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import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.Ledger
import OAI.MeasureTheory.FalconerCompletion.Angular.AngularHCongr
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFixedAngularBudget
namespace OAI
/-! Graph energies and pair estimates. -/
open _root_.OAI.PlanarFalconer
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer.TreeND
open Set MeasureTheory Attempt4Near Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {d : ℕ}
lemma graph_children_H_split
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
{a p : ℕ} (hap : a ≤ p) {X Y : CellPath d} (hX : X ∈ dyadicPaths a) (hY : Y ∈ dyadicPaths a)
{b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B)
(Ma₁ Ma₂ : ℝ) {Mp₁ Mp₂ : ℝ} (hMp₁ : Mp₁ ≠ 0) (hMp₂ : Mp₂ ≠ 0) (v : ℝ) (Ψ : ℝ → ℝ) :
sourceAngularHND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ
(fun z => sourceAngularTransformND ρ (treeCell X) Ma₁ b z.1 z.2)
(fun z => sourceAngularTransformND κ (treeCell Y) Ma₂ c z.1 z.2) =
sourceAngularHND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ
(fun z => (Mp₁/Ma₁) • ∑ P ∈ graphChildren ρ a p X, sourceAngularTransformND ρ (treeCell P) Mp₁ b z.1 z.2)
(fun z => (Mp₂/Ma₂) • ∑ Q ∈ graphChildren κ a p Y, sourceAngularTransformND κ (treeCell Q) Mp₂ c z.1 z.2) := by
apply sourceAngularHND_congr
· intro r
filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
exact graph_children_transform_split ρ hap hX hb hMp₁ r w hw
· intro r
filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
exact graph_children_transform_split κ hap hY hc hMp₂ r w hw
lemma graph_children_HF_split
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
{a p : ℕ} (hap : a ≤ p) {X Y : CellPath d} (hX : X ∈ dyadicPaths a) (hY : Y ∈ dyadicPaths a)
{b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B)
(Ma₁ Ma₂ : ℝ) {Mp₁ Mp₂ : ℝ} (hMp₁ : Mp₁ ≠ 0) (hMp₂ : Mp₂ ≠ 0)
(v : ℝ) (Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ) :
sourceAngularHFND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ ψ
(fun z => sourceAngularTransformND ρ (treeCell X) Ma₁ b z.1 z.2)
(fun z => sourceAngularTransformND κ (treeCell Y) Ma₂ c z.1 z.2) =
sourceAngularHFND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ ψ
(fun z => (Mp₁/Ma₁) • ∑ P ∈ graphChildren ρ a p X, sourceAngularTransformND ρ (treeCell P) Mp₁ b z.1 z.2)
(fun z => (Mp₂/Ma₂) • ∑ Q ∈ graphChildren κ a p Y, sourceAngularTransformND κ (treeCell Q) Mp₂ c z.1 z.2) := by
apply sourceAngularHFND_congr
· intro r
filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
exact graph_children_transform_split ρ hap hX hb hMp₁ r w hw
· intro r
filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
exact graph_children_transform_split κ hap hY hc hMp₂ r w hw
lemma graph_direct_product_identity (z : ℂ) (e₁ e₂ : ℝ) (u u' v v' : ℂ) :
z*((e₁:ℂ)*u*conj u')*((e₂:ℂ)*v*conj v') =
z*((e₁:ℂ)*(e₂:ℂ))*((u*v)*conj (u'*v')) := by
simp only [map_mul]
ring
lemma graph_crossed_product_identity (z : ℂ) (e₁ e₂ : ℝ) (s t u u' v v' : ℂ) :
z*((e₁:ℂ)*s*u*conj v')*((e₂:ℂ)*conj t*v*conj u') =
z*((e₁:ℂ)*(e₂:ℂ))*(((s*u)*v)*conj ((t*u')*v')) := by
simp only [map_mul]
ring
lemma graph_direct_cell_integrable
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
(M₁ M₂ : ℝ) {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B)
{Ψ : ℝ → ℝ} (v : ℝ) (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v)))
(δ : ℝ) (k₁ k₂ : Fin d → ℤ) (row col : CellPath d × CellPath d) :
Integrable (fun q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))) =>
(Ψ (q.1/(2:ℝ)^v):ℂ)*((tensorCapND δ k₁ q.2.1*tensorCapND δ k₂ q.2.2):ℂ)*
((sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1 *
sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2)*
conj (sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.1 *
sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.2)))
(volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))))) := by
have h := graph_angular_four_integrable (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)))
hΨ.ofReal (fun w => (tensorCapND δ k₁ w:ℂ)) (fun w => (tensorCapND δ k₂ w:ℂ))
(Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₁).continuous.measurable)
(Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₂).continuous.measurable)
(fun z => sourceAngularTransformND ρ (treeCell row.1) M₁ b z.1 z.2)
(fun z => sourceAngularTransformND ρ (treeCell col.1) M₁ b z.1 z.2)
(fun z => sourceAngularTransformND κ (treeCell row.2) M₂ c z.1 z.2)
(fun z => sourceAngularTransformND κ (treeCell col.2) M₂ c z.1 z.2)
(sourceAngularTransformND_measurable _ _ _ hb.measurable)
(sourceAngularTransformND_measurable _ _ _ hb.measurable)
(sourceAngularTransformND_measurable _ _ _ hc.measurable)
(sourceAngularTransformND_measurable _ _ _ hc.measurable)
(P₁:=1) (P₂:=1)
(B₁:=|M₁⁻¹| *ρ.real (treeCell row.1)*B) (B₂:=|M₁⁻¹| *ρ.real (treeCell col.1)*B)
(B₃:=|M₂⁻¹| *κ.real (treeCell row.2)*B) (B₄:=|M₂⁻¹| *κ.real (treeCell col.2)*B)
(by norm_num) (by norm_num) (by positivity [hb.nonneg]) (by positivity [hb.nonneg])
(by positivity [hc.nonneg]) (by positivity [hc.nonneg]) ?_
· apply h.congr
filter_upwards with q
exact graph_direct_product_identity _ _ _ _ _ _ _
· filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
refine ⟨?_,?_,fun r => ⟨?_,?_,?_,?_⟩⟩
· simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _)] using tensorCapND_le_one δ k₁ w
· simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _)] using tensorCapND_le_one δ k₂ w
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw)
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw)
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw)
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw)
lemma graph_crossed_cell_integrable
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
(M₁ M₂ : ℝ) {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B)
{Ψ : ℝ → ℝ} (v : ℝ) (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v)))
(ψ : EuclideanSpace ℝ (Fin d) → ℂ) (hmψ : Measurable ψ)
{P : ℝ} (hP : 0 ≤ P) (hψ : ∀ w, ‖w‖=1 → ‖ψ w‖ ≤ P)
(δ : ℝ) (k₁ k₂ : Fin d → ℤ) (row col : CellPath d × CellPath d) :
Integrable (fun q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))) =>
(Ψ (q.1/(2:ℝ)^v):ℂ)*((tensorCapND δ k₁ q.2.1*tensorCapND δ k₂ q.2.2):ℂ)*
(((ψ q.2.1*sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1) *
sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2)*
conj ((ψ q.2.2*sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.2) *
sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.1)))
(volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))))) := by
have h := graph_angular_four_integrable (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)))
hΨ.ofReal (fun w => (tensorCapND δ k₁ w:ℂ)*ψ w) (fun w => (tensorCapND δ k₂ w:ℂ)*conj (ψ w))
((Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₁).continuous.measurable).mul hmψ)
((Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₂).continuous.measurable).mul (Complex.continuous_conj.measurable.comp hmψ))
(fun z => sourceAngularTransformND ρ (treeCell row.1) M₁ b z.1 z.2)
(fun z => sourceAngularTransformND κ (treeCell col.2) M₂ c z.1 z.2)
(fun z => sourceAngularTransformND κ (treeCell row.2) M₂ c z.1 z.2)
(fun z => sourceAngularTransformND ρ (treeCell col.1) M₁ b z.1 z.2)
(sourceAngularTransformND_measurable _ _ _ hb.measurable)
(sourceAngularTransformND_measurable _ _ _ hc.measurable)
(sourceAngularTransformND_measurable _ _ _ hc.measurable)
(sourceAngularTransformND_measurable _ _ _ hb.measurable)
(P₁:=P) (P₂:=P)
(B₁:=|M₁⁻¹| *ρ.real (treeCell row.1)*B) (B₂:=|M₂⁻¹| *κ.real (treeCell col.2)*B)
(B₃:=|M₂⁻¹| *κ.real (treeCell row.2)*B) (B₄:=|M₁⁻¹| *ρ.real (treeCell col.1)*B)
hP hP (by positivity [hb.nonneg]) (by positivity [hc.nonneg])
(by positivity [hc.nonneg]) (by positivity [hb.nonneg]) ?_
· apply h.congr
filter_upwards with q
exact graph_crossed_product_identity _ _ _ _ _ _ _ _ _
· filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
refine ⟨?_,?_,fun r => ⟨?_,?_,?_,?_⟩⟩
· simpa only [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _),one_mul] using
mul_le_mul (tensorCapND_le_one δ k₁ w) (hψ w hw) (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 1)
· simpa only [norm_mul,Complex.norm_conj,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _),one_mul] using
mul_le_mul (tensorCapND_le_one δ k₂ w) (hψ w hw) (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 1)
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw)
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw)
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw)
· exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw)
end CurrentFalconer.TreeND
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer.TreeND
open Set MeasureTheory PlanarFalconer Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {d : ℕ}
attribute [local instance] Classical.propDecidable
lemma graphCapScale_pos {N a p : ℕ} {c₀ sig : ℝ} (hc₀ : 0 < c₀) :
0 < graphCapScale N a p c₀ sig := lt_min hc₀ (by positivity)
lemma graphCapScale_eq (N a p : ℕ) (c₀ sig : ℝ) :
graphCapScale N a p c₀ sig = min c₀ (((2:ℝ)^((p:ℝ)-a))⁻¹*((2:ℝ)^N)^(10*sig)) := by
simp only [graphCapScale,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2)]
lemma graph_pair_partition_ae {N a p : ℕ} {c₀ sig : ℝ}
(hc₀ : 0 < c₀) (hcsmall : (d+1:ℝ)*c₀ ≤ 1/2) :
let δ := graphCapScale N a p c₀ sig
∀ᵐ z ∂(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))),
(∀ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ),
0 ≤ tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2) ∧
∑ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ),
tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2=1 := by
dsimp only
filter_upwards [Measure.quasiMeasurePreserving_fst.ae (canonicalSphereMeasureND_ae_norm (d:=d)),
Measure.quasiMeasurePreserving_snd.ae (canonicalSphereMeasureND_ae_norm (d:=d))] with z hz₁ hz₂
exact sphereCap_pair_partition (graphCapScale_pos hc₀)
((mul_le_mul_of_nonneg_left (min_le_left _ _) (by positivity)).trans hcsmall) z ⟨hz₁,hz₂⟩
theorem source_graph_cap_degrees {sig c₀ : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) :
∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (ρ₁ ρ₂ : Measure (EuclideanSpace ℝ (Fin d))) (A₁ A₂ : ℕ → ℝ)
(C : ℝ) (J i₁ i₂ a g₁ g₂ p : ℕ) (X Y : CellPath d) (A B : Finset (CellPath d))
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ),
a ≤ g₁ → a ≤ g₂ → g₁ ≤ p → g₂ ≤ p →
(g₁:ℝ) ≤ (a:ℝ)+2*sig*N → (g₂:ℝ) ≤ (a:ℝ)+2*sig*N →
0 < J → i₁ ≤ J → i₂ ≤ J →
ImposesParallelTest ρ₁ g₁ p (testWidth N J i₁ p sig)
(((2:ℝ)^N)^(height A₁ g₁ p+C*sig)) b →
ImposesParallelTest ρ₂ g₂ p (testWidth N J i₂ p sig)
(((2:ℝ)^N)^(height A₂ g₂ p+C*sig)) c →
let δ := graphCapScale N a p c₀ sig
let τ := graphTolerance N p sig
let D := ((2:ℝ)^N)^(min (height A₁ a p) (height A₂ a p)+(C+32*(d:ℝ)+1)*sig)
∀ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ),
let Sv := eligibleCells ρ₁ a p X (tensorCapND δ k.1) b
let Sw := eligibleCells ρ₁ a p X (tensorCapND δ k.2) b
let Tv := eligibleCells ρ₂ a p Y (tensorCapND δ k.1) c
let Tw := eligibleCells ρ₂ a p Y (tensorCapND δ k.2) c
let v := sphereCapCenterND δ k.1
let w := sphereCapCenterND δ k.2
(∀ row,
(((A.product B).filter (retainedHGraph Sv Tw v w τ row)).card : ℝ) ≤ D ∧
(((A.product B).filter (retainedHFGraph Sv Sw Tv Tw v w τ row)).card : ℝ) ≤ D) ∧
(∀ col,
(((A.product B).filter (fun row => retainedHGraph Sv Tw v w τ row col)).card : ℝ) ≤ D ∧
(((A.product B).filter (fun row => retainedHFGraph Sv Sw Tv Tw v w τ row col)).card : ℝ) ≤ D) := by
obtain ⟨N₀,hN₀⟩ := source_graph_degrees (d:=d) hsig hc₀.le (by positivity : (0:ℝ) ≤ 2*(d+1:ℝ))
refine ⟨N₀,?_⟩
intro N hN ρ₁ ρ₂ A₁ A₂ C J i₁ i₂ a g₁ g₂ p X Y A B b c hag₁ hag₂ hg₁ hg₂ hga₁ hga₂ hJ hi₁ hi₂ ht₁ ht₂ δ τ D k hk
have hk' := Finset.mem_product.mp hk
apply hN₀ N hN ρ₁ ρ₂ A₁ A₂ C J i₁ i₂ a g₁ g₂ p X Y A B
(tensorCapND δ k.1) (tensorCapND δ k.2) b c
(sphereCapCenterND δ k.1) (sphereCapCenterND δ k.2)
hag₁ hag₂ hg₁ hg₂ hga₁ hga₂ hJ hi₁ hi₂ (sphereCapCenterND_norm hk'.1) (sphereCapCenterND_norm hk'.2)
?_ ?_ ht₁ ht₂
· intro w hw hn
rw [dist_comm]
exact sphereCapCenterND_dist (graphCapScale_pos hc₀) hk'.1 hw (subset_tsupport _ hn)
· intro w hw hn
rw [dist_comm]
exact sphereCapCenterND_dist (graphCapScale_pos hc₀) hk'.2 hw (subset_tsupport _ hn)
lemma graph_profile_ratio {N : ℕ} (_hN : 0 < N) (m : ℝ → ℝ) (a p : ℕ) :
((2:ℝ)^N)^(-m p)/((2:ℝ)^N)^(-m a) =
((2:ℝ)^N)^(-((adjustedProfile (d:=d) m N p-adjustedProfile (d:=d) m N a)+
(d:ℝ)/2*((p:ℝ)-a)/(N:ℝ))) := by
rw [←Real.rpow_sub (by positivity : (0:ℝ) < 2^N)]
congr 1
unfold adjustedProfile
ring
lemma graph_profile_label_exponent (m₁ m₂ : ℝ → ℝ) (N a p : ℕ) (e : ℝ) :
(m₁ p-m₁ a)+(m₂ p-m₂ a)+4*e =
(adjustedProfile (d:=d) m₁ N p-adjustedProfile (d:=d) m₁ N a)+
(adjustedProfile (d:=d) m₂ N p-adjustedProfile (d:=d) m₂ N a)+
(d:ℝ)*((p:ℝ)-a)/(N:ℝ)+4*e := by
unfold adjustedProfile
ring
lemma graphChildren_card_crude (ρ : Measure (EuclideanSpace ℝ (Fin d)))
{N a p : ℕ} (hp : p ≤ N) (X : CellPath d) :
((graphChildren ρ a p X).card:ℝ) ≤ ((2:ℝ)^N)^d := by
have hsub : graphChildren ρ a p X ⊆ dyadicPaths p := by
intro P hP
exact (mem_dyadicPaths _ _).mpr (graphChildren_length hP)
have hc := (Finset.card_le_card hsub).trans (dyadicPaths_card_le p)
have hmono : (2^d)^p ≤ (2^d)^N := Nat.pow_le_pow_right (by positivity) hp
have h := (Nat.cast_le (α:=ℝ)).mpr (hc.trans hmono)
simpa only [Nat.cast_pow,Nat.cast_ofNat,pow_right_comm] using h
lemma graphCellLaw_mass_crude (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsProbabilityMeasure ρ]
{R m : ℝ} (hR : 1 ≤ R) (hm : m ≤ d) (P : Set (EuclideanSpace ℝ (Fin d))) :
(graphCellLaw ρ P (R^(-m))).real univ ≤ R^d := by
rw [graphCellLaw_mass _ _ (by positivity)]
have hinv : (R^(-m))⁻¹=R^m := by rw [Real.rpow_neg (by linarith : 0 ≤ R),inv_inv]
rw [hinv]
calc
_ ≤ R^m*1 := mul_le_mul_of_nonneg_left (measureReal_le_one) (by positivity)
_ ≤ R^d := by simpa only [mul_one,Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le hR hm
lemma graph_profile_ratio_crude {R ma mp : ℝ} (hR : 1 ≤ R) (ha : ma ≤ d) (hp : 0 ≤ mp) :
0 ≤ R^(-mp)/R^(-ma) ∧ R^(-mp)/R^(-ma) ≤ R^d := by
constructor
· positivity
rw [←Real.rpow_sub (by linarith : 0 < R)]
simpa only [Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le hR (show -mp- -ma ≤ (d:ℝ) by linarith)
end CurrentFalconer.TreeND
end
end
end OAI
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