| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphChildSquare |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.ProfileChildCard |
| import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereShift |
|
|
| namespace OAI |
|
|
| /-! Graph energies and pair estimates. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace Attempt4Near |
| open Set MeasureTheory |
| open scoped ENNReal ComplexConjugate |
| variable {V : Type*} [MeasurableSpace V] |
|
|
| noncomputable def sourceAngularMomentND (μ : Measure V) (U : ℝ × V → ℂ) (r : ℝ) : ℝ := |
| ∫ w, ‖U (r,w)‖^2 ∂μ |
|
|
| noncomputable def sourceAngularHND (d : ℕ) (μ : Measure V) (a v : ℝ) |
| (Ψ : ℝ → ℝ) (U W : ℝ × V → ℂ) : ℝ := |
| (2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹* |
| ∫ r, Ψ (r/(2:ℝ)^v)*sourceAngularMomentND μ U r*sourceAngularMomentND μ W r |
|
|
| noncomputable def sourceAngularHFND (d : ℕ) (μ : Measure V) (a v : ℝ) |
| (Ψ : ℝ → ℝ) (ψ : V → ℂ) (U W : ℝ × V → ℂ) : ℝ := |
| (2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹* |
| ∫ r, Ψ (r/(2:ℝ)^v)*‖∫ w, ψ w*U (r,w)*conj (W (r,w)) ∂μ‖^2 |
|
|
| lemma graph_radial_normalizationND {Ω : ℝ} (hΩ : 0 < Ω) (Ψ f : ℝ → ℝ) : |
| Ω⁻¹*(∫ r, Ψ (r/Ω)*f r) = ∫ u, Ψ u*f (Ω*u) := by |
| have h := Measure.integral_comp_mul_left (fun r => Ψ (r/Ω)*f r) Ω |
| simp only [abs_of_pos (inv_pos.mpr hΩ),smul_eq_mul] at h |
| rw [← h] |
| apply integral_congr_ae |
| filter_upwards with u |
| rw [mul_div_cancel_left₀ u hΩ.ne'] |
|
|
| lemma sourceAngularHND_normalized (d : ℕ) (μ : Measure V) (a v : ℝ) |
| (Ψ : ℝ → ℝ) (U W : ℝ × V → ℂ) : |
| sourceAngularHND d μ a v Ψ U W = (2:ℝ)^((d:ℝ)*(v-a))* |
| ∫ u, Ψ u*sourceAngularMomentND μ U ((2:ℝ)^v*u)*sourceAngularMomentND μ W ((2:ℝ)^v*u) := by |
| dsimp [sourceAngularHND] |
| rw [mul_assoc] |
| congr 1 |
| simpa only [mul_assoc] using graph_radial_normalizationND (by positivity : (0:ℝ)<(2:ℝ)^v) |
| Ψ (fun r => sourceAngularMomentND μ U r*sourceAngularMomentND μ W r) |
|
|
| lemma sourceAngularHFND_normalized (d : ℕ) (μ : Measure V) (a v : ℝ) |
| (Ψ : ℝ → ℝ) (ψ : V → ℂ) (U W : ℝ × V → ℂ) : |
| sourceAngularHFND d μ a v Ψ ψ U W = (2:ℝ)^((d:ℝ)*(v-a))* |
| ∫ u, Ψ u*‖∫ w, ψ w*U ((2:ℝ)^v*u,w)*conj (W ((2:ℝ)^v*u,w)) ∂μ‖^2 := by |
| dsimp [sourceAngularHFND] |
| rw [mul_assoc,graph_radial_normalizationND (by positivity : (0:ℝ)<(2:ℝ)^v)] |
|
|
| lemma sourceAngularMomentND_measurable (μ : Measure V) [SFinite μ] |
| {U : ℝ × V → ℂ} (hU : Measurable U) : Measurable (sourceAngularMomentND μ U) := |
| (hU.norm.pow_const 2).stronglyMeasurable.integral_prod_right'.measurable |
|
|
| lemma sourceAngularMomentND_nonneg (μ : Measure V) (U : ℝ × V → ℂ) (r : ℝ) : |
| 0 ≤ sourceAngularMomentND μ U r := integral_nonneg (fun _ => sq_nonneg _) |
|
|
| lemma sourceAngularMomentND_bound (μ : Measure V) [IsFiniteMeasure μ] |
| (U : ℝ × V → ℂ) {C : ℝ} (r : ℝ) (hb : ∀ᵐ w ∂μ, ‖U (r,w)‖ ≤ C) : |
| sourceAngularMomentND μ U r ≤ C^2*μ.real univ := by |
| apply (le_abs_self _).trans |
| rw [← Real.norm_eq_abs] |
| apply norm_integral_le_of_norm_le_const |
| filter_upwards [hb] with w hw |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| exact pow_le_pow_left₀ (norm_nonneg _) hw 2 |
|
|
| lemma angular_square_integrableND (μ : Measure V) [IsFiniteMeasure μ] |
| {f : V → ℂ} (hf : Measurable f) {C : ℝ} (hb : ∀ᵐ w ∂μ, ‖f w‖ ≤ C) : |
| Integrable (fun w => ‖f w‖^2) μ := by |
| apply Integrable.of_bound (hf.norm.pow_const 2).aestronglyMeasurable (C^2) |
| filter_upwards [hb] with w hw |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| exact pow_le_pow_left₀ (norm_nonneg _) hw 2 |
|
|
| lemma graph_weighted_moments_integrableND (μ : Measure V) [IsFiniteMeasure μ] |
| {U W : ℝ × V → ℂ} (hU : Measurable U) (hW : Measurable W) |
| {C D : ℝ} (hUbound : ∀ r, ∀ᵐ w ∂μ, ‖U (r,w)‖ ≤ C) |
| (hWbound : ∀ r, ∀ᵐ w ∂μ, ‖W (r,w)‖ ≤ D) |
| {Ψ : ℝ → ℝ} (hΨ : Integrable Ψ) : |
| Integrable (fun r => Ψ r*sourceAngularMomentND μ U r*sourceAngularMomentND μ W r) := by |
| apply (hΨ.norm.mul_const ((C^2*μ.real univ)*(D^2*μ.real univ))).mono' |
| ((hΨ.aestronglyMeasurable.mul (sourceAngularMomentND_measurable μ hU).aestronglyMeasurable).mul |
| (sourceAngularMomentND_measurable μ hW).aestronglyMeasurable) |
| filter_upwards with r |
| change ‖Ψ r*sourceAngularMomentND μ U r*sourceAngularMomentND μ W r‖ ≤ _ |
| rw [norm_mul,norm_mul,Real.norm_eq_abs (sourceAngularMomentND μ U r), |
| Real.norm_eq_abs (sourceAngularMomentND μ W r), |
| abs_of_nonneg (sourceAngularMomentND_nonneg μ U r), |
| abs_of_nonneg (sourceAngularMomentND_nonneg μ W r),mul_assoc] |
| exact mul_le_mul_of_nonneg_left (mul_le_mul (sourceAngularMomentND_bound μ U r (hUbound r)) |
| (sourceAngularMomentND_bound μ W r (hWbound r)) (sourceAngularMomentND_nonneg μ W r) |
| (by positivity)) (norm_nonneg _) |
|
|
| lemma sourceAngularMomentND_pair (μ : Measure V) [SFinite μ] |
| (U W : ℝ × V → ℂ) (r : ℝ) : |
| (∫ w : V × V, ‖U (r,w.1)*W (r,w.2)‖^2 ∂μ.prod μ) = |
| sourceAngularMomentND μ U r*sourceAngularMomentND μ W r := by |
| simp_rw [norm_mul,mul_pow] |
| exact integral_prod_mul (fun w => ‖U (r,w)‖^2) (fun w => ‖W (r,w)‖^2) |
|
|
| end Attempt4Near |
| end GraphLiteralSupport |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace Attempt4Near |
| open scoped BigOperators ComplexConjugate |
| theorem finset_bipartite_schur {α β : Type*} (A : Finset α) (B : Finset β) |
| (G : α → β → Prop) [DecidableRel G] (u : α → ℝ) (v : β → ℝ) (d : ℝ) |
| (hrow : ∀ a ∈ A, ((B.filter (G a)).card : ℝ) ≤ d) |
| (hcol : ∀ b ∈ B, ((A.filter (fun a => G a b)).card : ℝ) ≤ d) : |
| (∑ a ∈ A, ∑ b ∈ B, if G a b then u a*v b else 0) ≤ |
| d/2*((∑ a ∈ A, u a^2)+(∑ b ∈ B, v b^2)) := by |
| have hp (a : α) (b : β) : 2*(if G a b then u a*v b else 0) ≤ |
| (if G a b then u a^2 else 0)+(if G a b then v b^2 else 0) := by |
| split_ifs |
| · nlinarith [sq_nonneg (u a-v b)] |
| · norm_num |
| have hs := Finset.sum_le_sum (s := A) (fun a _ => |
| Finset.sum_le_sum (s := B) (fun b _ => hp a b)) |
| simp only [Finset.sum_add_distrib,← Finset.mul_sum] at hs |
| have hR : (∑ a ∈ A, ∑ b ∈ B, if G a b then u a^2 else 0) ≤ d*(∑ a ∈ A, u a^2) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro a ha |
| simp only [← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul] |
| exact mul_le_mul_of_nonneg_right (hrow a ha) (sq_nonneg _) |
| have hC : (∑ a ∈ A, ∑ b ∈ B, if G a b then v b^2 else 0) ≤ d*(∑ b ∈ B, v b^2) := by |
| rw [Finset.sum_comm,Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro b hb |
| simp only [← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul] |
| exact mul_le_mul_of_nonneg_right (hcol b hb) (sq_nonneg _) |
| nlinarith |
|
|
| lemma graph_discarded_sumND {ι : Type*} (S : Finset ι) (G : ι → Prop) |
| [DecidablePred G] (K : ι → ℂ) {ε : ℝ} |
| (hε : 0 ≤ ε) (hK : ∀ i ∈ S, ¬G i → ‖K i‖ ≤ ε) : |
| ‖(∑ i ∈ S, K i)-(∑ i ∈ S.filter G, K i)‖ ≤ S.card*ε := by |
| classical |
| have he : (∑ i ∈ S, K i)-(∑ i ∈ S.filter G, K i) = |
| ∑ i ∈ S.filter (fun i => ¬G i), K i := by |
| have h := Finset.sum_filter_add_sum_filter_not S G K |
| linear_combination -h |
| rw [he] |
| calc |
| _ ≤ ∑ i ∈ S.filter (fun i => ¬G i), ‖K i‖ := norm_sum_le _ _ |
| _ ≤ ∑ _i ∈ S.filter (fun i => ¬G i), ε := Finset.sum_le_sum |
| (fun i hi => hK i (Finset.mem_filter.mp hi).1 (Finset.mem_filter.mp hi).2) |
| _ ≤ S.card*ε := by |
| simp only [Finset.sum_const,nsmul_eq_mul] |
| exact mul_le_mul_of_nonneg_right (by exact_mod_cast Finset.card_filter_le S _) hε |
|
|
| lemma graph_complex_schurND {α β : Type*} (A : Finset α) (B : Finset β) |
| (G : α → β → Prop) [DecidableRel G] (F : α → ℂ) (H : β → ℂ) (D : ℝ) |
| (hrow : ∀ a ∈ A, ((B.filter (G a)).card : ℝ) ≤ D) |
| (hcol : ∀ b ∈ B, ((A.filter (fun a => G a b)).card : ℝ) ≤ D) : |
| ‖∑ a ∈ A, ∑ b ∈ B, if G a b then F a*conj (H b) else 0‖ ≤ |
| D/2*((∑ a ∈ A, ‖F a‖^2)+(∑ b ∈ B, ‖H b‖^2)) := by |
| calc |
| _ ≤ ∑ a ∈ A, ∑ b ∈ B, if G a b then ‖F a‖*‖H b‖ else 0 := by |
| apply (norm_sum_le _ _).trans |
| apply Finset.sum_le_sum |
| intro a _ |
| apply (norm_sum_le _ _).trans |
| apply Finset.sum_le_sum |
| intro b _ |
| split_ifs <;> simp |
| _ ≤ _ := finset_bipartite_schur A B G (fun a => ‖F a‖) (fun b => ‖H b‖) D hrow hcol |
|
|
| lemma graph_partition_schurND {α β κ : Type*} (A : Finset α) (B : Finset β) |
| (C : Finset κ) (G : κ → α → β → Prop) [∀ k, DecidableRel (G k)] |
| (F : α → ℂ) (H : β → ℂ) (η : κ → ℝ) |
| (hη : ∀ k ∈ C, 0 ≤ η k) (hpart : ∑ k ∈ C, η k=1) (D : ℝ) |
| (hrow : ∀ k ∈ C, ∀ a ∈ A, ((B.filter (G k a)).card : ℝ) ≤ D) |
| (hcol : ∀ k ∈ C, ∀ b ∈ B, ((A.filter (fun a => G k a b)).card : ℝ) ≤ D) : |
| ‖∑ k ∈ C, (η k:ℂ)*(∑ a ∈ A, ∑ b ∈ B, if G k a b then F a*conj (H b) else 0)‖ ≤ |
| D/2*((∑ a ∈ A, ‖F a‖^2)+(∑ b ∈ B, ‖H b‖^2)) := by |
| calc |
| _ ≤ ∑ k ∈ C, η k*(D/2*((∑ a ∈ A, ‖F a‖^2)+(∑ b ∈ B, ‖H b‖^2))) := by |
| apply (norm_sum_le _ _).trans |
| apply Finset.sum_le_sum |
| intro k hk |
| rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hη k hk)] |
| exact mul_le_mul_of_nonneg_left (graph_complex_schurND A B (G k) F H D |
| (hrow k hk) (hcol k hk)) (hη k hk) |
| _ = _ := by rw [← Finset.sum_mul,hpart,one_mul] |
|
|
| lemma graph_product_square_sumND {α β : Type*} (A : Finset α) (B : Finset β) |
| (U : α → ℂ) (V : β → ℂ) : |
| (∑ z ∈ A.product B, ‖U z.1*V z.2‖^2) = |
| (∑ a ∈ A, ‖U a‖^2)*(∑ b ∈ B, ‖V b‖^2) := by |
| rw [Finset.product_eq_sprod,Finset.sum_product] |
| simp_rw [norm_mul,mul_pow] |
| rw [Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro a _ |
| rw [Finset.mul_sum] |
|
|
| lemma graph_partition_product_schurND {α β κ : Type*} (A : Finset α) (B : Finset β) |
| (C : Finset κ) (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] |
| (U₁ U₂ : α → ℂ) (V₁ V₂ : β → ℂ) (η : κ → ℝ) |
| (hη : ∀ k ∈ C, 0 ≤ η k) (hpart : ∑ k ∈ C, η k=1) (D : ℝ) |
| (hrow : ∀ k ∈ C, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D) |
| (hcol : ∀ k ∈ C, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D) : |
| ‖∑ k ∈ C, (η k:ℂ)*(∑ a ∈ A.product B, ∑ b ∈ A.product B, |
| if G k a b then (U₁ a.1*V₁ a.2)*conj (U₂ b.1*V₂ b.2) else 0)‖ ≤ |
| D/2*((∑ a ∈ A, ‖U₁ a‖^2)*(∑ b ∈ B, ‖V₁ b‖^2)+ |
| (∑ a ∈ A, ‖U₂ a‖^2)*(∑ b ∈ B, ‖V₂ b‖^2)) := by |
| simpa only [graph_product_square_sumND] using graph_partition_schurND |
| (A.product B) (A.product B) C G (fun z => U₁ z.1*V₁ z.2) (fun z => U₂ z.1*V₂ z.2) |
| η hη hpart D hrow hcol |
|
|
| lemma graph_pair_partitionND {κ : Type*} (C : Finset κ) (η₁ η₂ : κ → ℝ) |
| (h₁ : ∑ k ∈ C, η₁ k=1) (h₂ : ∑ k ∈ C, η₂ k=1) : |
| ∑ k ∈ C.product C, η₁ k.1*η₂ k.2=1 := by |
| rw [Finset.product_eq_sprod,Finset.sum_product,← Finset.sum_mul_sum,h₁,h₂,one_mul] |
|
|
| end Attempt4Near |
| end GraphLiteralSupport |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace Attempt4Near |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
| variable {V α β κ : Type*} [MeasurableSpace V] |
|
|
| noncomputable def graphChildSquareND (A : Finset α) (B : Finset β) |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) (r : ℝ) (z : V × V) : ℝ := |
| (∑ i ∈ A, ‖U i (r,z.1)‖^2)*(∑ j ∈ B, ‖W j (r,z.2)‖^2) |
|
|
| lemma graphChildSquareND_integrable (μ : Measure V) [IsFiniteMeasure μ] |
| (A : Finset α) (B : Finset β) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j)) |
| (C : α → ℝ) (D : β → ℝ) |
| (hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i) |
| (hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ D j) (r : ℝ) : |
| Integrable (graphChildSquareND A B U W r) (μ.prod μ) := by |
| unfold graphChildSquareND |
| apply Integrable.mul_prod (f := fun w => ∑ i ∈ A, ‖U i (r,w)‖^2) |
| (g := fun w => ∑ j ∈ B, ‖W j (r,w)‖^2) |
| · exact integrable_finsetSum A (fun i hi => angular_square_integrableND μ |
| ((hU i hi).comp (measurable_const.prodMk measurable_id)) (hbU i hi r)) |
| · exact integrable_finsetSum B (fun j hj => angular_square_integrableND μ |
| ((hW j hj).comp (measurable_const.prodMk measurable_id)) (hbW j hj r)) |
|
|
| lemma graphChildSquareND_integral (μ : Measure V) [IsFiniteMeasure μ] |
| (A : Finset α) (B : Finset β) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j)) |
| (C : α → ℝ) (D : β → ℝ) |
| (hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i) |
| (hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ D j) (r : ℝ) : |
| (∫ z, graphChildSquareND A B U W r z ∂μ.prod μ) = |
| ∑ i ∈ A, ∑ j ∈ B, sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by |
| dsimp [graphChildSquareND] |
| rw [integral_prod_mul (fun w => ∑ i ∈ A, ‖U i (r,w)‖^2) |
| (fun w => ∑ j ∈ B, ‖W j (r,w)‖^2),integral_finsetSum A (f := fun i w => ‖U i (r,w)‖^2) (fun i hi => angular_square_integrableND μ |
| ((hU i hi).comp (measurable_const.prodMk measurable_id)) (hbU i hi r)), |
| integral_finsetSum B (f := fun j w => ‖W j (r,w)‖^2) (fun j hj => angular_square_integrableND μ |
| ((hW j hj).comp (measurable_const.prodMk measurable_id)) (hbW j hj r))] |
| exact Finset.sum_mul_sum A B _ _ |
|
|
| noncomputable def graphRetainedCrossedND (A : Finset α) (B : Finset β) (K : Finset κ) |
| (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] |
| (η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (r : ℝ) (z : V × V) : ℂ := |
| ∑ k ∈ K, (η k z:ℂ)*(∑ a ∈ A.product B, ∑ b ∈ A.product B, |
| if G k a b then (U a.1 (r,z.1)*W a.2 (r,z.2))* |
| conj (U b.1 (r,z.2)*W b.2 (r,z.1)) else 0) |
|
|
| lemma graphRetainedCrossedND_angular_bound (μ : Measure V) [IsFiniteMeasure μ] |
| (A : Finset α) (B : Finset β) (K : Finset κ) |
| (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] |
| (η : κ → V × V → ℝ) |
| (hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1) |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j)) |
| (C : α → ℝ) (E : β → ℝ) |
| (hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i) |
| (hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ) |
| (hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D) |
| (hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D) |
| (r : ℝ) : |
| ‖∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ‖ ≤ |
| D*∑ i ∈ A, ∑ j ∈ B, sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by |
| have hi := graphChildSquareND_integrable μ A B U W hU hW C E hbU hbW r |
| have hb := norm_integral_le_of_norm_le ((hi.add hi.swap).const_mul (D/2)) |
| (show ∀ᵐ z ∂μ.prod μ, ‖graphRetainedCrossedND A B K G η U W r z‖ ≤ |
| D/2*(graphChildSquareND A B U W r z+graphChildSquareND A B U W r z.swap) from by |
| filter_upwards [hpart] with z hz |
| exact graph_partition_product_schurND A B K G |
| (fun i => U i (r,z.1)) (fun i => U i (r,z.2)) |
| (fun j => W j (r,z.2)) (fun j => W j (r,z.1)) (fun k => η k z) hz.1 hz.2 D hrow hcol) |
| simp only [Pi.add_apply,Function.comp_apply] at hb |
| rw [integral_const_mul] at hb |
| have hadd := integral_add hi hi.swap |
| simp only [Function.comp_apply] at hadd |
| rw [hadd,integral_prod_swap] at hb |
| rw [graphChildSquareND_integral μ A B U W hU hW C E hbU hbW] at hb |
| convert hb using 1 |
| ring |
|
|
| theorem graphRetainedCrossedND_integral_bound (μ : Measure V) [IsFiniteMeasure μ] |
| (A : Finset α) (B : Finset β) (K : Finset κ) |
| (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] |
| (η : κ → V × V → ℝ) |
| (hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1) |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j)) |
| (C : α → ℝ) (E : β → ℝ) |
| (hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i) |
| (hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ) |
| (hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D) |
| (hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D) |
| {Ψ : ℝ → ℝ} (hΨ : Integrable Ψ) (hΨpos : ∀ r, 0 ≤ Ψ r) : |
| ‖∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ‖ ≤ |
| D*∑ i ∈ A, ∑ j ∈ B, ∫ r, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by |
| have hi (i) (hi : i ∈ A) (j) (hj : j ∈ B) := |
| graph_weighted_moments_integrableND μ (hU i hi) (hW j hj) (hbU i hi) (hbW j hj) hΨ |
| have hs := (integrable_finsetSum A (fun i hiA => integrable_finsetSum B |
| (fun j hjB => hi i hiA j hjB))).const_mul D |
| have hh := norm_integral_le_of_norm_le hs |
| (show ∀ᵐ r : ℝ, ‖(Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ‖ ≤ |
| D*∑ i ∈ A, ∑ j ∈ B, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r from by |
| filter_upwards with r |
| rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hΨpos r)] |
| have hp := mul_le_mul_of_nonneg_left (graphRetainedCrossedND_angular_bound |
| μ A B K G η hpart U W hU hW C E hbU hbW D hrow hcol r) (hΨpos r) |
| simpa only [Finset.mul_sum,mul_assoc,mul_left_comm,mul_comm] using hp) |
| rw [integral_const_mul,integral_finsetSum A (fun i hiA => integrable_finsetSum B |
| (fun j hjB => hi i hiA j hjB))] at hh |
| apply hh.trans_eq |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i hiA |
| exact integral_finsetSum B (fun j hjB => hi i hiA j hjB) |
| end Attempt4Near |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
