| import OAI.MeasureTheory.Falconer.Graphs.RecursiveEnergyFamily |
| import OAI.MeasureTheory.Falconer.Recursion.WeightedBranchInduction |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ContDiff |
|
|
| theorem RecursiveEnvironment.energies_mono (e : RecursiveEnvironment) {K L : ℕ} |
| (hKL : K ≤ L) (Ψ : ℝ → ℝ) {s : BranchState} {x : ℝ} |
| (hx : e.energies K Ψ s x) : e.energies L Ψ s x := by |
| cases s with |
| | D a t i => exact hx |
| | H a₀ t₀ a v i => |
| obtain ⟨X,Y,hX,hY,k,hk,h,hh,hhb,hx⟩ := hx |
| exact ⟨X,Y,hX,hY,k,fun l => (hk l).trans hKL,h,hh,hhb,hx⟩ |
|
|
| theorem uniform_actual_profile_with_overhead {lam sig kap theta : ℝ} |
| (hlam : 0 ≤ lam) (hsig : 0 < sig) (hkap : 0 < kap) (hkap1 : kap ≤ 1) |
| (hgap : sig < kap/1000) (hloss : 4000*(lam+sig) ≤ 1) |
| (htheta : 0 < theta) (htheta1 : theta ≤ 1) (K₀ : ℕ) |
| {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hpos : ∀ x, 0 ≤ Ψ x) |
| (hs : ∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) |
| (hone : ∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) : |
| ∃ N₀ : ℕ, 0 < N₀ ∧ ∀ e : RecursiveEnvironment, |
| e.lam = lam → e.sig = sig → e.q = sourcePolicyBlock e.N kap → |
| e.J = sourceSmoothingLevels sig kap → N₀ ≤ e.N → e.Q.side = 1 → e.ρ e.Q.carrierᶜ = 0 → |
| ∀ B : ℕ, ∀ δ : ℝ, RegularizedProfileData e.Q e.ρ e.N B δ e.lam e.mass → |
| ∀ root : BranchState, root.Valid e.N e.sig → root.index = 0 → |
| root.start+10*e.q ≤ root.finish → ∀ x, e.energies K₀ Ψ root x → |
| x ≤ ((e.N:ℝ)+4)^(⌈20/kap+1/sig⌉₊+1) * |
| (((2:ℝ)^e.N)^(2*e.profile root.start-potential e.profile root.start root.finish (10*e.q) |
| +500*kap+10^6*(lam+sig)/kap+theta)+((2:ℝ)^e.N)^(-10:ℝ)) := by |
| let Dstar : ℝ := 1+20/kap+1/sig |
| have hDstar : 1 ≤ Dstar := by |
| have hk : 0 ≤ 20/kap := by positivity |
| have hs : 0 ≤ 1/sig := by positivity |
| dsimp only [Dstar] |
| linarith |
| have hDpos : 0 < Dstar := by linarith |
| let eps := theta/(2*Dstar) |
| have heps : 0 < eps := div_pos htheta (by positivity) |
| have hepstau : eps ≤ theta/2 := by |
| apply (div_le_iff₀ (by positivity : 0 < 2*Dstar)).mpr |
| nlinarith |
| have heps1 : eps ≤ 1 := by linarith |
| let T : ℝ := 1000*(1/sig+1/kap)+1001 |
| have hT : 0 < T := by dsimp [T]; positivity |
| have hTlarge : 1000*(1/sig+1/kap)+1000 < T := by dsimp [T]; linarith |
| obtain ⟨K,hK,N₁,hN₁,h₁⟩ := uniform_recursive_energy_rules hlam hsig hkap hgap heps |
| (half_pos htheta) hT K₀ hΨ hpos hs hone |
| obtain ⟨N₂,_,h₂⟩ := uniform_branch_successor_validity hsig hkap hgap |
| obtain ⟨N₃,_,h₃⟩ := uniform_source_branch_depth hsig hkap hkap1 hgap |
| obtain ⟨N₄,_,h₄⟩ := uniform_source_weighted_terminal hlam hsig hkap hkap1 hgap htheta |
| obtain ⟨N₅,_,h₅⟩ := uniform_source_weighted_error hsig hkap hkap1 hgap hloss heps1 hTlarge |
| refine ⟨max N₁ (max N₂ (max N₃ (max N₄ N₅))),by omega,?_⟩ |
| intro e hdlam hdsig hdq hdJ hN hQ hcarry B δ hprof root hroot hi hwide x hx |
| have hNp : 0 < e.N := by omega |
| have hqp := sourcePolicyBlock_pos hNp hkap |
| have hA := hprof.discrete_lipschitz |
| have hrules := h₁ e hdlam hdsig hdq hdJ (by omega) hQ hcarry B δ hprof |
| let D : ℕ := ⌈20/kap+1/sig⌉₊ |
| let L := ((2:ℝ)^e.N)^(2*e.profile root.start-potential e.profile root.start root.finish (10*e.q) |
| +500*kap+10^6*(lam+sig)/kap+theta) |
| let terminal := ((2:ℝ)^e.N)^(200*(e.q:ℝ)/e.N+12*lam+theta/2) |
| let floor := ((2:ℝ)^e.N)^(12*lam+eps) |
| have hfloor : floor ≤ terminal := by |
| apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) |
| have hnonneg : 0 ≤ 200*(e.q:ℝ)/e.N := by positivity |
| linarith |
| have hdepth : ∀ n, ∀ b : PolicyBranch e.profile e.N (sourcePolicyBlock e.N kap) sig n, |
| b.state 0 = root → n ≤ D := by |
| intro n b hb |
| have hidx : (b.state 0).index = 0 := by rw [hb]; exact hi |
| have hn := (h₃ e.N (by omega) e.profile n b hidx).2.2.1 |
| exact Nat.cast_le.mp (hn.trans (Nat.le_ceil _)) |
| have htermWeight : ∀ n, ∀ b : PolicyBranch e.profile e.N (sourcePolicyBlock e.N kap) sig n, |
| b.state 0 = root → b.weight lam eps*terminal ≤ L := by |
| intro n b hb |
| have hw : (b.state 0).start+10*sourcePolicyBlock e.N kap ≤ (b.state 0).finish := by |
| simpa only [hb,hdq] using hwide |
| have hh := h₄ e.N (by omega) e.profile hA n b hw |
| simpa only [hb,← hdq] using hh |
| have hind := weighted_branch_energy_induction e.profile e.N (sourcePolicyBlock e.N kap) D |
| sig lam eps ((e.N:ℝ)+3) terminal floor (((2:ℝ)^e.N)^(-T)) L (((2:ℝ)^e.N)^(-10:ℝ)) |
| root (e.energies K Ψ) hNp hqp hsig |
| (by linarith [Nat.cast_nonneg (α := ℝ) e.N]) (by positivity) (by positivity) |
| (fun s u l hv hstep => (h₂ e.N (by omega) e.profile s u l hv hstep).1) |
| hdepth |
| (by |
| intro s hv ht y hy |
| exact hrules.1 s (by simpa only [hdsig] using hv) (by simpa only [hdq] using ht) y hy) |
| (by |
| intro s hv hr hw M hM hfloorM hchildren y hy |
| obtain ⟨n,b,hb,hbs⟩ := hr |
| have hidx : (b.state 0).index = 0 := by rw [hb]; exact hi |
| have hindex := ((h₃ e.N (by omega) e.profile n b hidx).2.2.2 n le_rfl).2 |
| rw [hbs,← hdJ] at hindex |
| apply hrules.2 s (by simpa only [hdsig] using hv) hindex.le |
| (by simpa only [hdq] using hw) M hM hfloorM _ y hy |
| intro u l hstep z hz |
| exact hchildren u l (by simpa only [hdq,hdsig] using hstep) z hz) |
| htermWeight |
| (fun n b hb => (mul_le_mul_of_nonneg_left hfloor (b.weight_pos lam eps).le).trans (htermWeight n b hb)) |
| (by |
| intro n b hb |
| have hidx : (b.state 0).index = 0 := by rw [hb]; exact hi |
| exact h₅ e.N (by omega) e.profile hA n b hidx) |
| let b : PolicyBranch e.profile e.N (sourcePolicyBlock e.N kap) sig 0 := |
| ⟨fun _ => root,fun _ => .low,fun _ _ => by simpa only [hdsig] using hroot,by omega⟩ |
| have hh := hind 0 b rfl x (e.energies_mono hK Ψ hx) |
| simpa only [PolicyBranch.weight_zero,one_mul,Nat.sub_zero,show (e.N:ℝ)+3+1 = (e.N:ℝ)+4 by ring] using hh |
|
|
| theorem recursion_overhead_absorption (D : ℕ) {eta theta : ℝ} |
| (heta : 0 < eta) (htheta : theta ≤ eta/4) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ g : ℝ, -4 ≤ g → |
| ((N:ℝ)+4)^(D+1)*(((2:ℝ)^N)^g+((2:ℝ)^N)^(-10:ℝ)) ≤ |
| ((2:ℝ)^N)^(g+eta-theta) := by |
| have he : 0 < eta-theta := by linarith |
| obtain ⟨N₁,h₁⟩ := fixed_polynomial_subpower (D+2) he |
| refine ⟨max 2 N₁,?_⟩ |
| intro N hN g hg |
| have hNR : 2 ≤ (N:ℝ) := by exact_mod_cast (show 2 ≤ N by omega) |
| have hRp : 0 < (2:ℝ)^N := by positivity |
| have herr : ((2:ℝ)^N)^(-10:ℝ) ≤ ((2:ℝ)^N)^g := |
| Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) (by linarith) |
| have hpol : 2*((N:ℝ)+4)^(D+1) ≤ (4*(N:ℝ)+1)^(D+2) := by |
| calc |
| _ ≤ 2*(4*(N:ℝ)+1)^(D+1) := mul_le_mul_of_nonneg_left |
| (pow_le_pow_left₀ (by linarith) (by linarith) (D+1)) (by norm_num) |
| _ ≤ (4*(N:ℝ)+1)^(D+1)*(4*(N:ℝ)+1) := by nlinarith [pow_nonneg (by linarith : 0 ≤ 4*(N:ℝ)+1) (D+1)] |
| _ = _ := (pow_succ _ (D+1)).symm |
| calc |
| _ ≤ (2*((N:ℝ)+4)^(D+1))*((2:ℝ)^N)^g := by nlinarith [pow_nonneg (by linarith : 0 ≤ (N:ℝ)+4) (D+1)] |
| _ ≤ ((2:ℝ)^N)^(eta-theta)*((2:ℝ)^N)^g := |
| mul_le_mul_of_nonneg_right (hpol.trans (h₁ N (by omega))) (by positivity) |
| _ = _ := by rw [← Real.rpow_add hRp]; congr 1; ring |
| |
| theorem uniform_actual_recursive_profile {lam sig kap eta : ℝ} |
| (hlam : 0 ≤ lam) (hsig : 0 < sig) (hkap : 0 < kap) (hkap1 : kap ≤ 1) |
| (hgap : sig < kap/1000) (hloss : 4000*(lam+sig) ≤ 1) (heta : 0 < eta) (K₀ : ℕ) |
| {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hpos : ∀ x, 0 ≤ Ψ x) |
| (hs : ∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) |
| (hone : ∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) : |
| ∃ N₀ : ℕ, 0 < N₀ ∧ ∀ e : RecursiveEnvironment, |
| e.lam = lam → e.sig = sig → e.q = sourcePolicyBlock e.N kap → |
| e.J = sourceSmoothingLevels sig kap → N₀ ≤ e.N → e.Q.side = 1 → e.ρ e.Q.carrierᶜ = 0 → |
| ∀ B : ℕ, ∀ δ : ℝ, RegularizedProfileData e.Q e.ρ e.N B δ e.lam e.mass → |
| ∀ root : BranchState, root.Valid e.N e.sig → root.index = 0 → |
| root.start+10*e.q ≤ root.finish → ∀ x, e.energies K₀ Ψ root x → |
| x ≤ ((2:ℝ)^e.N)^(2*e.profile root.start-potential e.profile root.start root.finish (10*e.q) |
| +500*kap+10^6*(lam+sig)/kap+eta) := by |
| let theta := min (eta/4) 1 |
| have htheta : 0 < theta := lt_min (by positivity) (by norm_num) |
| obtain ⟨N₁,hN₁,h₁⟩ := uniform_actual_profile_with_overhead hlam hsig hkap hkap1 hgap hloss |
| htheta (min_le_right _ _) K₀ hΨ hpos hs hone |
| obtain ⟨N₂,h₂⟩ := recursion_overhead_absorption ⌈20/kap+1/sig⌉₊ heta (min_le_left (eta/4) 1) |
| refine ⟨max N₁ N₂,by omega,?_⟩ |
| intro e hdlam hdsig hdq hdJ hN hQ hcarry B δ hprof root hv hi hw x hx |
| have hNp : 0 < e.N := by omega |
| have hq : 0 < e.q := by rw [hdq]; exact sourcePolicyBlock_pos hNp hkap |
| have hA := hprof.discrete_lipschitz |
| have hzero : e.profile 0 = 0 := by simp only [RecursiveEnvironment.profile,Nat.cast_zero,hprof.zero,zero_div,sub_zero] |
| have hrootdepth := BranchState.valid_depths hv |
| have hbase := source_profile_exponent_lower hA hNp hzero (by omega : 0 < 10*e.q) hw hrootdepth.2 |
| change -4 ≤ 2*e.profile root.start-potential e.profile root.start root.finish (10*e.q) at hbase |
| have hg : -4 ≤ 2*e.profile root.start-potential e.profile root.start root.finish (10*e.q) |
| +500*kap+10^6*(lam+sig)/kap+theta := by |
| have hloss0 : 0 ≤ 10^6*(lam+sig)/kap := by positivity |
| linarith |
| have hh := h₁ e hdlam hdsig hdq hdJ (by omega) hQ hcarry B δ hprof root hv hi hw x hx |
| have habs := h₂ e.N (by omega) _ hg |
| have hh := hh.trans habs |
| apply hh.trans_eq |
| congr 1 |
| dsimp only [theta] |
| ring |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
