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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.Ledger
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphCapScaleInvCrude
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 PlanarFalconer Attempt4FullFar Attempt4Near
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {d : ℕ}
theorem graph_HF_finite_jet_estimate {sig c₀ Ggeom : ℝ}
(hsig : 0 < sig) (hc₀ : 0 < c₀) (hcsmall : (d+1:ℝ)*c₀ ≤ 1/2) (hGgeom : 0 ≤ Ggeom) (T : ℝ) :
∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) (B₀ : ℕ) (s₁ s₂ : ℝ) (m₁ m₂ : ℝ → ℝ),
RegularizedProfileData ρ N B₀ s₁ sig m₁ → RegularizedProfileData κ N B₀ s₂ sig m₂ →
∀ (a p : ℕ) (v L Ctest K B Pcut : ℝ) (J i₁ i₂ g₁ g₂ : ℕ)
(X Y : CellPath d) (Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ)
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ),
a < p → (p:ℝ) ≤ v → v ≤ N →
(p:ℝ)-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 10*sig*N ≤ v-p →
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 →
X ∈ dyadicPaths a → Y ∈ dyadicPaths a → 0 < ρ (treeCell X) → 0 < κ (treeCell Y) →
‖treeCenter X-treeCenter Y‖ ≤ Ggeom*(2:ℝ)^(-(a:ℝ)) →
ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 → (∀ r, 0 ≤ Ψ r) →
(∫ u : ℝ, ‖(Ψ u:ℂ)‖) ≤ K → (∀ u, ‖iteratedDeriv m (fun u => (Ψ u:ℂ)) u‖ ≤ K) → 0 ≤ K →
let R := (2:ℝ)^N
let A₁ := adjustedProfile (d:=d) m₁ N
let A₂ := adjustedProfile (d:=d) m₂ N
let S := R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))
GraphSymbolJet b m S B → GraphSymbolJet c m S B →
0 ≤ Pcut → ContDiff ℝ ∞ ψ → (∀ w, ‖w‖=1 → ‖ψ w‖ ≤ 1) →
(∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ w‖ ≤ Pcut) →
ImposesParallelTest ρ g₁ p (testWidth N J i₁ p sig) (R^(height A₁ g₁ p+Ctest*sig)) b →
ImposesParallelTest κ g₂ p (testWidth N J i₂ p sig) (R^(height A₂ g₂ p+Ctest*sig)) c →
sourceAngularHFND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ ψ
(fun z => sourceAngularTransformND ρ (treeCell X) (R^(-m₁ a)) b z.1 z.2)
(fun z => sourceAngularTransformND κ (treeCell Y) (R^(-m₂ a)) c z.1 z.2) ≤
R^(-((A₁ p-A₁ a)+(A₂ p-A₂ a))+min (height A₁ a p) (height A₂ a p)+(Ctest+36*(d:ℝ)+5)*sig)*
graphSourceChildMax ρ κ a p v X Y (R^(-m₁ p)) (R^(-m₂ p)) Ψ b c + C*K*Pcut^2*B^4*R^(-T) := by
classical
obtain ⟨m,C,hC,N₁,hdec⟩ := graph_crossed_nonedge_decay (d:=d) hsig hc₀ hGgeom (T+15*(d:ℝ))
obtain ⟨N₂,hdeg⟩ := source_graph_cap_degrees (d:=d) hsig hc₀
refine ⟨m,((9*(c₀⁻¹+1))^d)^2*C,by positivity,max 1 (max N₁ N₂),?_⟩
intro N hN ρ κ B₀ s₁ s₂ m₁ m₂ hρ hκ a p v L Ctest K B Pcut J i₁ i₂ g₁ g₂ X Y Ψ ψ b c
hap hpv hvN hh hL hgap hag₁ hag₂ hg₁ hg₂ hga₁ hga₂ hJ hi₁ hi₂ hX hY hoccX hoccY hcenters
hΨ hsΨ hΨ0 hK hdΨ hK0 R A₁ A₂ S hb hc hPcut hψ hψ1 hdψ ht₁ ht₂
let := hρ.probability
let := hκ.probability
have hNpos : 0 < N := lt_of_lt_of_le Nat.zero_lt_one (le_trans (le_max_left _ _) hN)
have hNN₁ : N₁ ≤ N := le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) hN
have hNN₂ : N₂ ≤ N := le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) hN
have hpN : p ≤ N := by exact_mod_cast hpv.trans hvN
have haN := hap.le.trans hpN
have hR : 1 ≤ R := one_le_pow₀ (by norm_num)
have hRp : 0 < R := by positivity
have hc₀1 : c₀ ≤ 1 := by nlinarith [show (0:ℝ) ≤ d by positivity]
let A := graphChildren ρ a p X
let Z := graphChildren κ a p Y
let δ := graphCapScale N a p c₀ sig
let τ := graphTolerance N p sig
let Caps := (sphereCapGridND d δ).product (sphereCapGridND d δ)
let G := fun k : (Fin d → ℤ)×(Fin d → ℤ) =>
retainedHFGraph (eligibleCells ρ a p X (tensorCapND δ k.1) b)
(eligibleCells ρ a p X (tensorCapND δ k.2) b)
(eligibleCells κ a p Y (tensorCapND δ k.1) c)
(eligibleCells κ a p Y (tensorCapND δ k.2) c)
(sphereCapCenterND δ k.1) (sphereCapCenterND δ k.2) τ
let η := fun k : (Fin d → ℤ)×(Fin d → ℤ) => fun z : EuclideanSpace ℝ (Fin d)×EuclideanSpace ℝ (Fin d) =>
tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2
let U := fun (P : CellPath d) (z : ℝ × EuclideanSpace ℝ (Fin d)) => sourceAngularTransformND ρ (treeCell P) (R^(-m₁ p)) b z.1 z.2
let W := fun (Q : CellPath d) (z : ℝ × EuclideanSpace ℝ (Fin d)) => sourceAngularTransformND κ (treeCell Q) (R^(-m₂ p)) c z.1 z.2
let lam₁ := R^(-m₁ p)/R^(-m₁ a)
let lam₂ := R^(-m₂ p)/R^(-m₂ a)
let D := R^(min (height A₁ a p) (height A₂ a p)+(Ctest+32*(d:ℝ)+1)*sig)
let ε := C*K*Pcut^2*B^4*(R^d)^4*R^(-(T+15*(d:ℝ)))*(2:ℝ)^v
have hAB : (A.product Z).Nonempty := Finset.nonempty_product.mpr
⟨graphChildren_nonempty ρ hap.le hX hoccX,graphChildren_nonempty κ hap.le hY hoccY⟩
have hlabels : (A.card:ℝ)*Z.card ≤ R^((A₁ p-A₁ a)+(A₂ p-A₂ a)+(d:ℝ)*((p:ℝ)-a)/(N:ℝ)+4*(d+1:ℝ)*sig) := by
have h := graphChildren_pair_profile_card ρ κ hap.le hX hY hRp
(fun P hP => hρ.graph_child_lower hpN hP) (fun Q hQ => hκ.graph_child_lower hpN hQ)
(hρ.graph_parent_upper haN hX hoccX) (hκ.graph_parent_upper haN hY hoccY)
rw [graph_profile_label_exponent (d:=d) m₁ m₂ N a p ((d+1:ℝ)*sig)] at h
convert h using 1
congr 1
dsimp only [A₁,A₂]
ring
have hdegree := hdeg N hNN₂ ρ κ A₁ A₂ Ctest J i₁ i₂ a g₁ g₂ p X Y A Z b c
hag₁ hag₂ hg₁.le hg₂.le hga₁ hga₂ hJ hi₁ hi₂ ht₁ ht₂
have hiΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v)) :=
(hΨ.continuous.integrable_of_hasCompactSupport
(IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport Ψ) hsΨ)).comp_div (by positivity)
have hbounds₁ : ∀ P ∈ A, ∀ r, ∀ᵐ w ∂canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)),
‖U P (r,w)‖ ≤ |(R^(-m₁ p))⁻¹| * ρ.real (treeCell P)*B := by
intro P _ r
filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw)
have hbounds₂ : ∀ Q ∈ Z, ∀ r, ∀ᵐ w ∂canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)),
‖W Q (r,w)‖ ≤ |(R^(-m₂ p))⁻¹| * κ.real (treeCell Q)*B := by
intro Q _ r
filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw)
have hne : ∀ k ∈ Caps, ∀ row ∈ A.product Z, ∀ col ∈ A.product Z, ¬G k row col →
‖graphCrossedCellTerm ρ κ (R^(-m₁ p)) (R^(-m₂ p)) b c Ψ ψ v δ k.1 k.2 row col‖ ≤ ε := by
intro k hk row hr col hcol hn
have hk' := Finset.mem_product.mp hk
have hd := hdec N hNN₁ a p v L hh hL (sub_nonneg.mpr hpv) ρ κ X Y row col
(R^(-m₁ p)) (R^(-m₂ p)) (R^d) K B Pcut Ψ ψ b c k.1 k.2 hX hY hcenters hr hcol
(by positivity) (by positivity) (by positivity)
(graphCellLaw_mass_crude ρ hR (hρ.profile_le_dim hNpos hpN) _)
(graphCellLaw_mass_crude ρ hR (hρ.profile_le_dim hNpos hpN) _)
(graphCellLaw_mass_crude κ hR (hκ.profile_le_dim hNpos hpN) _)
(graphCellLaw_mass_crude κ hR (hκ.profile_le_dim hNpos hpN) _)
rw [←graphCapScale_eq] at hd
exact hd hk'.1 hk'.2 hΨ hsΨ hK hdΨ hK0 hPcut hψ hdψ hb hc hn
have hmain := graphFiniteHFND_to_child_max d a p v lam₁ lam₂ hNpos
(A₁ p-A₁ a) (A₂ p-A₂ a)
(min (height A₁ a p) (height A₂ a p)+(Ctest+32*(d:ℝ)+1)*sig) (4*(d+1:ℝ)*sig) 1
(graph_profile_ratio (d:=d) hNpos m₁ a p) (graph_profile_ratio (d:=d) hNpos m₂ a p)
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) A Z Caps hAB hlabels G η
(graph_pair_partition_ae hc₀ hcsmall) U W
(fun P _ => sourceAngularTransformND_measurable _ _ _ hb.measurable)
(fun Q _ => sourceAngularTransformND_measurable _ _ _ hc.measurable)
(fun P => |(R^(-m₁ p))⁻¹| * ρ.real (treeCell P)*B)
(fun Q => |(R^(-m₂ p))⁻¹| * κ.real (treeCell Q)*B) hbounds₁ hbounds₂ (D:=D) (by positivity)
(by simp only [one_mul]; exact le_rfl)
(fun k hk row _ => ((hdegree k hk).1 row).2)
(fun k hk col _ => ((hdegree k hk).2 col).2) hiΨ hΨ0 hψ.continuous.measurable (by
filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw
exact hψ1 w hw)
(fun k _ row _ col _ => by
simpa only [η,U,W,Complex.ofReal_mul] using
graph_crossed_cell_integrable ρ κ (R^(-m₁ p)) (R^(-m₂ p)) hb hc v hiΨ
ψ hψ.continuous.measurable (by norm_num : (0:ℝ) ≤ 1) hψ1 δ k.1 k.2 row col)
(show 0 ≤ ε by positivity) (by
intro k hk row hr col hc hn
simpa only [graphCrossedCellTerm,η,U,W,Complex.ofReal_mul] using hne k hk row hr col hc hn)
rw [←graph_children_HF_split ρ κ hap.le hX hY hb hc (R^(-m₁ a)) (R^(-m₂ a))
(by positivity : R^(-m₁ p) ≠ 0) (by positivity : R^(-m₂ p) ≠ 0) v Ψ ψ] at hmain
have hid : graphSourceChildMax ρ κ a p v X Y (R^(-m₁ p)) (R^(-m₂ p)) Ψ b c =
graphChildMaxND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) p v Ψ A Z hAB U W := by
unfold graphSourceChildMax
split_ifs with hx
· rfl
· exact (hx hAB).elim
rw [←hid,one_mul] at hmain
have hexp : -((A₁ p-A₁ a)+(A₂ p-A₂ a))+
(min (height A₁ a p) (height A₂ a p)+(Ctest+32*(d:ℝ)+1)*sig)+4*(d+1:ℝ)*sig =
-((A₁ p-A₁ a)+(A₂ p-A₂ a))+min (height A₁ a p) (height A₂ a p)+(Ctest+36*(d:ℝ)+5)*sig := by ring
rw [hexp] at hmain
apply hmain.trans
apply add_le_add le_rfl
have herr := graph_source_error_sum (T:=T) ρ κ hρ hκ hNpos haN hpN hvN
hc₀ hc₀1 hsig.le (by positivity : 0 ≤ C*K*Pcut^2*B^4) X Y
convert herr using 1
ring
end CurrentFalconer.TreeND
end
end
end OAI
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