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7f37411 | 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 | 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
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