openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /ContinuousCoefficientMeasurableSquare.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.InverseEndpointCoefficientEq | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.PairMaskProduct | |
| import OAI.MeasureTheory.FalconerCompletion.Shells.AngularCorrelation | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open Set MeasureTheory | |
| lemma continuous_coefficient_measurable_squareND {α : Type*} [MeasurableSpace α] | |
| (ν : Measure α) [SFinite ν] (μ : Measure ℝ) [IsFiniteMeasure μ] | |
| (c : α → ℂ) (hc : Measurable c) (hic : Integrable c ν) | |
| (F : α × ℝ → ℂ) (hF : Measurable F) {C : ℝ} (hb : ∀ p, ‖F p‖ ≤ C) : | |
| Measurable (fun r => ∫ ξ, c ξ*F (ξ,r) ∂ν) ∧ | |
| Integrable (fun r => ‖∫ ξ, c ξ*F (ξ,r) ∂ν‖^2) μ := by | |
| have hm : Measurable (fun r => ∫ ξ, c ξ*F (ξ,r) ∂ν) := | |
| ((hc.comp measurable_fst).mul hF).stronglyMeasurable.integral_prod_left'.measurable | |
| refine ⟨hm,?_⟩ | |
| have hnorm (r : ℝ) : ‖∫ ξ, c ξ*F (ξ,r) ∂ν‖ ≤ (∫ ξ, ‖c ξ‖ ∂ν)*C := by | |
| calc | |
| _ ≤ ∫ ξ, ‖c ξ*F (ξ,r)‖ ∂ν := norm_integral_le_integral_norm _ | |
| _ ≤ ∫ ξ, ‖c ξ‖*C ∂ν := by | |
| apply integral_mono_of_nonneg (Filter.Eventually.of_forall (fun _ => norm_nonneg _)) | |
| (hic.norm.mul_const C) | |
| filter_upwards with ξ | |
| rw [norm_mul] | |
| exact mul_le_mul_of_nonneg_left (hb (ξ,r)) (norm_nonneg _) | |
| _ = _ := integral_mul_const _ _ | |
| apply (integrable_const (((∫ ξ, ‖c ξ‖ ∂ν)*C)^2)).mono' | |
| (hm.norm.pow_const 2).aestronglyMeasurable | |
| filter_upwards with r | |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] | |
| exact pow_le_pow_left₀ (norm_nonneg _) (hnorm r) 2 | |
| theorem full_HF_shell_mixtureND {ι α V : Type*} [MeasurableSpace α] [MeasurableSpace V] | |
| (n : ℕ) (ν : Measure α) [SFinite ν] (μ : Measure V) [IsFiniteMeasure μ] | |
| (a v : ℝ) (L : List ι) (c : ι → α → ℂ) (m : ι → ℝ → ℂ) | |
| (ψ : V → ℂ) (U W : ι → α × (ℝ × V) → ℂ) | |
| (H : ℝ → ℂ) (A C D : ι → ℝ) {P B E : ℝ} | |
| (hA : ∀ i ∈ L, 0 ≤ A i) (hc : ∀ i ∈ L, Measurable (c i)) | |
| (hic : ∀ i ∈ L, Integrable (c i) ν) (hm : ∀ i ∈ L, Measurable (m i)) | |
| (hbM : ∀ i ∈ L, ∀ r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v), ‖m i r‖ ≤ A i) | |
| (hψ : Measurable ψ) (hU : ∀ i ∈ L, Measurable (U i)) | |
| (hW : ∀ i ∈ L, Measurable (W i)) (hP : 0 ≤ P) (hC : ∀ i ∈ L, 0 ≤ C i) | |
| (hbψ : ∀ w, ‖ψ w‖ ≤ P) (hbU : ∀ i ∈ L, ∀ ξ r w, ‖U i (ξ,(r,w))‖ ≤ C i) | |
| (hbW : ∀ i ∈ L, ∀ ξ r w, ‖W i (ξ,(r,w))‖ ≤ D i) {Ψ : ℝ → ℝ} | |
| (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) (hΨpos : ∀ r, 0 ≤ Ψ r) | |
| (hone : ∀ r ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ r) | |
| (hB : ∀ i ∈ L, ∀ ξ, sourceAngularHFND (n+1) μ a v Ψ ψ | |
| (fun z => U i (ξ,z)) (fun z => W i (ξ,z)) ≤ B) | |
| (hH : AEStronglyMeasurable H (volume.restrict (Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v)))) | |
| (hE : ∀ r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v), | |
| ‖H r-(((2:ℝ)^((n:ℝ)*(v-a)/2):ℝ):ℂ)*(L.map (fun i => m i r* | |
| ∫ ξ, c i ξ*sourceAngularCorrelationND μ ψ | |
| (fun z => U i (ξ,z)) (fun z => W i (ξ,z)) r ∂ν)).sum‖ ≤ E) : | |
| Integrable (fun r => ‖H 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), ‖H r‖^2) ≤ | |
| 2*(L.map (fun i => A i*(∫ ξ, ‖c i ξ‖ ∂ν))).sum^2*B + 3*(2:ℝ)^(v-a)*E^2 := by | |
| let F : ι → ℝ → ℂ := fun i r => ∫ ξ, c i ξ*sourceAngularCorrelationND μ ψ | |
| (fun z => U i (ξ,z)) (fun z => W i (ξ,z)) r ∂ν | |
| let μshell := volume.restrict (Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v)) | |
| have hFm (i : ι) (hi : i ∈ L) : Measurable (F i) ∧ Integrable (fun r => ‖F i r‖^2) μshell := by | |
| apply continuous_coefficient_measurable_squareND ν μshell (c i) (hc i hi) (hic i hi) | |
| (fun p => sourceAngularCorrelationND μ ψ (fun z => U i (p.1,z)) (fun z => W i (p.1,z)) p.2) | |
| (sourceAngularCorrelationND_parameter_measurable μ hψ (hU i hi) (hW i hi)) | |
| (C := P*C i*D i*μ.real univ) | |
| intro p | |
| exact sourceAngularCorrelationND_bound μ ψ _ _ hP (hC i hi) hbψ | |
| (hbU i hi p.1) (hbW i hi p.1) p.2 | |
| have hmix (i : ι) (hi : i ∈ L) := shell_square_multiplierND μshell | |
| ((hm i hi).aestronglyMeasurable.mul (hFm i hi).1.aestronglyMeasurable) (hFm i hi).2 | |
| (show ∀ᵐ r ∂μshell, ‖m i r‖ ≤ A i from by | |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr | |
| exact hbM i hi r hr) | |
| apply normalized_list_shell_remainderND n a v L (fun i r => m i r*F i r) | |
| (fun i => A i*(∫ ξ, ‖c i ξ‖ ∂ν)) H | |
| · intro i hi | |
| exact mul_nonneg (hA i hi) (integral_nonneg (fun _ => norm_nonneg _)) | |
| · intro i hi | |
| exact (hmix i hi).1 | |
| · intro i hi | |
| exact ((hm i hi).mul (hFm i hi).1).aestronglyMeasurable | |
| · intro i hi | |
| have henergy := continuous_HF_shell_boundND (n+1) ν μ a v (c i) (hc i hi) | |
| (hic i hi) hψ (hU i hi) (hW i hi) hP (hC i hi) hbψ | |
| (hbU i hi) (hbW i hi) hΨ hΨpos hone (hB i hi) | |
| exact (hmix i hi).2.trans ((mul_le_mul_of_nonneg_left henergy (sq_nonneg (A i))).trans_eq (by ring)) | |
| · exact hH | |
| · exact hE | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open Set | |
| noncomputable def shellRadialMultiplierND (n j : ℕ) (v r : ℝ) : ℂ := | |
| (((r/(2:ℝ)^v)^((n:ℝ)/2):ℝ):ℂ)*((r/(2:ℝ)^v:ℝ):ℂ)^(-(j:ℤ)) | |
| lemma positive_shell_ratioND {v r : ℝ} | |
| (hr : r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v)) : | |
| (1/2:ℝ) ≤ r/(2:ℝ)^v ∧ r/(2:ℝ)^v ≤ 2 := by | |
| have hR : 0 < (2:ℝ)^v := Real.rpow_pos_of_pos (by norm_num) _ | |
| constructor | |
| · apply (le_div_iff₀ hR).mpr | |
| linarith [hr.1] | |
| · exact (div_le_iff₀ hR).mpr hr.2 | |
| lemma shellRadialMultiplierND_bound (n j : ℕ) {v r : ℝ} | |
| (hr : r ∈ Icc ((2:ℝ)^v/2) (2*(2:ℝ)^v)) : | |
| ‖shellRadialMultiplierND n j v r‖ ≤ (2:ℝ)^((n:ℝ)/2)*(2:ℝ)^j := by | |
| obtain ⟨hl,hu⟩ := positive_shell_ratioND hr | |
| have hs : 0 < r/(2:ℝ)^v := by linarith | |
| have hsn : 0 ≤ (r/(2:ℝ)^v)^((n:ℝ)/2) := Real.rpow_nonneg hs.le _ | |
| have hpow : (r/(2:ℝ)^v)^((n:ℝ)/2) ≤ (2:ℝ)^((n:ℝ)/2) := | |
| Real.rpow_le_rpow hs.le hu (by positivity) | |
| have hi : (r/(2:ℝ)^v)⁻¹ ≤ 2 := (inv_le_comm₀ hs (by norm_num : (0:ℝ)<2)).mpr (by simpa only [one_div] using hl) | |
| simp only [shellRadialMultiplierND,norm_mul,zpow_neg,zpow_natCast,norm_inv,norm_pow, | |
| Complex.norm_real,Real.norm_eq_abs,abs_of_pos hs,abs_of_nonneg hsn,← inv_pow] | |
| exact mul_le_mul hpow (pow_le_pow_left₀ (inv_nonneg.mpr hs.le) hi j) | |
| (by positivity) (by positivity) | |
| lemma shellRadialMultiplierND_measurable (n j : ℕ) (v : ℝ) : | |
| Measurable (shellRadialMultiplierND n j v) := by | |
| unfold shellRadialMultiplierND | |
| fun_prop | |
| lemma positive_inverse_coefficient_splitND {t u : ℝ} (ht : 0 < t) (hu : 0 < u) | |
| (e : ℝ) (j : ℕ) : | |
| (((t*u)^e:ℝ):ℂ)*((t*u:ℝ):ℂ)^(-(j:ℤ)) = | |
| ((t^e:ℝ):ℂ)*(((u^e:ℝ):ℂ)*(u:ℂ)^(-(j:ℤ)))*((t⁻¹:ℝ):ℂ)^j := by | |
| rw [Real.mul_rpow ht.le hu.le] | |
| simp only [Complex.ofReal_mul,zpow_neg,zpow_natCast,Complex.ofReal_inv,inv_pow] | |
| ring | |
| lemma dyadic_inverse_coefficient_splitND (n j : ℕ) {a v r : ℝ} (hr : 0 < r) : | |
| (((r/(2:ℝ)^a)^((n:ℝ)/2):ℝ):ℂ)*((r/(2:ℝ)^a:ℝ):ℂ)^(-(j:ℤ)) = | |
| (((2:ℝ)^((n:ℝ)*(v-a)/2):ℝ):ℂ)* | |
| (shellRadialMultiplierND n j v r * (((2:ℝ)^(-(v-a)):ℝ):ℂ)^j) := by | |
| have hp : 0 < (2:ℝ)^v := Real.rpow_pos_of_pos (by norm_num) _ | |
| have hq : 0 < (2:ℝ)^(v-a) := Real.rpow_pos_of_pos (by norm_num) _ | |
| have he : r/(2:ℝ)^a = (2:ℝ)^(v-a)*(r/(2:ℝ)^v) := by | |
| rw [Real.rpow_sub (by norm_num : (0:ℝ)<2)] | |
| field_simp | |
| rw [he,positive_inverse_coefficient_splitND hq (div_pos hr hp)] | |
| rw [← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| have hring : (v-a)*((n:ℝ)/2) = (n:ℝ)*(v-a)/2 := by ring | |
| rw [hring,← Real.rpow_neg (by norm_num : (0:ℝ)≤2)] | |
| simp only [shellRadialMultiplierND] | |
| ring | |
| 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 | |
| lemma sum_range_flatMapND {Q A : Type*} [AddCommMonoid A] | |
| (m : ℕ) (L : ℕ → List Q) (F : ℕ → Q → A) : | |
| ((((List.range m).flatMap (fun j => (L j).map (fun x => (j,x)))).map | |
| (fun p => F p.1 p.2)).sum) = ∑ j ∈ Finset.range m, ((L j).map (F j)).sum := by | |
| induction m with | |
| | zero => simp | |
| | succ m ih => | |
| simp only [List.range_succ,List.flatMap_append,List.flatMap_singleton, | |
| List.map_append,List.sum_append,Finset.sum_range_succ,List.map_map,Function.comp_def] | |
| rw [ih] | |
| 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 inverseTermListND (J : ℕ) (S : Finset ι) (T : Finset κ) : List (ℕ × Assignment) := | |
| (List.range (J+1)).flatMap (fun j => (inverseAngularTermsND j (S.disjSum T)).map (fun a => (j,a))) | |
| lemma sum_inverseTermListND {A : Type*} [AddCommMonoid A] | |
| (J : ℕ) (S : Finset ι) (T : Finset κ) (F : ℕ × Assignment → A) : | |
| ((inverseTermListND J S T).map F).sum = | |
| ∑ j ∈ Finset.range (J+1), ((inverseAngularTermsND j (S.disjSum T)).map (fun p => F (j,p))).sum := by | |
| simpa only [inverseTermListND, Prod.mk.eta] using sum_range_flatMapND (J+1) (fun j => inverseAngularTermsND j (S.disjSum T)) (fun j p => F (j,p)) | |
| lemma mem_inverseTermListND (J : ℕ) (S : Finset ι) (T : Finset κ) (p : ℕ × Assignment) : | |
| p ∈ inverseTermListND J S T ↔ p.1 ≤ J ∧ p.2 ∈ inverseAngularTermsND p.1 (S.disjSum T) := by | |
| rcases p with ⟨j,a⟩ | |
| simp only [inverseTermListND,List.mem_flatMap,List.mem_range,List.mem_map,Prod.mk.injEq] | |
| constructor | |
| · rintro ⟨k,hk,b,hb,hbj⟩ | |
| rcases hbj with ⟨rfl,rfl⟩ | |
| exact ⟨Nat.le_of_lt_succ hk,hb⟩ | |
| · rintro ⟨hj,ha⟩ | |
| exact ⟨j,Nat.lt_succ_of_le hj,a,ha,rfl,rfl⟩ | |
| noncomputable def spatialEndpointTransformND (scale : ℝ) (ρ : Measure V) (X : Set V) (M : ℝ) | |
| (b : V → V → ℂ) (p : V × (ℝ × V)) : ℂ := | |
| sourceAngularTransformND ρ X M | |
| (fun x w => SpatialFourierND.spatialFourierPhase (n+1) scale p.1 x*b x w) p.2.1 p.2.2 | |
| omit [DecidableEq ι] [DecidableEq κ] in | |
| lemma spatialEndpointTransformND_measurable (scale : ℝ) (ρ : Measure V) [SFinite ρ] | |
| (X : Set V) (M : ℝ) {b : V → V → ℂ} (hb : Measurable (Function.uncurry b)) : | |
| Measurable (spatialEndpointTransformND scale ρ X M b) := by | |
| have hm : Measurable (fun z : (V × (ℝ × V)) × V => | |
| spherePlaneWaveND z.1.2.1 z.2 z.1.2.2 * | |
| (SpatialFourierND.spatialFourierPhase (n+1) scale z.1.1 z.2*b z.2 z.1.2.2)) := by | |
| apply Measurable.mul (by unfold spherePlaneWaveND; fun_prop) | |
| exact ((SpatialFourierND.spatialFourierPhase_continuous (n+1) scale).measurable.comp | |
| (measurable_fst.fst.prodMk measurable_snd)).mul | |
| (hb.comp (measurable_snd.prodMk measurable_fst.snd.snd)) | |
| exact (hm.stronglyMeasurable.integral_prod_right' (ν := ρ.restrict X)).measurable.const_smul (M⁻¹) | |
| omit [DecidableEq ι] [DecidableEq κ] in | |
| lemma spatialEndpointTransformND_bound (scale : ℝ) (ρ : Measure V) [IsFiniteMeasure ρ] | |
| (X : Set V) (M : ℝ) (b : V → V → ℂ) {B : ℝ} | |
| (hb : ∀ x w, ‖b x w‖ ≤ B) (p : V × (ℝ × V)) : | |
| ‖spatialEndpointTransformND scale ρ X M b p‖ ≤ |M⁻¹| *(B*(ρ.restrict X).real univ) := by | |
| unfold spatialEndpointTransformND sourceAngularTransformND | |
| rw [norm_smul,Real.norm_eq_abs] | |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) | |
| apply norm_integral_le_of_norm_le_const | |
| filter_upwards with x | |
| simpa only [norm_mul,spherePlaneWaveND_norm,SpatialFourierND.spatialFourierPhase_norm,one_mul] | |
| using hb x p.2.2 | |
| lemma all_mask_shell_factorizationND (J : ℕ) (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) | |
| (δ K : ℝ) (hδ : 0 < δ) (hK : 0 < K) | |
| (ρ ν : Measure V) (X Y : Set V) (M N a v r : ℝ) (hr : 0 < r) (ψ : V → ℂ) : | |
| (∑ j ∈ Finset.range (J+1), | |
| ((((r/(2:ℝ)^a)^((n:ℝ)/2):ℝ):ℂ)*((r/(2:ℝ)^a:ℝ):ℂ)^(-(j:ℤ))) * | |
| inverseEndpointMixtureND j S T f g F G D E δ K hδ hK ((2:ℝ)^a)⁻¹ ρ ν X Y M N r ψ) = | |
| (((2:ℝ)^((n:ℝ)*(v-a)/2):ℝ):ℂ)* | |
| ((inverseTermListND J S T).map (fun p => shellRadialMultiplierND n p.1 v r* | |
| ∫ ξ, paidSpatialCoefficientND p.1 S T D E ((2:ℝ)^(-(v-a))) δ K hδ hK p.2 ξ* | |
| sourceAngularCorrelationND (canonicalSphereMeasureND V) ψ | |
| (fun z => spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ρ X M | |
| (inverseLeftEndpointND S D f F p.2) (ξ,z)) | |
| (fun z => spatialEndpointTransformND ((2:ℝ)^a)⁻¹ ν Y N | |
| (inverseRightEndpointND T E g G p.2) (ξ,z)) r)).sum := by | |
| rw [sum_inverseTermListND,Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro j hj | |
| rw [dyadic_inverse_coefficient_splitND n j hr] | |
| unfold inverseEndpointMixtureND | |
| rw [← List.sum_map_mul_left,← List.sum_map_mul_left] | |
| apply congrArg List.sum | |
| apply List.map_congr_left | |
| intro p hp | |
| simp only [paidSpatialCoefficientND,sourceAngularCorrelationND,spatialEndpointTransformND, | |
| mul_assoc,integral_const_mul] | |
| ring | |
| lemma inverseTermListND_paid_cost (J : ℕ) (S : Finset ι) (T : Finset κ) | |
| (D : ι → ℝ) (E : κ → ℝ) {M t : ℝ} | |
| (hM : 1 ≤ M) (ht : 0 ≤ t) (hpaid : t*M^2 ≤ 1) | |
| (hD₀ : ∀ i ∈ S, 0 ≤ D i) (hE₀ : ∀ k ∈ T, 0 ≤ E k) | |
| (hD : ∀ i ∈ S, D i ≤ M) (hE : ∀ k ∈ T, E k ≤ M) | |
| (δ K : ℝ) (hδ : 0 < δ) (hK : 0 < K) : | |
| (((inverseTermListND (n := n) J S T).map (fun p => | |
| ((2:ℝ)^((n:ℝ)/2)*(2:ℝ)^p.1)* | |
| ∫ ξ, ‖paidSpatialCoefficientND p.1 S T D E t δ K hδ hK p.2 ξ‖)).sum) ≤ | |
| fullInverseCoefficientCostND n J δ K hδ hK* | |
| (max 1 ((S.card+T.card:ℕ):ℝ))^(2*J) := by | |
| rw [sum_inverseTermListND] | |
| exact full_paidSpatialCoefficientND_cost J S T D E hM ht hpaid hD₀ hE₀ hD hE δ K hδ hK | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |