 := by | |
| change (((2:ℝ)^N)^(-m₁ 0))=1 | |
| rw [hd₁.zero,neg_zero,Real.rpow_zero] | |
| have hM₂ : (((2:ℝ)^e₂.N)^(-e₂.mass 0))=1 := by | |
| change (((2:ℝ)^N)^(-m₂ 0))=1 | |
| rw [hd₂.zero,neg_zero,Real.rpow_zero] | |
| have hmass : |((((2:ℝ)^e₁.N)^(-e₁.mass 0)*((2:ℝ)^e₂.N)^(-e₂.mass 0))⁻¹)| * | |
| ((ρ.restrict X).prod (ν.restrict Y)).real univ ≤ R^(0:ℝ) := by | |
| simpa only [hM₁,hM₂,one_mul,inv_one,abs_one,Real.rpow_zero] using assembly_pair_mass ρ ν X Y | |
| have hcompact : HasCompactSupport Ψ := | |
| IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport Ψ) (hsupp.trans Ioo_subset_Icc_self) | |
| have hiΨ : Integrable (fun r => Ψ (r/(2:ℝ)^(N:ℝ))) := by | |
| simpa only [Real.rpow_natCast] using | |
| (hΨ.continuous.integrable_of_hasCompactSupport hcompact).comp_div (by positivity : (2:ℝ)^N≠0) | |
| have hcut' := assembly_cutoff_transfer ρ ν hX hY hδ | |
| (fun x hx y hy => (hsep x hx y hy).1) hφeq hcut | |
| choose BL BR hBL hBR using (fun p : PrefixAssignment n => assemblyInverse_bounds e₁ e₂ c hfn hgn p) | |
| have hHF : ∀ j≤J,∀ p∈Attempt4PrefixShell.inverseAngularTermsND j (S.disjSum T),∀ ξ : V, | |
| Attempt4PrefixShell.sourceAngularHFND (n+1) (Attempt4PrefixShell.canonicalSphereMeasureND V) | |
| 0 (N:ℝ) Ψ φ | |
| (fun z => Attempt4PrefixShell.spatialEndpointTransformND 1 ρ X (((2:ℝ)^e₁.N)^(-e₁.mass 0)) | |
| (assemblyInverseLeft e₁ c f p) (ξ,z)) | |
| (fun z => Attempt4PrefixShell.spatialEndpointTransformND 1 ν Y (((2:ℝ)^e₂.N)^(-e₂.mass 0)) | |
| (assemblyInverseRight e₂ c g p) (ξ,z)) ≤ BHF := by | |
| intro j hj p hp ξ | |
| rw [assembly_HF_transform e₁ e₂ c hX hY hXroot hYroot f g p ξ Ψ φ] | |
| have hbudget := (assemblyInverse_total S T j hp).trans (Nat.mul_le_mul_left 2 hj) | |
| apply hroot N (by omega) ρ ν B₀ m₁ m₂ hd₁ hd₂ hroot₁ hroot₂ | |
| (prefixLeftWords p) (prefixRightWords p) (assemblySpatialFactor X f ξ) (assemblySpatialFactor Y g ξ) | |
| recursiveUnit recursiveUnit | |
| (assemblySpatialFactor_measurable hX hf ξ) (assemblySpatialFactor_measurable hY hg ξ) | |
| (assemblySpatialFactor_norm hfn ξ) (assemblySpatialFactor_norm hgn ξ) | |
| (fun b _ => prefixInverseVocabulary_mem J j hj _ hp (Sum.inl b)) | |
| (fun b _ => prefixInverseVocabulary_mem J j hj _ hp (Sum.inr b)) | |
| (by change (∑ b∈S,(prefixLeftWords p b).length)≤2*J;omega) | |
| (by change (∑ b∈T,(prefixRightWords p b).length)≤2*J;omega) | |
| (fun l _ w _ => recursiveUnit_jets l w) (fun l _ w _ => recursiveUnit_jets l w) | |
| have hreg₁ : 20*sig*(N:ℝ) ≤ (N:ℝ)-0 := by | |
| have hs : 20*sig ≤ 1 := by linarith only [hsigkap,hkap1] | |
| simpa only [sub_zero,one_mul,mul_assoc] using mul_le_mul_of_nonneg_right hs hNR.le | |
| have hreg₂ : ((N/2:ℕ):ℝ) ≤ ((N:ℝ)-0)/2+sig*N := by | |
| have hh : ((N/2:ℕ):ℝ) ≤ (N:ℝ)/2 := by | |
| have hn : 2*(N/2)≤N := by omega | |
| have hn' : (2:ℝ)*(N/2:ℕ)≤N := by exact_mod_cast hn | |
| linarith only [hn'] | |
| have := mul_nonneg hsig.le hNR.le | |
| linarith only [hh,this] | |
| have hFG : ∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), ‖f z.1*conj (g z.2)‖ ≤ 1 := by | |
| apply Filter.Eventually.of_forall | |
| intro z | |
| rw [norm_mul,Complex.norm_conj] | |
| simpa only [one_mul] using mul_le_mul (hfn z.1) (hgn z.2) (norm_nonneg _) zero_le_one | |
| have hshell := hsh N (by omega) S T (assemblyEndpointSmooth e₁) (assemblyEndpointSmooth e₂) f g | |
| (fun b => sourceMaskScaleND (descriptorLength b) R sig) | |
| (fun b => sourceMaskScaleND (descriptorLength b) R sig) | |
| (assemblyLists_card e₁ e₂ (by change 1≤N;omega) rfl hq hq hcN) ρ ν X Y hX hY (N/2) | |
| (((2:ℝ)^e₁.N)^(-e₁.mass 0)) (((2:ℝ)^e₂.N)^(-e₂.mass 0)) 0 N | |
| hreg₁ hreg₂ (by linarith only [hNR]) hmass | |
| (fun b hb => assemblyDescriptor_scale e₁ hq hc hb) | |
| (fun b hb => assemblyDescriptor_scale e₂ hq hc hb) | |
| (fun x _ b _ w hw => assemblyEndpoint_norm_le e₁ x w b hw) | |
| (fun x _ b _ w hw => assemblyEndpoint_norm_le e₂ x w b hw) | |
| (fun x _ b _ j hj w _ => hderiv _ e₁ x b j hj w) | |
| (fun x _ b _ j hj w _ => hderiv _ e₂ x b j hj w) | |
| (by simpa only [Real.rpow_zero,mul_one] using hsep) | |
| hcut' hFG (assembly_pairMask_measurable e₁ e₂ c hf hg) P hP (fun _ => BL) (fun _ => BR) (fun _ _ p _ => assemblyInverseLeft_measurable e₁ c hf p) (fun _ _ p _ => assemblyInverseRight_measurable e₂ c hg p) (fun _ _ p _ x w => hBL p x w) (fun _ _ p _ x w => hBR p x w) Ψ hiΨ hΨ0 hΨ1 BHF hBHF (by simpa only [Real.rpow_zero,inv_one,assemblyInverseLeft,assemblyInverseRight,S,T,R,e₁,e₂] using hHF) | |
| have htarget : (∫ r in Icc ((2:ℝ)^N/2) (2*(2:ℝ)^N), | |
| ‖assemblyDistanceTransform e₁ e₂ c X Y f g r‖^2) ≤ R^(eta/4)*BHF+R^(-2:ℝ) := by | |
| simpa only [neg_zero,Real.rpow_zero,one_mul,Real.rpow_natCast,assemblyDistanceTransform] | |
| using hshell.2 | |
| refine ⟨?_,htarget.trans ?_⟩ | |
| · simpa only [Real.rpow_natCast,assemblyDistanceTransform] using hshell.1 | |
| · have herr : (-2:ℝ) ≤ G+eta/2 := by | |
| have hb : beta<1/10 := hbetasm.trans_le (min_le_right _ _) | |
| have hp : 0≤Cd*(kap+sig/kap) := by positivity | |
| dsimp only [G] | |
| linarith only [hb,hp,heta] | |
| have htwo : (2:ℝ)≤R^(eta/2) := hconst N (by omega) | |
| calc | |
| R^(eta/4)*BHF+R^(-2:ℝ) = R^(G+eta/2)+R^(-2:ℝ) := by | |
| dsimp only [BHF] | |
| rw [←Real.rpow_add hRp] | |
| congr 2 | |
| ring | |
| _ ≤ R^(G+eta/2)+R^(G+eta/2) := add_le_add le_rfl | |
| (Real.rpow_le_rpow_of_exponent_le hR herr) | |
| _ = 2*R^(G+eta/2) := by ring | |
| _ ≤ R^(eta/2)*R^(G+eta/2) := mul_le_mul_of_nonneg_right htwo (Real.rpow_nonneg hRp.le _) | |
| _ = R^(G+eta) := by rw [←Real.rpow_add hRp];congr 1;ring | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |