openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /ShellEffectiveScale2.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.AllMaskProductAdmissible2 | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.MaskScale2 | |
| import OAI.MeasureTheory.FalconerCompletion.Shells.ShellEffectiveScale | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open Filter Real | |
| open scoped Topology | |
| noncomputable def shellEffectiveScaleND (l : ℕ) (R σ : ℝ) : ℝ := | |
| R^σ*sourceMaskScaleND l R σ | |
| lemma dyadic_half_sqND (q : ℝ) : ((2:ℝ)^(q/2))^2=(2:ℝ)^q := by | |
| rw [← Real.rpow_natCast ((2:ℝ)^(q/2)) 2,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| congr 1 | |
| norm_num | |
| lemma shell_effective_scale_upperND {R σ q : ℝ} (l : ℕ) (hR : 0<R) | |
| (hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) : | |
| shellEffectiveScaleND l R σ ≤ max (R^σ) ((2:ℝ)^(q/2)*R^(-48*σ)) := by | |
| unfold shellEffectiveScaleND sourceMaskScaleND | |
| rw [mul_max_of_nonneg _ _ (Real.rpow_nonneg hR.le _),mul_one] | |
| apply max_le_max_left | |
| calc | |
| _ ≤ R^σ*(((2:ℝ)^(q/2)*R^σ)*R^(-50*σ)) := by gcongr | |
| _ = (2:ℝ)^(q/2)*(R^σ*R^σ*R^(-50*σ)) := by ring | |
| _ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hR]; congr 2; ring | |
| lemma shell_effective_scale_sqND {R σ q : ℝ} (l : ℕ) (hR : 0<R) | |
| (hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) : | |
| (shellEffectiveScaleND l R σ)^2 ≤ max (R^(2*σ)) ((2:ℝ)^q*R^(-96*σ)) := by | |
| have hu := shell_effective_scale_upperND l hR hl | |
| have hn : 0 ≤ shellEffectiveScaleND l R σ := | |
| mul_nonneg (Real.rpow_nonneg hR.le _) (sourceMaskScaleND_pos l R σ).le | |
| have he₁ : (R^σ)^2=R^(2*σ) := by | |
| rw [← Real.rpow_natCast (R^σ) 2,← Real.rpow_mul hR.le] | |
| congr 1 | |
| norm_num | |
| ring | |
| have he₂ : ((2:ℝ)^(q/2)*R^(-48*σ))^2=(2:ℝ)^q*R^(-96*σ) := by | |
| rw [mul_pow,dyadic_half_sqND,← Real.rpow_natCast (R^(-48*σ)) 2,← Real.rpow_mul hR.le] | |
| congr 2 | |
| norm_num | |
| ring | |
| rcases le_total (R^σ) ((2:ℝ)^(q/2)*R^(-48*σ)) with h | h | |
| · rw [max_eq_right h] at hu | |
| exact (pow_le_pow_left₀ hn hu 2).trans (by rw [he₂]; exact le_max_right _ _) | |
| · rw [max_eq_left h] at hu | |
| exact (pow_le_pow_left₀ hn hu 2).trans (by rw [he₁]; exact le_max_left _ _) | |
| lemma shell_scale_regimeND {R σ q c Λ : ℝ} (l : ℕ) | |
| (hR : 1 ≤ R) (hσ : 0 ≤ σ) (hσ₁ : σ ≤ 1) (hc : 0<c) | |
| (hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) | |
| (hqlo : R^(20*σ) ≤ (2:ℝ)^q) (hqup : (2:ℝ)^q ≤ R^2) | |
| (hΛ : c*(2:ℝ)^q ≤ Λ) : | |
| 1 ≤ shellEffectiveScaleND l R σ ∧ shellEffectiveScaleND l R σ ≤ R^2 ∧ | |
| c*R^(20*σ) ≤ Λ ∧ (shellEffectiveScaleND l R σ)^2/Λ ≤ c⁻¹*R^(-18*σ) ∧ | |
| (2:ℝ)^(-q)*(sourceMaskScaleND l R σ)^2 ≤ 1 := by | |
| have hRp : 0<R := zero_lt_one.trans_le hR | |
| have hQp : 0<(2:ℝ)^q := Real.rpow_pos_of_pos (by norm_num) _ | |
| have hΛp : 0<Λ := (mul_pos hc hQp).trans_le hΛ | |
| have hRs : 1≤R^σ := Real.one_le_rpow hR hσ | |
| have hu := shell_effective_scale_upperND l hRp hl | |
| have hsquare := shell_effective_scale_sqND l hRp hl | |
| have hhalf : (2:ℝ)^(q/2) ≤ R := by | |
| nlinarith [dyadic_half_sqND q, Real.rpow_nonneg (by norm_num : (0:ℝ)≤2) (q/2)] | |
| have hle (u : ℝ) (hu : u ≤ 0) : R^u ≤ 1 := by | |
| simpa using Real.rpow_le_rpow_of_exponent_le hR hu | |
| have hnorm : (shellEffectiveScaleND l R σ)^2 ≤ (2:ℝ)^q*R^(-18*σ) := by | |
| apply hsquare.trans | |
| apply max_le | |
| · calc | |
| R^(2*σ) = R^(20*σ)*R^(-18*σ) := by rw [← Real.rpow_add hRp]; congr 1; ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_right hqlo (Real.rpow_nonneg hRp.le _) | |
| · exact mul_le_mul_of_nonneg_left | |
| (Real.rpow_le_rpow_of_exponent_le hR (by linarith)) hQp.le | |
| refine ⟨?_,?_,?_,?_,?_⟩ | |
| · exact one_le_mul_of_one_le_of_one_le hRs (sourceMaskScaleND_one_le l R σ) | |
| · apply hu.trans | |
| apply max_le | |
| · simpa using Real.rpow_le_rpow_of_exponent_le hR (show σ ≤ (2:ℝ) by linarith) | |
| · calc | |
| _ ≤ (2:ℝ)^(q/2) := mul_le_of_le_one_right (by positivity) (hle (-48*σ) (by linarith)) | |
| _ ≤ R := hhalf | |
| _ ≤ R^2 := by nlinarith | |
| · exact (mul_le_mul_of_nonneg_left hqlo hc.le).trans hΛ | |
| · calc | |
| _ ≤ ((2:ℝ)^q*R^(-18*σ))/(c*(2:ℝ)^q) := | |
| div_le_div₀ (by positivity) hnorm (mul_pos hc hQp) hΛ | |
| _ = _ := by field_simp | |
| · have hQle : sourceMaskScaleND l R σ ≤ shellEffectiveScaleND l R σ := by | |
| exact le_mul_of_one_le_left (sourceMaskScaleND_pos l R σ).le hRs | |
| have hsq : (sourceMaskScaleND l R σ)^2 ≤ (2:ℝ)^q := | |
| (pow_le_pow_left₀ (sourceMaskScaleND_pos l R σ).le hQle 2).trans | |
| (hnorm.trans (mul_le_of_le_one_right hQp.le (hle (-18*σ) (by linarith)))) | |
| rw [Real.rpow_neg (by norm_num : (0:ℝ)≤2)] | |
| exact (inv_mul_le_one₀ hQp).mpr hsq | |
| theorem shell_scale_regime_eventuallyND {σ c : ℝ} (hσ : 0<σ) (hσ₁ : σ≤1) (hc : 0<c) : | |
| ∀ᶠ R : ℝ in atTop, ∀ (l : ℕ) (q Λ : ℝ), | |
| (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ → R^(20*σ) ≤ (2:ℝ)^q → (2:ℝ)^q ≤ R^2 → | |
| c*(2:ℝ)^q ≤ Λ → | |
| 1 ≤ shellEffectiveScaleND l R σ ∧ shellEffectiveScaleND l R σ ≤ R^2 ∧ | |
| R^σ ≤ Λ ∧ (shellEffectiveScaleND l R σ)^2/Λ ≤ R^(-σ) ∧ | |
| (2:ℝ)^(-q)*(sourceMaskScaleND l R σ)^2 ≤ 1 := by | |
| filter_upwards [eventually_ge_atTop (1:ℝ), | |
| (tendsto_rpow_atTop (by linarith : (0:ℝ)<17*σ)).eventually_ge_atTop c⁻¹] with R hR hlarge | |
| intro l q Λ hl hqlo hqup hΛ | |
| have hRp : 0<R := zero_lt_one.trans_le hR | |
| obtain ⟨h₁,h₂,h₃,h₄,h₅⟩ := shell_scale_regimeND l hR hσ.le hσ₁ hc hl hqlo hqup hΛ | |
| refine ⟨h₁,h₂,?_,?_,h₅⟩ | |
| · have hcR : 1 ≤ c*R^(17*σ) := by simpa only [mul_comm] using (inv_le_iff_one_le_mul₀ hc).mp hlarge | |
| have hRs : R^σ ≤ R^(3*σ) := Real.rpow_le_rpow_of_exponent_le hR (by linarith) | |
| apply le_trans _ h₃ | |
| calc | |
| _ ≤ R^(3*σ) := hRs | |
| _ ≤ (c*R^(17*σ))*R^(3*σ) := le_mul_of_one_le_left (Real.rpow_nonneg hRp.le _) hcR | |
| _ = _ := by rw [mul_assoc,← Real.rpow_add hRp]; congr 2; ring | |
| · apply h₄.trans | |
| calc | |
| _ ≤ R^(17*σ)*R^(-18*σ) := mul_le_mul_of_nonneg_right hlarge (Real.rpow_nonneg hRp.le _) | |
| _ = _ := by rw [← Real.rpow_add hRp]; congr 1; ring | |
| lemma shell_source_exponentsND {N l : ℕ} {σ q : ℝ} | |
| (hl : (l:ℝ) ≤ q/2+σ*N) (hqlo : 20*σ*N ≤ q) (hqup : q ≤ 2*N) : | |
| (2:ℝ)^l ≤ (2:ℝ)^(q/2)*((2:ℝ)^N)^σ ∧ | |
| ((2:ℝ)^N)^(20*σ) ≤ (2:ℝ)^q ∧ (2:ℝ)^q ≤ ((2:ℝ)^N)^2 := by | |
| have h₁ : ((2:ℝ)^N)^σ=(2:ℝ)^((N:ℝ)*σ) := by | |
| rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| have h₂ : ((2:ℝ)^N)^(20*σ)=(2:ℝ)^((N:ℝ)*(20*σ)) := by | |
| rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| have h₃ : ((2:ℝ)^N)^2=(2:ℝ)^((N:ℝ)*2) := by | |
| rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_natCast ((2:ℝ)^(N:ℝ)) 2, | |
| ← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| norm_num | |
| rw [h₁,h₂,h₃,← Real.rpow_add (by norm_num : (0:ℝ)<2),← Real.rpow_natCast (2:ℝ) l] | |
| exact ⟨Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith), | |
| Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith), | |
| Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith)⟩ | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open Set | |
| lemma shell_pair_frequency_lowerND (a v r D δ : ℝ) | |
| (hr : (2:ℝ)^v/2 ≤ r) (hD : δ ≤ D*(2:ℝ)^a) (hδ : 0<δ) : | |
| (δ/2)*(2:ℝ)^(v-a) ≤ r*D := by | |
| have ha : 0<(2:ℝ)^a := Real.rpow_pos_of_pos (by norm_num) _ | |
| have hv : 0<(2:ℝ)^v := Real.rpow_pos_of_pos (by norm_num) _ | |
| have hDp : 0<D := (mul_pos_iff_of_pos_right ha).mp (hδ.trans_le hD) | |
| calc | |
| _ = ((2:ℝ)^v/2)*(δ/(2:ℝ)^a) := by | |
| rw [Real.rpow_sub (by norm_num : (0:ℝ)<2)]; ring | |
| _ ≤ ((2:ℝ)^v/2)*D := mul_le_mul_of_nonneg_left | |
| ((div_le_iff₀ ha).mpr hD) (by positivity) | |
| _ ≤ r*D := mul_le_mul_of_nonneg_right hr hDp.le | |
| 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 source_weighted_shellND (ψ : V → ℂ) (hψ : ContDiff ℝ ∞ ψ) | |
| (kCard : ℕ) (Cderiv : ℕ → ℝ) (σ ε T₀ Bmass Cmask δ K : ℝ) | |
| (hσ : 0<σ) (hσ₁ : σ≤1) (hCderiv : ∀ m, 0≤Cderiv m) | |
| (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), | |
| 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 → ∀ | |
| (L : ℕ) (M N a v : ℝ), 20*σ*kN ≤ v-a → (L:ℝ) ≤ (v-a)/2+σ*kN → v-a ≤ 2*kN → | |
| |(M*N)⁻¹| * ((ρ.restrict X).prod (ν.restrict Y)).real univ ≤ R^Bmass → | |
| (∀ i ∈ S, 1 ≤ D i ∧ D i ≤ sourceMaskScaleND L R σ) → | |
| (∀ k ∈ T, 1 ≤ E k ∧ E k ≤ sourceMaskScaleND L R σ) → | |
| (∀ x ∈ X, ∀ i ∈ S, ∀ w, ‖w‖=1 → ‖(f x i).1 w‖ ≤ 1) → | |
| (∀ y ∈ Y, ∀ k ∈ T, ∀ w, ‖w‖=1 → ‖(g y k).1 w‖ ≤ 1) → | |
| (∀ x ∈ X, ∀ i ∈ S, ∀ j ≤ 2*(J+1)+n+3, ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ j (f x i).1 w‖ ≤ Cderiv (2*(J+1)+n+3)*(D i)^j) → | |
| (∀ y ∈ Y, ∀ k ∈ T, ∀ j ≤ 2*(J+1)+n+3, ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ j (g y k).1 w‖ ≤ Cderiv (2*(J+1)+n+3)*(E k)^j) → | |
| (∀ x ∈ X, ∀ y ∈ Y, δ ≤ dist x y*(2:ℝ)^a ∧ dist x y*(2:ℝ)^a ≤ K) → | |
| (∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), | |
| 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)) → | |
| (∀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,hMain⟩ := weighted_all_mask_shell_asymptoticND (ι := ι) (κ := κ) | |
| ψ hψ kCard σ 1 ε T₀ Bmass Cmask δ K hσ (by norm_num) hε hδ hK | |
| refine ⟨J,?_⟩ | |
| filter_upwards [hMain, | |
| pair_mask_product_admissibleND (ι := ι) (κ := κ) (n := n) | |
| (2*(J+1)+n+3) kCard (Cderiv (2*(J+1)+n+3)) Cmask σ (hCderiv _) hσ, | |
| shell_dyadic_tendstoND.eventually (shell_scale_regime_eventuallyND hσ hσ₁ | |
| (show 0<δ/2 by linarith))] with kN hMainN hproduct hscale | |
| dsimp only at hMainN ⊢ | |
| intro S T f g F G D E hcard ρ ν hρ hν X Y hX hY L M N a v hqlo hl hqup | |
| hmass hD hE hf0 hg0 hf hg hsep hcut hFG hsym P hP B C hb hc hB hC | |
| Ψ hΨ hΨpos hone BHF hBHF hHF | |
| obtain ⟨hl',hqlo',hqup'⟩ := shell_source_exponentsND hl hqlo hqup | |
| have hA := hscale L (v-a) ((δ/2)*(2:ℝ)^(v-a)) hl' hqlo' hqup' le_rfl | |
| apply hMainN S T f g F G D E | |
| (fun i hi => zero_lt_one.trans_le (hD i hi).1) | |
| (fun i hi => zero_lt_one.trans_le (hE i hi).1) hcard ρ ν X Y hX hY M N | |
| (shellEffectiveScaleND L ((2:ℝ)^kN) σ) (sourceMaskScaleND L ((2:ℝ)^kN) σ) | |
| a v hA.1 hA.2.1 (sourceMaskScaleND_one_le _ _ _) hqup' hmass hA.2.2.2.2 | |
| (fun i hi => (hD i hi).2) (fun i hi => (hE i hi).2) hsep ?_ | |
| hFG hsym P hP B C hb hc hB hC Ψ hΨ hΨpos hone BHF hBHF hHF | |
| intro r hr | |
| have hxy : ∀ᵐ z ∂(ρ.restrict X).prod (ν.restrict Y), 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⟩) | |
| filter_upwards [hxy,hcut] with z hz hc | |
| have hfreq := shell_pair_frequency_lowerND a v r (dist z.1 z.2) δ hr.1 | |
| (hsep z.1 hz.1 z.2 hz.2).1 hδ | |
| have hsa := hscale L (v-a) (r*dist z.1 z.2) hl' hqlo' hqup' hfreq | |
| refine ⟨hsa.2.2.1,hsa.2.2.2.1,hc.1,hc.2,?_⟩ | |
| simpa only [one_mul,shellEffectiveScaleND] using | |
| hproduct S T f g D E (sourceMaskScaleND L ((2:ℝ)^kN) σ) z | |
| ((le_max_right _ _).trans hcard) hD hE (sourceMaskScaleND_pos _ _ _).le | |
| (hf0 z.1 hz.1) (hg0 z.2 hz.2) (hf z.1 hz.1) (hg z.2 hz.2) | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |