openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /TwoProfileOuterAngularBound.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Estimates.ActualOuterMasks | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.Ledger | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.AssemblyRootGraph | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| 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 | |
| variable {n : ℕ} | |
| theorem actual_two_profile_outer_angular_bound (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 → | |
| ∀ (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (B total : ℕ), (∀ L∈W,L.length≤B) → | |
| ∀ (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) → | |
| ∃ M : ℕ,∀ Cq : ℝ,0≤Cq → ∃ 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 | |
| ∀ (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ) | |
| (Q P : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))), | |
| Measurable f → Measurable g → (∀ x,‖f x‖≤1) → (∀ x,‖g x‖≤1) → | |
| (∀ b∈assemblyDescriptorList e₁.profile N e₁.q c,L₁ b∈W) → | |
| (∀ b∈assemblyDescriptorList e₂.profile N e₂.q c,L₂ b∈W) → | |
| (∑ b∈assemblyDescriptorList e₁.profile N e₁.q c,(L₁ b).length)≤total → | |
| (∑ b∈assemblyDescriptorList e₂.profile N e₂.q c,(L₂ b).length)≤total → | |
| (∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) → | |
| (∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j P.1 w‖≤Cq) → | |
| assemblyRootEnergy e₁ e₂ c L₁ L₂ f g Q P Ψ ψ ≤ | |
| ((2:ℝ)^N)^(-beta+Cd*(kap+sig/kap)+eta) | |
| := by | |
| let Dloss : ℝ := |Ctest|+100*(n+2:ℝ) | |
| let Cd : ℝ := profileDimLoss n Ctest+Dloss | |
| have hDloss : 0≤Dloss := by dsimp [Dloss]; positivity | |
| have hCd : 0≤Cd := add_nonneg (profileDimLoss_nonneg n Ctest) hDloss | |
| refine ⟨Cd,hCd,?_⟩ | |
| intro chi beta sig kap eta hchi hbeta hbetasm hkap hkap1 hsig hsigkap hsmall heta | |
| W B total hW Ψ hΨ hsupp hΨ0 hΨ1 ψ hψ hψ1 | |
| have hbeta1 : beta<1/10 := hbetasm.trans_le (min_le_right _ _) | |
| have hkap1' : kap≤1 := by linarith only [hkap1] | |
| have hsig1 : sig≤1/100 := by linarith only [hsigkap,hkap1] | |
| obtain ⟨D,hD,N₁,hN₁,hchild⟩ := uniform_assembly_child_max hn Ctest hsig hkap hkap1' hsigkap | |
| (show 0<eta/4 by positivity) W B total hW Ψ hΨ hsupp hΨ0 hΨ1 | |
| obtain ⟨M,hgraph⟩ := uniform_assembly_root_graph Ctest hsig hsig1 | |
| (sourceSmoothingLevels_pos hsig hkap) W B total hW Ψ hΨ hsupp hΨ0 ψ hψ hψ1 | |
| refine ⟨M,?_⟩ | |
| intro Cq hCq | |
| obtain ⟨Cerr,hCerr,N₂,hgraph⟩ := hgraph Cq hCq | |
| obtain ⟨N₃,hconst⟩ := fixed_constant_absorbed (show 0<eta/2 by positivity) ((Cq*D^B)^4+Cerr) | |
| obtain ⟨N₄,hN₄⟩ := exists_nat_ge ((n+1:ℝ)/(eta/4)) | |
| obtain ⟨N₅,hN₅⟩ := exists_nat_ge ((n+1:ℝ)/beta) | |
| refine ⟨max 1000 (max N₁ (max N₂ (max N₃ (max N₄ N₅)))),?_⟩ | |
| intro N hN ρ ν B₀ m₁ m₂ hd₁ hd₂ hroot₁ hroot₂ e₁ e₂ c | |
| L₁ L₂ f g Q P hf hg hfn hgn hL₁ hL₂ ht₁ ht₂ hQ hP | |
| 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 heN : e₂.N=e₁.N := rfl | |
| have hentry := assembly_entry_estimates hn e₁ e₂ heN (by change 1000≤N; omega) | |
| hd₁ hd₂ hchi hbeta hbeta1 hsmall hkap hkap1 rfl rfl | |
| change ⌊beta*(N:ℝ)/(n+1:ℝ)⌋₊≤c ∧ (c:ℝ)<(N:ℝ)/4 ∧ c+100*e₁.q<N ∧ _ at hentry | |
| have hfloor : 1≤⌊beta*(N:ℝ)/(n+1:ℝ)⌋₊ := by | |
| apply (Nat.le_floor_iff (by positivity : 0≤beta*(N:ℝ)/(n+1:ℝ))).mpr | |
| have hN5 : (N₅:ℝ)≤N := by exact_mod_cast (show N₅≤N by omega) | |
| have hh := (div_le_iff₀ hbeta).mp (hN₅.trans hN5) | |
| apply (le_div_iff₀ (by positivity : 0<(n+1:ℝ))).mpr | |
| norm_num | |
| nlinarith only [hh] | |
| have hcpos : 0<c := by omega | |
| have hQ0 : ∀ w,‖w‖=1 → ‖Q.1 w‖≤Cq := by | |
| intro w hw | |
| simpa only [norm_iteratedFDeriv_zero] using hQ 0 (Nat.zero_le _) w hw | |
| have hP0 : ∀ w,‖w‖=1 → ‖P.1 w‖≤Cq := by | |
| intro w hw | |
| simpa only [norm_iteratedFDeriv_zero] using hP 0 (Nat.zero_le _) w hw | |
| have hch := hchild e₁ e₂ (by change N₁≤N;omega) rfl rfl rfl rfl rfl rfl rfl rfl rfl | |
| B₀ ((n+1:ℝ)/2+chi) ((n+1:ℝ)/2+chi) hd₁ hd₂ c hentry.2.2.1 | |
| L₁ L₂ f g Q P Cq hf hg hfn hgn hL₁ hL₂ ht₁ ht₂ hQ0 hP0 | |
| have hgr := hgraph e₁ e₂ (by change N₂≤N;omega) rfl rfl rfl rfl rfl rfl rfl hq hq | |
| B₀ ((n+1:ℝ)/2+chi) ((n+1:ℝ)/2+chi) hd₁ hd₂ hroot₁ hroot₂ | |
| c hcpos hentry.2.1 L₁ L₂ f g Q P hf hg hfn hgn hL₁ hL₂ ht₁ ht₂ hQ hP | |
| let R : ℝ := (2:ℝ)^N | |
| let u : ℝ := -e₁.profile c-e₂.profile c+min (height e₁.profile 0 c) (height e₂.profile 0 c)+ | |
| (Ctest+36*(n+1:ℝ)+5)*sig | |
| let v : ℝ := (assemblyProfileExponent e₁ c+assemblyProfileExponent e₂ c)/2+ | |
| profileDimLoss n Ctest*(kap+sig/kap)+eta/4 | |
| let G : ℝ := -beta+Cd*(kap+sig/kap) | |
| have hR : 1≤R := one_le_pow₀ (by norm_num) | |
| have hRp : 0<R := by dsimp [R]; positivity | |
| have hcorr : (n+1:ℝ)/(N:ℝ)≤eta/4 := by | |
| apply (div_le_iff₀ hNR).mpr | |
| have hN4 : (N₄:ℝ)≤N := by exact_mod_cast (show N₄≤N by omega) | |
| have hh := (div_le_iff₀ (show 0<eta/4 by positivity)).mp (hN₄.trans hN4) | |
| nlinarith only [hh] | |
| have hsigloss : sig≤kap+sig/kap := by | |
| have hh : sig ≤ sig/kap := by | |
| apply (le_div_iff₀ hkap).mpr | |
| nlinarith only [hkap1',hsig] | |
| linarith only [hh,hkap] | |
| have hcoefficient : (n+4:ℝ)+Ctest+36*(n+1:ℝ)+5≤Dloss := by | |
| have hn0 := Nat.cast_nonneg (α:=ℝ) n | |
| have hC := le_abs_self Ctest | |
| dsimp only [Dloss] | |
| linarith only [hn0,hC] | |
| have hcf : ((n+4:ℝ)+Ctest+36*(n+1:ℝ)+5)*sig≤Dloss*(kap+sig/kap) := | |
| (mul_le_mul_of_nonneg_right hcoefficient hsig.le).trans (mul_le_mul_of_nonneg_left hsigloss hDloss) | |
| have huv : u+v≤G+eta/2 := by | |
| have he := hentry.2.2.2 | |
| change -e₁.profile c-e₂.profile c+min (height e₁.profile 0 c) (height e₂.profile 0 c)+ | |
| (assemblyProfileExponent e₁ c+assemblyProfileExponent e₂ c)/2≤-beta+(n+4:ℝ)*sig+(n+1:ℝ)/N at he | |
| dsimp only [u,v,G,Cd] | |
| nlinarith only [he,hcorr,hcf] | |
| have herr : (-2:ℝ)≤G+eta/2 := by | |
| have hpos : 0≤Cd*(kap+sig/kap) := mul_nonneg hCd (by positivity) | |
| dsimp only [G] | |
| linarith only [hbeta1,hpos,heta] | |
| have hnum : (Cq*D^B)^4+Cerr≤R^(eta/2) := hconst N (by omega) | |
| change assemblyRootEnergy e₁ e₂ c L₁ L₂ f g Q P Ψ ψ≤R^(G+eta) | |
| change assemblyRootEnergy e₁ e₂ c L₁ L₂ f g Q P Ψ ψ≤R^u*_+Cerr*R^(-2:ℝ) at hgr | |
| change _≤(Cq*D^B)^4*R^v at hch | |
| calc | |
| assemblyRootEnergy e₁ e₂ c L₁ L₂ f g Q P Ψ ψ | |
| ≤R^u*((Cq*D^B)^4*R^v)+Cerr*R^(-2:ℝ) := | |
| hgr.trans (add_le_add (mul_le_mul_of_nonneg_left hch (Real.rpow_nonneg hRp.le _)) le_rfl) | |
| _=(Cq*D^B)^4*R^(u+v)+Cerr*R^(-2:ℝ) := by rw [Real.rpow_add hRp u v]; ring | |
| _≤(Cq*D^B)^4*R^(G+eta/2)+Cerr*R^(G+eta/2) := | |
| add_le_add (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hR huv) (by positivity)) | |
| (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hR herr) hCerr) | |
| _=((Cq*D^B)^4+Cerr)*R^(G+eta/2) := by ring | |
| _≤R^(eta/2)*R^(G+eta/2) := mul_le_mul_of_nonneg_right hnum (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 | |