| import Mathlib |
| 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 |
|
|
