| import Mathlib |
| import OAI.MeasureTheory.Falconer.Estimates.ActualDirectBuffer |
| import OAI.MeasureTheory.Falconer.Estimates.ActualOuterMasks |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.Falconer.Recursion.BranchTransitions |
| import OAI.MeasureTheory.Falconer.Recursion.InteriorPolicy |
| import OAI.MeasureTheory.Falconer.Recursion.Potential |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.RecursiveSuffixRule |
|
|
| namespace OAI |
|
|
| /-! Recursive policies and quantitative depth 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 uniform_recursive_rules (hn : 1≤n) {sig kap eps T : ℝ} |
| (hsig : 0<sig) (hkap : 0<kap) (hkap1 : kap≤1) (hgap : sig<kap/1000) |
| (heps : 0<eps) (hT : 0<T) (ell : ℕ) (hK : 1000*(n+2:ℝ)<(2:ℝ)^ell) |
| (Ψ : ℝ → ℝ) (hΨ : ContDiff ℝ ∞ Ψ) (hsupp : tsupport Ψ⊆Ioo (1/4:ℝ) 4) |
| (hΨ0 : ∀ r,0≤Ψ r) (hΨ1 : ∀ r∈Icc (1/2:ℝ) 2,1≤Ψ r) (hiΨ : Integrable Ψ) : |
| ∃ Order : ℕ,∀ (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) |
| (B total : ℕ), (∀ L∈W,L.length≤B) → prefixInverseVocabulary n Order⊆W → 2*Order≤total → |
| ∃ H E : ℝ,1≤H ∧ 0≤E ∧ ∃ N₀ : ℕ,0<N₀ ∧ ∀ e : RecursiveEnvironmentND n, |
| N₀≤e.N → e.sig=sig → e.q=sourcePolicyBlock e.N kap → e.J=sourceSmoothingLevels sig kap → |
| ∀ B₀ s,RegularizedProfileData e.ρ e.N B₀ s sig e.mass → |
| ∀ state : BranchState,state.Valid e.N sig → state.index<e.J → state.start+100*e.q<state.finish → |
| ∀ M : ℝ,0≤M → ((2:ℝ)^e.N)^(4*(n+2:ℝ)*sig)≤M → |
| (∀ u l,DimensionalPolicy.BranchStep e.profile e.N e.q (ell+2) sig state l u → |
| ∀ x,e.energies ((2:ℝ)^ell) W total Ψ u x → |
| ((2:ℝ)^e.N)^(DimensionalPolicy.branchExponentND e.profile (recursiveLoss n e.C) sig sig eps state l u)*x≤M) → |
| ∀ x,e.energies ((2:ℝ)^ell) W total Ψ state x → |
| x≤H*((e.N:ℝ)+1)*M+E*((2:ℝ)^e.N)^(-T) := by |
| have hsig1 : sig≤1 := by linarith |
| have hK' : 32*(n+2:ℝ)≤(2:ℝ)^ell := by nlinarith [(Nat.cast_nonneg n : (0:ℝ)≤(n:ℝ))] |
| have hJ : 0<sourceSmoothingLevels sig kap := sourceSmoothingLevels_pos hsig hkap |
| obtain ⟨Order,hpref⟩ := uniform_recursive_prefix_rule hn hsig hsig1 hK' _ hJ eps T heps hT Ψ hΨ hsupp hΨ0 hΨ1 hiΨ |
| obtain ⟨Hs,hHs,Ns,hsuff⟩ := uniform_recursive_suffix_rule (n:=n) hsig hJ |
| obtain ⟨Nc,hNc,hceil⟩ := uniform_source_ceiling_slack hsig hgap |
| refine ⟨Order,?_⟩ |
| intro W B total hW hV htotal |
| obtain ⟨Hp,Ep,hHp,hEp,Np,hpref⟩ := hpref W B total hW hV htotal |
| obtain ⟨Hd,Ed,hHd,hEd,Nd,hdir⟩ := uniform_recursive_direct_rule hsig T _ hJ Ψ hΨ hsupp hΨ0 hiΨ W B total hW |
| obtain ⟨Hf,hHf,Nf,hnf⟩ := uniform_recursive_near_far_rule Ψ hΨ hsupp hΨ0 hsig hsig1 W B total hW hJ T ell hK' |
| let H := 1+Hs+Hp+Hd+Hf |
| let E := 1+Ep+Ed |
| have hH : 1≤H := by dsimp only [H]; linarith |
| have hE : 0≤E := by dsimp only [E]; positivity |
| refine ⟨H,E,hH,hE,max Nc (max Ns (max Np (max Nd Nf))),by omega,?_⟩ |
| intro e hN he hqeq hJe B₀ s hd state hv hi hw M hM hfloor hchildren x hx |
| let := hd.probability |
| have hq : 0<e.q := by rw [hqeq]; exact sourcePolicyBlock_pos (by omega) hkap |
| have hcl := hceil e.N (by omega) |
| have hslack10 : 10*sig*(e.N:ℝ)≤e.q := by |
| rw [hqeq] |
| exact (Nat.le_ceil _).trans (Nat.cast_le.mpr hcl.1.le) |
| have hslack : sig*(e.N:ℝ)≤e.q := by nlinarith [mul_nonneg hsig.le (Nat.cast_nonneg e.N : (0:ℝ)≤(e.N:ℝ))] |
| have hcl' : ⌈10*sig*(e.N:ℝ)⌉₊<e.q := by simpa only [hqeq] using hcl.1 |
| have hN0 : 0≤(e.N:ℝ) := Nat.cast_nonneg e.N |
| have hpow : 0≤((2:ℝ)^e.N)^(-T) := by positivity |
| have hfH : ∀ A : ℝ,0≤A → A≤H → A*M≤H*((e.N:ℝ)+1)*M := by |
| intro A hA hAH |
| have h₁ := mul_le_mul_of_nonneg_right hAH hM |
| have h₂ : 0≤H*(e.N:ℝ)*M := by positivity |
| nlinarith only [h₁,h₂] |
| have hfHE : ∀ A F : ℝ,0≤A → A≤H → F≤E → |
| A*((e.N:ℝ)+1)*M+F*((2:ℝ)^e.N)^(-T)≤H*((e.N:ℝ)+1)*M+E*((2:ℝ)^e.N)^(-T) := by |
| intro A F hA hAH hFE |
| exact add_le_add (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hAH (by positivity)) hM) |
| (mul_le_mul_of_nonneg_right hFE hpow) |
| cases state with |
| | D a t i => |
| by_cases hp : 2*policyDepth e.profile e.N e.q a t≤a+t |
| · have hh := hpref e (by omega) he hJe B₀ s hd hq hslack a t i (ell+2) hi hv.2 hw hp M hM hfloor hchildren x hx |
| exact hh.trans (hfHE Hp Ep hHp (by dsimp only [H]; linarith) (by dsimp only [E]; linarith)) |
| · have hh := hsuff e (by omega) he hJe (show IsFiniteMeasure e.ρ from inferInstance) hq _ hK W total (ell+2) a t i Ψ eps heps.le hi hv.2 hw (by omega) M |
| (fun y hy => hchildren _ _ (.suffix hw (by omega)) y hy) x hx |
| exact hh.trans ((hfH Hs hHs (by dsimp only [H]; linarith)).trans (le_add_of_nonneg_right (mul_nonneg hE hpow))) |
| | H a₀ t₀ a v i => |
| by_cases hnear : a<(BranchState.H a₀ t₀ a v i).nearDepth e.profile e.N e.q sig |
| · have hh := hnf e (by omega) he hJe B₀ s hd hq a₀ t₀ a v i hv hw hnear eps M heps.le hM hchildren x hx |
| have hb := hfHE Hf 1 hHf (by dsimp only [H]; linarith) (by dsimp only [E]; linarith) |
| simpa only [one_mul] using hh.trans (by simpa only [one_mul] using hb) |
| · have hh := hdir e (by omega) he hJe B₀ s hd hq hcl' hslack hslack10 a₀ t₀ a v i (ell+2) hi hv hw (by omega) _ eps M heps.le hM hchildren x hx |
| exact hh.trans (add_le_add (hfH Hd (by linarith) (by dsimp only [H]; linarith)) |
| (mul_le_mul_of_nonneg_right (by dsimp only [E]; linarith) hpow)) |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
| |
| 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} |
| |
| lemma recursiveLoss_dim_self (n : ℕ) (C : ℝ) : 4*(n+2:ℝ)≤recursiveLoss n C := by |
| unfold recursiveLoss |
| nlinarith [abs_nonneg C,(Nat.cast_nonneg n : (0:ℝ)≤(n:ℝ))] |
| |
| lemma RegularizedProfileData.environment_lipschitz (e : RecursiveEnvironmentND n) |
| {B : ℕ} {s sig : ℝ} (h : RegularizedProfileData e.ρ e.N B s sig e.mass) : |
| DimensionalPolicy.ProfileLipschitzND e.profile e.N ((n+1:ℝ)/2) := by |
| intro i hi j hj |
| have hh := h.adjusted (j:ℝ) (i:ℝ) (Nat.cast_nonneg _) (Nat.cast_le.mpr hj) |
| (Nat.cast_nonneg _) (Nat.cast_le.mpr hi) |
| simpa only [RecursiveEnvironmentND.profile,adjustedProfile,Nat.cast_add,Nat.cast_one] using hh |
| |
| lemma profile_exponent_lowerND {A : ℕ → ℝ} {N a t Q : ℕ} {L : ℝ} |
| (hA : DimensionalPolicy.ProfileLipschitzND A N L) (hN : 0<N) (hL : 0≤L) |
| (hQ : 0<Q) (hw : a+Q≤t) (ht : t≤N) : |
| -2*L≤2*A a-potential A a t Q := by |
| have ha : a≤t := by omega |
| have hpot := DimensionalPolicy.potential_le_final_startND hN hL hA hQ hw ht (le_refl a) ha |
| have hQN : (Q:ℝ)≤N := Nat.cast_le.mpr (by omega) |
| have hfrac : (Q:ℝ)/(N:ℝ)≤1 := (div_le_one (Nat.cast_pos.mpr hN)).mpr hQN |
| have hh := mul_le_mul_of_nonneg_left hfrac (show 0≤2*L by positivity) |
| rw [←mul_div_assoc] at hh |
| linarith |
| |
| theorem recursion_overhead_absorptionND (D : ℕ) (H E : ℝ) |
| {eta theta B : ℝ} (heta : 0<eta) (htheta : theta≤eta/4) |
| (hH : 1≤H) (hE : 0≤E) : |
| ∃ N₀ : ℕ,∀ N≥N₀,∀ g : ℝ,-B≤g → |
| (H*((N:ℝ)+1)+1)^(D+1)*(H*((2:ℝ)^N)^g+E*((2:ℝ)^N)^(-B))≤ |
| ((2:ℝ)^N)^(g+eta-theta) := by |
| obtain ⟨N₁,h₁⟩ := fixed_polynomial_subpower (d:=2) (2*(D+1)+2) (show 0<eta-theta by linarith) |
| refine ⟨max 2 (max N₁ (Nat.ceil (max H E))),?_⟩ |
| intro N hN g hg |
| have hNR : 2≤(N:ℝ) := Nat.cast_le.mpr (show 2≤N by omega) |
| have hCE : max H E≤(N:ℝ) := (Nat.le_ceil _).trans (Nat.cast_le.mpr (by omega)) |
| have hHN : H≤(N:ℝ) := (le_max_left _ _).trans hCE |
| have hEN : E≤(N:ℝ) := (le_max_right _ _).trans hCE |
| have hp : 0<((2:ℝ)^N)^g := by positivity |
| have herr : ((2:ℝ)^N)^(-B)≤((2:ℝ)^N)^g := |
| Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) hg |
| have hsum : H*((2:ℝ)^N)^g+E*((2:ℝ)^N)^(-B)≤(2*(N:ℝ))*((2:ℝ)^N)^g := by |
| have hh := mul_le_mul_of_nonneg_left herr hE |
| have hh₁ := mul_le_mul_of_nonneg_right hHN hp.le |
| have hh₂ := mul_le_mul_of_nonneg_right hEN hp.le |
| linarith only [hh,hh₁,hh₂] |
| have hbase : H*((N:ℝ)+1)+1≤(4*(N:ℝ)+1)^2 := by |
| have hh := mul_le_mul_of_nonneg_right hHN (show 0≤(N:ℝ)+1 by linarith) |
| nlinarith only [hh,hNR] |
| have hpoly : (H*((N:ℝ)+1)+1)^(D+1)*(2*(N:ℝ))≤(4*(N:ℝ)+1)^(2*(D+1)+2) := by |
| calc |
| _≤((4*(N:ℝ)+1)^2)^(D+1)*(2*(N:ℝ)) := |
| mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (by positivity) hbase _) (by positivity) |
| _≤((4*(N:ℝ)+1)^2)^(D+1)*(4*(N:ℝ)+1)^2 := |
| mul_le_mul_of_nonneg_left (by nlinarith only [hNR]) (by positivity) |
| _=_ := by rw [←pow_mul,←pow_add] |
| calc |
| _≤(H*((N:ℝ)+1)+1)^(D+1)*((2*(N:ℝ))*((2:ℝ)^N)^g) := |
| mul_le_mul_of_nonneg_left hsum (by positivity) |
| _≤(4*(N:ℝ)+1)^(2*(D+1)+2)*((2:ℝ)^N)^g := by |
| rw [←mul_assoc]; exact mul_le_mul_of_nonneg_right hpoly hp.le |
| _≤((2:ℝ)^N)^(eta-theta)*((2:ℝ)^N)^g := |
| mul_le_mul_of_nonneg_right (by have hh := h₁ N (by omega); norm_num at hh; exact hh) hp.le |
| _=_ := by rw [←Real.rpow_add (by positivity : 0<(2:ℝ)^N)]; congr 1; ring |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
| |
| end OAI |
| |
