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d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | 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
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