openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /WeightedMaskTransform.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.OutgoingWeightedKernelNorm | |
| import OAI.MeasureTheory.FalconerCompletion.Shells.ContinuousCoefficientMeasurableSquare | |
| import OAI.MeasureTheory.FalconerCompletion.Shells.ShellPolynomialSubpower | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate Topology ContDiff | |
| variable {n : ℕ} {ι κ : Type*} [DecidableEq ι] [DecidableEq κ] | |
| local notation "V" => EuclideanSpace ℝ (Fin (n+1)) | |
| local notation "Assignment" => ℂ × ((ι ⊕ κ) → List (SphereAmbientND n →L[ℝ] SphereAmbientND n)) | |
| noncomputable def weightedMaskTransformND (S : Finset ι) (T : Finset κ) | |
| (f : V → ι → SmoothFunctionND V) (g : V → κ → SmoothFunctionND V) | |
| (F G : V → ℂ) (ρ ν : Measure V) (X Y : Set V) (M N a r : ℝ) : ℂ := | |
| ((M*N)⁻¹:ℝ) • (∫ z, (F z.1*conj (G z.2))* | |
| outgoingWeightedKernelND r ((2:ℝ)^a) (pairMaskProductND S T f g) z | |
| ∂(ρ.restrict X).prod (ν.restrict Y)) | |
| theorem weighted_all_mask_shellND (ψ : V → ℂ) (hψ : ContDiff ℝ ∞ ψ) | |
| (η T₀ C₀ : ℝ) (hη : 0 < η) (hC₀ : 0 ≤ C₀) : | |
| ∃ J : ℕ, ∃ Error : ℝ, 0 ≤ Error ∧ ∀ | |
| (S : Finset ι) (T : Finset κ) | |
| (f : V → ι → SmoothFunctionND V) (g : V → κ → SmoothFunctionND V) | |
| (F G : V → ℂ) (D : ι → ℝ) (E : κ → ℝ), | |
| (∀ i ∈ S, 0 < D i) → (∀ k ∈ T, 0 < E k) → ∀ | |
| (δ K : ℝ) (hδ : 0 < δ) (hK : 0 < K) | |
| (ρ ν : Measure V) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] | |
| (X Y : Set V), MeasurableSet X → MeasurableSet Y → ∀ | |
| (M N R A Q a v : ℝ), 1 ≤ R → 1 ≤ A → A ≤ R^2 → 1 ≤ Q → | |
| (2:ℝ)^(-(v-a))*Q^2 ≤ 1 → | |
| (∀ i ∈ S, D i ≤ Q) → (∀ k ∈ T, E k ≤ Q) → | |
| (∀ x ∈ X, ∀ y ∈ Y, δ ≤ dist x y*(2:ℝ)^a ∧ dist x y*(2:ℝ)^a ≤ K) → | |
| (∀ r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v), ∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), | |
| R^η ≤ r*dist z.1 z.2 ∧ A^2/(r*dist z.1 z.2) ≤ R^(-η) ∧ | |
| ψ =ᶠ[𝓝 ((dist z.1 z.2)⁻¹ • (z.1-z.2))] (fun _ => 1) ∧ | |
| ψ =ᶠ[𝓝 (-((dist z.1 z.2)⁻¹ • (z.1-z.2)))] (fun _ => 0) ∧ | |
| (∀ k ≤ 2*(J+1)+n+3, ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ k (pairMaskProductND S T f g z).1 w‖ ≤ C₀*A^k)) → | |
| (∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), ‖F z.1*conj (G z.2)‖ ≤ 1) → | |
| Measurable (fun z => (F z.1*conj (G z.2))* | |
| (pairMaskProductND S T f g z).1 ((dist z.1 z.2)⁻¹ • (z.1-z.2))) → | |
| ∀ (P : ℝ), (∀ w, ‖ψ w‖ ≤ P) → ∀ (B C : ℕ → Assignment → ℝ), | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), | |
| Measurable (Function.uncurry (inverseLeftEndpointND S D f F p))) → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), | |
| Measurable (Function.uncurry (inverseRightEndpointND T E g G p))) → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), ∀ x w, | |
| ‖inverseLeftEndpointND S D f F p x w‖ ≤ B j p) → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), ∀ y w, | |
| ‖inverseRightEndpointND T E g G p y w‖ ≤ C j p) → | |
| ∀ (Ψ : ℝ → ℝ), Integrable (fun r => Ψ (r/(2:ℝ)^v)) → | |
| (∀ r, 0 ≤ Ψ r) → (∀ r ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ r) → ∀ (BHF : ℝ), 0 ≤ BHF → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), ∀ ξ, | |
| sourceAngularHFND (n+1) (canonicalSphereMeasureND V) a v Ψ ψ | |
| (fun z => spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ρ X M | |
| (inverseLeftEndpointND S D f F p) (ξ,z)) | |
| (fun z => spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ν Y N | |
| (inverseRightEndpointND T E g G p) (ξ,z)) ≤ BHF) → | |
| Integrable (fun r => ‖weightedMaskTransformND S T f g F G ρ ν X Y M N a r‖^2) | |
| (volume.restrict (Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v))) ∧ | |
| (2:ℝ)^(-a)*(∫ r in Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v), | |
| ‖weightedMaskTransformND S T f g F G ρ ν X Y M N a r‖^2) ≤ | |
| 2*(fullInverseCoefficientCostND n J δ K hδ hK* | |
| (max 1 ((S.card+T.card:ℕ):ℝ))^(2*J))^2*BHF + | |
| 3*(2:ℝ)^(v-a)* | |
| (|(M*N)⁻¹| * ((δ^(-(n:ℝ)/2)*Error*R^(-T₀))* | |
| ((ρ.restrict X).prod (ν.restrict Y)).real univ))^2 := by | |
| obtain ⟨J,Error,hError,hInv⟩ := all_mask_weighted_inverseND (ι := ι) (κ := κ) ψ hψ η T₀ C₀ hη hC₀ | |
| refine ⟨J,Error,hError,?_⟩ | |
| intro S T f g F G D E hD hE δ K hδ hK ρ ν hρ hν X Y hX hY | |
| M N R A Q a v hR hA hup hQ hpaid hDQ hEQ hsep hgeom hFG hsym | |
| P hP B C hb hc hB hC Ψ hΨ hΨpos hone BHF hBHF hHF | |
| let μ := (ρ.restrict X).prod (ν.restrict Y) | |
| let L := inverseTermListND (n := n) J S T | |
| let cc : (ℕ × Assignment) → V → ℂ := fun p => | |
| paidSpatialCoefficientND p.1 S T D E ((2:ℝ)^(-(v-a))) δ K hδ hK p.2 | |
| let mm : (ℕ × Assignment) → ℝ → ℂ := fun p => shellRadialMultiplierND n p.1 v | |
| let U : (ℕ × Assignment) → V × (ℝ × V) → ℂ := fun p => | |
| spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ρ X M (inverseLeftEndpointND S D f F p.2) | |
| let W : (ℕ × Assignment) → V × (ℝ × V) → ℂ := fun p => | |
| spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ν Y N (inverseRightEndpointND T E g G p.2) | |
| let AA : (ℕ × Assignment) → ℝ := fun p => (2:ℝ)^((n:ℝ)/2)*(2:ℝ)^p.1 | |
| let BB : (ℕ × Assignment) → ℝ := fun p => |M⁻¹| * (|B p.1 p.2| *(ρ.restrict X).real univ) | |
| let CC : (ℕ × Assignment) → ℝ := fun p => |N⁻¹| * (|C p.1 p.2| *(ν.restrict Y).real univ) | |
| let H := weightedMaskTransformND S T f g F G ρ ν X Y M N a | |
| have hs : 0 < (2:ℝ)^a := Real.rpow_pos_of_pos (by norm_num) _ | |
| have hxy : ∀ᵐ z ∂μ, z.1 ∈ X ∧ z.2 ∈ Y := | |
| (Measure.ae_prod_iff_ae_ae (hX.prod hY)).mpr (by | |
| filter_upwards [ae_restrict_mem hX] with x hx | |
| filter_upwards [ae_restrict_mem hY] with y hy | |
| exact ⟨hx,hy⟩) | |
| have hδs : ∀ᵐ z ∂μ, δ ≤ dist z.1 z.2*(2:ℝ)^a := by | |
| filter_upwards [hxy] with z hz | |
| exact (hsep z.1 hz.1 z.2 hz.2).1 | |
| have hraw (r : ℝ) (hr : r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v)) : | |
| Integrable (fun z => (F z.1*conj (G z.2))* | |
| outgoingWeightedKernelND r ((2:ℝ)^a) (pairMaskProductND S T f g) z) μ := by | |
| apply outgoingWeightedKernelND_integrable ((2:ℝ)^a) hs _ _ μ r hδ hδs hsym | |
| filter_upwards [hδs,hgeom r hr,hFG] with z hz hg hfg | |
| have hd : 0 < dist z.1 z.2 := (mul_pos_iff_of_pos_right hs).mp (hδ.trans_le hz) | |
| have he : ‖(dist z.1 z.2)⁻¹ • (z.1-z.2)‖=1 := by | |
| rw [norm_smul,Real.norm_eq_abs,abs_inv,abs_of_pos hd,← dist_eq_norm,inv_mul_cancel₀ hd.ne'] | |
| have hb₀ := hg.2.2.2.2 0 (Nat.zero_le _) _ he | |
| simp only [norm_iteratedFDeriv_zero,pow_zero,mul_one] at hb₀ | |
| rw [norm_mul] | |
| exact (mul_le_mul hfg hb₀ (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 1)).trans_eq (one_mul _) | |
| have hH : Measurable H := weighted_pair_integral_measurableND ((2:ℝ)^a) | |
| (pairMaskProductND S T f g) (fun z => F z.1*conj (G z.2)) μ ((M*N)⁻¹) hsym | |
| have happrox (r : ℝ) (hr : r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v)) : | |
| ‖H r-(((2:ℝ)^((n:ℝ)*(v-a)/2):ℝ):ℂ)* | |
| (L.map (fun p => mm p r*∫ ξ, cc p ξ*sourceAngularCorrelationND | |
| (canonicalSphereMeasureND V) ψ (fun z => U p (ξ,z)) (fun z => W p (ξ,z)) r)).sum‖ ≤ | |
| |(M*N)⁻¹| * ((δ^(-(n:ℝ)/2)*Error*R^(-T₀))*μ.real univ) := by | |
| have hrp : 0 < r := (div_pos (Real.rpow_pos_of_pos (by norm_num) _) (by norm_num)).trans_le hr.1 | |
| have hh := hInv S T f g F G D E (fun i hi => (hD i hi).ne') (fun k hk => (hE k hk).ne') | |
| δ K hδ hK ρ ν X Y hX hY M N R A r ((2:ℝ)^a) hrp hs hR hA hup hsep | |
| (hgeom r hr) hFG (hraw r hr) P hP B C hb hc hB hC | |
| rw [all_mask_shell_factorizationND J S T f g F G D E δ K hδ hK ρ ν X Y M N a v r hrp ψ] at hh | |
| exact hh | |
| have hmemb (p : ℕ × Assignment) (hp : p ∈ L) := (mem_inverseTermListND J S T p).mp hp | |
| have hU (p : ℕ × Assignment) (hp : p ∈ L) : Measurable (U p) := | |
| spatialEndpointTransformND_measurable _ _ _ _ (hb p.1 (hmemb p hp).1 p.2 (hmemb p hp).2) | |
| have hW (p : ℕ × Assignment) (hp : p ∈ L) : Measurable (W p) := | |
| spatialEndpointTransformND_measurable _ _ _ _ (hc p.1 (hmemb p hp).1 p.2 (hmemb p hp).2) | |
| have hbU (p : ℕ × Assignment) (hp : p ∈ L) (ξ : V) (r : ℝ) (w : V) : | |
| ‖U p (ξ,(r,w))‖ ≤ BB p := | |
| spatialEndpointTransformND_bound _ _ _ _ _ | |
| (fun x w => (hB p.1 (hmemb p hp).1 p.2 (hmemb p hp).2 x w).trans (le_abs_self _)) _ | |
| have hbW (p : ℕ × Assignment) (hp : p ∈ L) (ξ : V) (r : ℝ) (w : V) : | |
| ‖W p (ξ,(r,w))‖ ≤ CC p := | |
| spatialEndpointTransformND_bound _ _ _ _ _ | |
| (fun x w => (hC p.1 (hmemb p hp).1 p.2 (hmemb p hp).2 x w).trans (le_abs_self _)) _ | |
| have hh := full_HF_shell_mixtureND n volume (canonicalSphereMeasureND V) a v L cc mm ψ U W H AA BB CC | |
| (fun _ _ => by dsimp [AA]; positivity) | |
| (fun p _ => paidSpatialCoefficientND_measurable p.1 S T D E _ δ K hδ hK p.2) | |
| (fun p _ => paidSpatialCoefficientND_integrable p.1 S T D E _ δ K hδ hK p.2) | |
| (fun p _ => shellRadialMultiplierND_measurable n p.1 v) | |
| (fun p _ _ hr => shellRadialMultiplierND_bound n p.1 hr) | |
| hψ.continuous.measurable hU hW (abs_nonneg P) | |
| (fun _ _ => by dsimp [BB]; positivity) (fun w => (hP w).trans (le_abs_self P)) hbU hbW | |
| hΨ hΨpos hone (fun p hp ξ => hHF p.1 (hmemb p hp).1 p.2 (hmemb p hp).2 ξ) | |
| hH.aestronglyMeasurable happrox | |
| refine ⟨hh.1,hh.2.trans ?_⟩ | |
| apply add_le_add _ le_rfl | |
| apply mul_le_mul_of_nonneg_right _ hBHF | |
| apply mul_le_mul_of_nonneg_left _ (by norm_num : (0:ℝ) ≤ 2) | |
| apply pow_le_pow_left₀ | |
| · apply List.sum_nonneg | |
| intro x hx | |
| obtain ⟨p,_,rfl⟩ := List.mem_map.mp hx | |
| exact mul_nonneg (by dsimp [AA]; positivity) (integral_nonneg (fun _ => norm_nonneg _)) | |
| · exact inverseTermListND_paid_cost J S T D E hQ (by positivity) hpaid | |
| (fun i hi => (hD i hi).le) (fun k hk => (hE k hk).le) hDQ hEQ δ K hδ hK | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open Filter Real | |
| open scoped Topology | |
| theorem shell_polynomial_subpowerND (k : ℕ) (C ε : ℝ) (hε : 0<ε) : | |
| ∀ᶠ N : ℕ in atTop, C*(N:ℝ)^k ≤ ((2:ℝ)^N)^ε := by | |
| have hr : 1<(2:ℝ)^ε := Real.one_lt_rpow (by norm_num) hε | |
| have ht := (tendsto_pow_const_div_const_pow_of_one_lt k hr).const_mul C | |
| have hh : ∀ᶠ N : ℕ in atTop, C*((N:ℝ)^k/((2:ℝ)^ε)^N) ≤ 1 := by | |
| have hc := ht.eventually_lt_const (show C*0 < (1:ℝ) by simp) | |
| exact hc.mono (fun _ h => h.le) | |
| filter_upwards [hh] with N hN | |
| have he : ((2:ℝ)^ε)^N=((2:ℝ)^N)^ε := by | |
| rw [← Real.rpow_natCast ((2:ℝ)^ε) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2), | |
| ← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2),mul_comm ε] | |
| rw [← mul_div_assoc] at hN | |
| exact ((div_le_one (by positivity)).mp hN).trans_eq he | |
| theorem shell_mask_cost_subpowerND (J k : ℕ) (D C ε : ℝ) (hD : 0≤D) | |
| (hε : 0<ε) : | |
| ∀ᶠ N : ℕ in atTop, ∀ P : ℝ, 0≤P → P≤C*(N:ℝ)^k → | |
| 2*(D*P^(2*J))^2 ≤ ((2:ℝ)^N)^ε := by | |
| filter_upwards [shell_polynomial_subpowerND (k*(4*J)) (2*D^2*C^(4*J)) ε hε] with N hN | |
| intro P hP hPC | |
| calc | |
| _ ≤ 2*(D*(C*(N:ℝ)^k)^(2*J))^2 := by gcongr | |
| _ = (2*D^2*C^(4*J))*(N:ℝ)^(k*(4*J)) := by | |
| simp only [mul_pow,← pow_mul] | |
| have he : 2*J*2=4*J := by omega | |
| simp only [Nat.mul_assoc,he] | |
| ring | |
| _ ≤ _ := hN | |
| lemma shell_dyadic_tendstoND : Tendsto (fun N : ℕ => (2:ℝ)^N) atTop atTop := | |
| tendsto_pow_atTop_atTop_of_one_lt (by norm_num) | |
| lemma shell_eventually_const_mul_rpowND {a b C : ℝ} (hab : a < b) : | |
| ∀ᶠ R : ℝ in atTop, C*R^a ≤ R^b := by | |
| have ht : Tendsto (fun R : ℝ => C*R^(a-b)) atTop (𝓝 0) := by | |
| have hh := (tendsto_rpow_neg_atTop (sub_pos.mpr hab)).const_mul C | |
| simpa only [neg_sub,mul_zero] using hh | |
| filter_upwards [ht.eventually_lt_const (by norm_num : (0:ℝ)<1), | |
| eventually_gt_atTop (0:ℝ)] with R hR hRp | |
| have he : C*R^a = (C*R^(a-b))*R^b := by | |
| rw [mul_assoc,← Real.rpow_add hRp] | |
| congr 2 | |
| ring | |
| rw [he] | |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right hR.le (Real.rpow_nonneg hRp.le _) | |
| theorem shell_mass_error_decayND (B T C : ℝ) (hC : 0≤C) : | |
| ∀ᶠ R : ℝ in atTop, ∀ (q H : ℝ), 0≤H → H≤R^B → (2:ℝ)^q ≤ R^2 → | |
| 3*(2:ℝ)^q*(C*R^(-(T/2+B+2))*H)^2 ≤ R^(-T) := by | |
| filter_upwards [shell_eventually_const_mul_rpowND (C := 3*C^2) | |
| (show -T-2 < -T by linarith),eventually_gt_atTop (0:ℝ)] with R hR hRp | |
| intro q H hH hHB hq | |
| calc | |
| _ ≤ 3*R^2*(C*R^(-(T/2+B+2))*R^B)^2 := by gcongr | |
| _ = (3*C^2)*R^(-T-2) := by | |
| rw [mul_pow,mul_pow,← Real.rpow_natCast (R^(-(T/2+B+2))) 2, | |
| ← Real.rpow_mul hRp.le,← Real.rpow_natCast (R^B) 2,← Real.rpow_mul hRp.le, | |
| ← Real.rpow_natCast R 2] | |
| calc | |
| _ = (3*C^2)*(R^(2:ℝ)*(R^(-(T/2+B+2)*(2:ℝ))*R^(B*(2:ℝ)))) := by ring_nf | |
| _ = _ := by rw [← Real.rpow_add hRp,← Real.rpow_add hRp]; congr 2; ring | |
| _ ≤ _ := hR | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate Topology ContDiff | |
| variable {n : ℕ} {ι κ : Type*} [DecidableEq ι] [DecidableEq κ] | |
| local notation "V" => EuclideanSpace ℝ (Fin (n+1)) | |
| local notation "Assignment" => ℂ × ((ι ⊕ κ) → List (SphereAmbientND n →L[ℝ] SphereAmbientND n)) | |
| theorem weighted_all_mask_shell_asymptoticND (ψ : V → ℂ) (hψ : ContDiff ℝ ∞ ψ) | |
| (kCard : ℕ) (η C₀ ε T₀ Bmass Cmask δ K : ℝ) (hη : 0 < η) (hC₀ : 0 ≤ C₀) | |
| (hε : 0<ε) (hδ : 0<δ) (hK : 0<K) : | |
| ∃ J : ℕ, ∀ᶠ kN : ℕ in Filter.atTop, let R := (2:ℝ)^kN; ∀ | |
| (S : Finset ι) (T : Finset κ) | |
| (f : V → ι → SmoothFunctionND V) (g : V → κ → SmoothFunctionND V) | |
| (F G : V → ℂ) (D : ι → ℝ) (E : κ → ℝ), | |
| (∀ i ∈ S, 0 < D i) → (∀ k ∈ T, 0 < E k) → | |
| max 1 ((S.card+T.card:ℕ):ℝ) ≤ Cmask*(kN:ℝ)^kCard → ∀ (ρ ν : Measure V) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] | |
| (X Y : Set V), MeasurableSet X → MeasurableSet Y → ∀ | |
| (M N A Q a v : ℝ), 1 ≤ A → A ≤ R^2 → 1 ≤ Q → | |
| (2:ℝ)^(v-a) ≤ R^2 → | |
| |(M*N)⁻¹| * ((ρ.restrict X).prod (ν.restrict Y)).real univ ≤ R^Bmass → | |
| (2:ℝ)^(-(v-a))*Q^2 ≤ 1 → | |
| (∀ i ∈ S, D i ≤ Q) → (∀ k ∈ T, E k ≤ Q) → | |
| (∀ x ∈ X, ∀ y ∈ Y, δ ≤ dist x y*(2:ℝ)^a ∧ dist x y*(2:ℝ)^a ≤ K) → | |
| (∀ r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v), ∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), | |
| R^η ≤ r*dist z.1 z.2 ∧ A^2/(r*dist z.1 z.2) ≤ R^(-η) ∧ | |
| ψ =ᶠ[𝓝 ((dist z.1 z.2)⁻¹ • (z.1-z.2))] (fun _ => 1) ∧ | |
| ψ =ᶠ[𝓝 (-((dist z.1 z.2)⁻¹ • (z.1-z.2)))] (fun _ => 0) ∧ | |
| (∀ k ≤ 2*(J+1)+n+3, ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ k (pairMaskProductND S T f g z).1 w‖ ≤ C₀*A^k)) → | |
| (∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), ‖F z.1*conj (G z.2)‖ ≤ 1) → | |
| Measurable (fun z => (F z.1*conj (G z.2))* | |
| (pairMaskProductND S T f g z).1 ((dist z.1 z.2)⁻¹ • (z.1-z.2))) → | |
| ∀ (P : ℝ), (∀ w, ‖ψ w‖ ≤ P) → ∀ (B C : ℕ → Assignment → ℝ), | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), | |
| Measurable (Function.uncurry (inverseLeftEndpointND S D f F p))) → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), | |
| Measurable (Function.uncurry (inverseRightEndpointND T E g G p))) → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), ∀ x w, | |
| ‖inverseLeftEndpointND S D f F p x w‖ ≤ B j p) → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), ∀ y w, | |
| ‖inverseRightEndpointND T E g G p y w‖ ≤ C j p) → | |
| ∀ (Ψ : ℝ → ℝ), Integrable (fun r => Ψ (r/(2:ℝ)^v)) → | |
| (∀ r, 0 ≤ Ψ r) → (∀ r ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ r) → ∀ (BHF : ℝ), 0 ≤ BHF → | |
| (∀ j ≤ J, ∀ p ∈ inverseAngularTermsND j (S.disjSum T), ∀ ξ, | |
| sourceAngularHFND (n+1) (canonicalSphereMeasureND V) a v Ψ ψ | |
| (fun z => spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ρ X M | |
| (inverseLeftEndpointND S D f F p) (ξ,z)) | |
| (fun z => spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ν Y N | |
| (inverseRightEndpointND T E g G p) (ξ,z)) ≤ BHF) → | |
| Integrable (fun r => ‖weightedMaskTransformND S T f g F G ρ ν X Y M N a r‖^2) | |
| (volume.restrict (Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v))) ∧ | |
| (2:ℝ)^(-a)*(∫ r in Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v), | |
| ‖weightedMaskTransformND S T f g F G ρ ν X Y M N a r‖^2) ≤ | |
| R^ε*BHF+R^(-T₀) := by | |
| obtain ⟨J,Error,hError,hMain⟩ := weighted_all_mask_shellND (ι := ι) (κ := κ) | |
| ψ hψ η (T₀/2+Bmass+2) C₀ hη hC₀ | |
| refine ⟨J,?_⟩ | |
| filter_upwards [shell_mask_cost_subpowerND J kCard (fullInverseCoefficientCostND n J δ K hδ hK) | |
| Cmask ε (fullInverseCoefficientCostND_nonneg n J δ K hδ hK) hε, | |
| shell_dyadic_tendstoND.eventually (shell_mass_error_decayND Bmass T₀ | |
| (δ^(-(n:ℝ)/2)*Error) (mul_nonneg (Real.rpow_nonneg hδ.le _) hError))] | |
| with kN hcost herror | |
| dsimp only | |
| intro S T f g F G D E hD hE hcard ρ ν hρ hν X Y hX hY M N A Q a v | |
| hA hup hQ hq hmass hpaid hDQ hEQ hsep hgeom hFG hsym P hP B C hb hc hB hC | |
| Ψ hΨ hΨpos hone BHF hBHF hHF | |
| have hR : 1 ≤ (2:ℝ)^kN := one_le_pow₀ (by norm_num) | |
| obtain ⟨hint,hbound⟩ := hMain S T f g F G D E hD hE δ K hδ hK ρ ν X Y hX hY | |
| M N ((2:ℝ)^kN) A Q a v hR hA hup hQ hpaid hDQ hEQ hsep hgeom hFG hsym | |
| P hP B C hb hc hB hC Ψ hΨ hΨpos hone BHF hBHF hHF | |
| refine ⟨hint,hbound.trans (add_le_add ?_ ?_)⟩ | |
| · exact mul_le_mul_of_nonneg_right (hcost _ (le_trans zero_le_one (le_max_left _ _)) hcard) hBHF | |
| · simpa only [mul_assoc,mul_left_comm,mul_comm] using herror (v-a) (|(M*N)⁻¹| * ((ρ.restrict X).prod (ν.restrict Y)).real univ) | |
| (mul_nonneg (abs_nonneg _) (measureReal_nonneg)) hmass hq | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |