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import OAI.MeasureTheory.FalconerCompletion.Estimates.EuclideanSourceComparableFarRapid
namespace OAI
/-! Recursive policies and quantitative depth estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4FullFar
open Set MeasureTheory Filter Real Attempt4FarGrid
open scoped ContDiff ComplexConjugate Topology ENNReal
attribute [local instance] Classical.propDecidable
def farOriginalSetND {d : ℕ} (a h : ℕ) (X Y : Fin d → ℤ) :
Set ((EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d)) ×
(EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d))) :=
((ambientGridCell a X ×ˢ ambientGridCell a X) ×ˢ
(ambientGridCell a Y ×ˢ ambientGridCell a Y)) ∩
{x | 1/(4*(2:ℝ)^h)≤ max (dist x.1.1 x.1.2) (dist x.2.1 x.2.2)}
lemma gridPairMeasureND_mono_parent {d a k : ℕ}
(ρ : Measure (EuclideanSpace ℝ (Fin d))) {X : Fin d → ℤ}
{p : (Fin d → ℤ) × (Fin d → ℤ)}
(hp : ambientGridCell k p.1⊆ambientGridCell a X ∧ ambientGridCell k p.2⊆ambientGridCell a X) :
gridPairMeasureND ρ k p≤gridPairMeasureND ρ a (X,X) :=
Measure.restrict_mono_set _ (Set.prod_mono hp.1 hp.2)
theorem euclidean_source_full_far_branchND {n : ℕ} {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ)
(hs : ∀ t, Ψ t≠0 → t∈Ioo (1/4:ℝ) 4) (C : ℝ) (hC : 0≤C)
{σ : ℝ} (hσ : 0<σ) (hσ₁ : σ≤1) (err T D : ℝ) (ell : ℕ)
(hK : 32*(n+2:ℝ)≤(2:ℝ)^ell) :
∃ J : ℕ, ∃ H : ℝ, 0≤H ∧ ∀ᶠ R : ℝ in atTop,
∀ (N : ℕ), 0<N → R=(2:ℝ)^N →
∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))), ∀ [IsFiniteMeasure ρ] [IsFiniteMeasure κ],
∀ (a L h : ℕ) (X Y : Fin (n+1) → ℤ)
(v ma : ℝ) (mb : ℕ → ℝ)
(b c : (EuclideanSpace ℝ (Fin (n+1)) × EuclideanSpace ℝ (Fin (n+1))) → EuclideanSpace ℝ (Fin (n+1)) → ℂ)
(U W : (EuclideanSpace ℝ (Fin (n+1)) × EuclideanSpace ℝ (Fin (n+1))) → Set (EuclideanSpace ℝ (Fin (n+1)))),
∀ S : Set ((EuclideanSpace ℝ (Fin (n+1)) × EuclideanSpace ℝ (Fin (n+1))) ×
(EuclideanSpace ℝ (Fin (n+1)) × EuclideanSpace ℝ (Fin (n+1)))),
MeasurableSet S → S⊆farOriginalSetND a h X Y → a≤h → v-a≤N →
((h+ell+2:ℕ):ℝ)≤v-2*(max L (Nat.ceil (10*σ*N)):ℕ)+D →
(∀ k∈Finset.Icc (a+1) (h+ell+2), ∀ i∈occupiedDescendantsND ρ a k X,
R^(-mb k-err)≤(ρ (ambientGridCell k i)).toReal) →
(∀ k∈Finset.Icc (a+1) (h+ell+2), ∀ j∈occupiedDescendantsND κ a k Y,
R^(-mb k-err)≤(κ (ambientGridCell k j)).toReal) →
(ρ (ambientGridCell a X)).toReal≤R^(-ma+err) →
(κ (ambientGridCell a Y)).toReal≤R^(-ma+err) →
Measurable (Function.uncurry b) → Measurable (Function.uncurry c) →
MeasurableSet {x | euclideanPairDirectionND x∈U x} →
MeasurableSet {x | euclideanPairDirectionND x∈W x} →
(∀ᵐ x ∂gridPairMeasureND ρ a (X,X),
ContDiff ℝ ∞ (b x) ∧ (∀ w∈U x, -w∈U x) ∧ tsupport (b x) ⊆ U x ∧
(∀ j≤2*(J+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (b x) w‖≤C*(shellEffectiveScaleND L R σ)^j)) →
(∀ᵐ x ∂gridPairMeasureND κ a (Y,Y),
ContDiff ℝ ∞ (c x) ∧ (∀ w∈W x, -w∈W x) ∧ tsupport (c x) ⊆ W x ∧
(∀ j≤2*(J+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (c x) w‖≤C*(shellEffectiveScaleND L R σ)^j)) →
((2:ℝ)^((n+1:ℝ)*(v-a))/(R^(-ma))^4)*
‖∫ x in S, euclideanFarKernelND Ψ v b c x ∂(ρ.prod ρ).prod (κ.prod κ)‖ ≤
H*(∑ k∈Finset.Icc (a+1) (h+ell+2),
R^(-2*((mb k-(n+1:ℝ)*k/(2*N))-(ma-(n+1:ℝ)*a/(2*N)))+4*err)*
(farChildMaximumND ρ κ k (occupiedFarPairsND ρ a k ((2:ℝ)^ell) X)
(occupiedFarPairsND κ a k ((2:ℝ)^ell) Y) v (R^(-mb k)) U W).toReal)+
R^(-T) := by
obtain ⟨Jc,H,hH,hcomp⟩ := euclidean_source_comparable_far_rapidND hΨ hs C hC hσ hσ₁ err (T+1) D ell hK
obtain ⟨Jn,hnon⟩ := euclidean_source_noncomparable_regionND hΨ hs C hC hσ hσ₁ err (T+1) D
refine ⟨max Jc Jn,H,hH,?_⟩
filter_upwards [hcomp,hnon,eventually_ge_atTop (2:ℝ)] with R hcomp hnon hR
intro N hN hRN ρ κ _ _ a L h X Y v ma mb b c U W S hS hSG hah hav hdepth
hl₁ hl₂ hu₁ hu₂ hmb hmc hU hW hb hc
let μ := (ρ.prod ρ).prod (κ.prod κ)
let P := (ambientGridCell a X ×ˢ ambientGridCell a X) ×ˢ (ambientGridCell a Y ×ˢ ambientGridCell a Y)
let Sc := S∩farComparableSetND a h X Y
let Sn := S\farComparableSetND a h X Y
let F := euclideanFarKernelND Ψ v b c
let A := (2:ℝ)^((n+1:ℝ)*(v-a))/(R^(-ma))^4
have hA : 0≤A := by dsimp [A]; positivity
have hSP : S⊆P := fun _ hx => (hSG hx).1
have hSn : MeasurableSet Sn := hS.diff (farComparableSetND_measurable a h X Y)
have hSc : MeasurableSet Sc := hS.inter (farComparableSetND_measurable a h X Y)
have hJc : 2*(Jc+1)+n+3≤2*(max Jc Jn+1)+n+3 := by have hh := le_max_left Jc Jn; omega
have hJn : 2*(Jn+1)+n+3≤2*(max Jc Jn+1)+n+3 := by have hh := le_max_right Jc Jn; omega
have hbs : ∀ᵐ z ∂μ.restrict S,
ContDiff ℝ ∞ (b z.1) ∧ (∀ w∈U z.1, -w∈U z.1) ∧ tsupport (b z.1) ⊆ U z.1 ∧
(∀ j≤2*(max Jc Jn+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (b z.1) w‖≤C*(shellEffectiveScaleND L R σ)^j) := by
apply ae_mono (Measure.restrict_mono_set μ hSP)
rw [show μ.restrict P=(gridPairMeasureND ρ a (X,X)).prod (gridPairMeasureND κ a (Y,Y)) from (Measure.prod_restrict _ _).symm]
exact Measure.quasiMeasurePreserving_fst.ae hb
have hcs : ∀ᵐ z ∂μ.restrict S,
ContDiff ℝ ∞ (c z.2) ∧ (∀ w∈W z.2, -w∈W z.2) ∧ tsupport (c z.2) ⊆ W z.2 ∧
(∀ j≤2*(max Jc Jn+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (c z.2) w‖≤C*(shellEffectiveScaleND L R σ)^j) := by
apply ae_mono (Measure.restrict_mono_set μ hSP)
rw [show μ.restrict P=(gridPairMeasureND ρ a (X,X)).prod (gridPairMeasureND κ a (Y,Y)) from (Measure.prod_restrict _ _).symm]
exact Measure.quasiMeasurePreserving_snd.ae hc
have hFi : IntegrableOn F S μ := by
apply euclideanFarKernelND_integrable hΨ.continuous hs v b c hmb hmc _ hC
filter_upwards [hbs,hcs] with z hz hz'
constructor
· intro w hw
simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hz.2.2.2 0 (by omega) w hw
· intro w hw
simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hz'.2.2.2 0 (by omega) w hw
have hbc (k : ℕ) (hk : k∈Finset.Icc (a+1) (h+ell+2))
(p) (hp : p∈occupiedFarPairsND ρ a k ((2:ℝ)^ell) X) :
∀ᵐ x ∂gridPairMeasureND ρ k p,
ContDiff ℝ ∞ (b x) ∧ (∀ w∈U x, -w∈U x) ∧ tsupport (b x) ⊆ U x ∧
(∀ j≤2*(Jc+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (b x) w‖≤C*(shellEffectiveScaleND L R σ)^j) := by
have hak : a≤k := by have hh := (Finset.mem_Icc.mp hk).1; omega
filter_upwards [ae_mono (gridPairMeasureND_mono_parent ρ (occupiedFarPairsND_geometry hak ρ hp).1) hb] with x hx
exact ⟨hx.1,hx.2.1,hx.2.2.1,fun j hj => hx.2.2.2 j (hj.trans hJc)⟩
have hcc (k : ℕ) (hk : k∈Finset.Icc (a+1) (h+ell+2))
(q) (hq : q∈occupiedFarPairsND κ a k ((2:ℝ)^ell) Y) :
∀ᵐ x ∂gridPairMeasureND κ k q,
ContDiff ℝ ∞ (c x) ∧ (∀ w∈W x, -w∈W x) ∧ tsupport (c x) ⊆ W x ∧
(∀ j≤2*(Jc+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (c x) w‖≤C*(shellEffectiveScaleND L R σ)^j) := by
have hak : a≤k := by have hh := (Finset.mem_Icc.mp hk).1; omega
filter_upwards [ae_mono (gridPairMeasureND_mono_parent κ (occupiedFarPairsND_geometry hak κ hq).1) hc] with x hx
exact ⟨hx.1,hx.2.1,hx.2.2.1,fun j hj => hx.2.2.2 j (hj.trans hJc)⟩
have hcb := hcomp N hN hRN ρ κ a L h X Y v ma mb b c U W Sc hSc Set.inter_subset_right
hav hdepth hl₁ hl₂ hu₁ hu₂ hmb hmc hU hW hbc hcc
have hNP : P∩Sn=Sn := inter_eq_right.mpr (Set.sdiff_subset.trans hSP)
have hND : ∀ᵐ z ∂μ.restrict Sn,
(2:ℝ)^(-(h:ℝ))/4≤ max (dist z.1.1 z.1.2) (dist z.2.1 z.2.2) ∧
2*min (dist z.1.1 z.1.2) (dist z.2.1 z.2.2)≤ max (dist z.1.1 z.1.2) (dist z.2.1 z.2.2) ∧
ContDiff ℝ ∞ (b z.1) ∧ ContDiff ℝ ∞ (c z.2) ∧
(∀ j≤2*(Jn+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (b z.1) w‖≤C*(shellEffectiveScaleND L R σ)^j) ∧
(∀ j≤2*(Jn+1)+n+3, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (c z.2) w‖≤C*(shellEffectiveScaleND L R σ)^j) := by
filter_upwards [ae_restrict_mem (μ := μ) hSn,
ae_mono (Measure.restrict_mono_set μ (show Sn⊆S from Set.sdiff_subset)) hbs,
ae_mono (Measure.restrict_mono_set μ (show Sn⊆S from Set.sdiff_subset)) hcs] with z hz hzb hzc
have hp := hSG hz.1
have hne : ¬ max (dist z.1.1 z.1.2) (dist z.2.1 z.2.2)≤2*min (dist z.1.1 z.1.2) (dist z.2.1 z.2.2) :=
fun hcomp' => hz.2 ⟨hp.1,hp.2,hcomp'⟩
have hlo : (2:ℝ)^(-(h:ℝ))/4=1/(4*(2:ℝ)^h) := by
rw [Real.rpow_neg (by norm_num : (0:ℝ)≤2),Real.rpow_natCast]
ring
exact ⟨by simpa only [hlo,Set.mem_ofPred_eq] using hp.2,(lt_of_not_ge hne).le,hzb.1,hzc.1,
fun j hj => hzb.2.2.2 j (hj.trans hJn),fun j hj => hzc.2.2.2 j (hj.trans hJn)⟩
have hhh : (h:ℝ)≤((h+ell+2:ℕ):ℝ) := by exact_mod_cast (by omega : h≤h+ell+2)
have hnb := hnon N hN hRN ρ κ (ambientGridCell a X) (ambientGridCell a Y) Sn a v ma L h b c
hav (by exact_mod_cast hah) (hhh.trans hdepth) hu₁ hu₂ hmb hmc (by
change ∀ᵐ z ∂μ.restrict (P∩Sn), _
rw [hNP]
exact hND)
change Integrable F (μ.restrict (P∩Sn)) ∧ A*‖∫ z, F z ∂μ.restrict (P∩Sn)‖≤R^(-(T+1)) at hnb
rw [hNP] at hnb
have heq : (∫ z in Sc, F z ∂μ)+(∫ z in Sn, F z ∂μ)=(∫ z in S, F z ∂μ) :=
integral_inter_add_sdiff (farComparableSetND_measurable a h X Y) hFi
have hRpos : 0<R := lt_of_lt_of_le (by norm_num : (0:ℝ)<2) hR
have herr : 2*R^(-(T+1))≤R^(-T) := by
calc
_ ≤ R*R^(-(T+1)) := mul_le_mul_of_nonneg_right hR (by positivity)
_ = R^(1+(-(T+1))) := by rw [Real.rpow_add hRpos,Real.rpow_one]
_ = _ := by congr 1; ring
calc
A*‖∫ z in S, F z ∂μ‖ = A*‖(∫ z in Sc, F z ∂μ)+(∫ z in Sn, F z ∂μ)‖ := by rw [heq]
_ ≤ A*‖∫ z in Sc, F z ∂μ‖+A*‖∫ z in Sn, F z ∂μ‖ := by
simpa only [mul_add] using mul_le_mul_of_nonneg_left (norm_add_le _ _) hA
_ ≤ _ := by
have hh := add_le_add hcb hnb.2
linarith
end Attempt4FullFar
end GraphLiteralSupport
end
end
end OAI
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