| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphDegrees |
|
|
| namespace OAI |
|
|
| /-! Graph energies and pair estimates. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| 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 |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace Attempt4Near |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
| variable {V α β κ : Type*} [MeasurableSpace V] |
| noncomputable def graphRetainedDirectND (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.1)*W b.2 (r,z.2)) else 0) |
|
|
| lemma graphRetainedDirectND_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, graphRetainedDirectND 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.const_mul D) |
| (show ∀ᵐ z ∂μ.prod μ, ‖graphRetainedDirectND A B K G η U W r z‖ ≤ |
| D*graphChildSquareND A B U W r z from by |
| filter_upwards [hpart] with z hz |
| have hh := graph_partition_product_schurND A B K G |
| (fun i => U i (r,z.1)) (fun i => U i (r,z.1)) |
| (fun j => W j (r,z.2)) (fun j => W j (r,z.2)) (fun k => η k z) hz.1 hz.2 D hrow hcol |
| convert hh using 1 |
| · rfl |
| · dsimp [graphChildSquareND] |
| ring) |
| rw [integral_const_mul,graphChildSquareND_integral μ A B U W hU hW C E hbU hbW] at hb |
| exact hb |
|
|
| theorem graphRetainedDirectND_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, graphRetainedDirectND 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, graphRetainedDirectND 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 (graphRetainedDirectND_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 |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace Attempt4Near |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
| variable {V α β κ : Type*} [MeasurableSpace V] |
|
|
| lemma graph_four_sumND (A : Finset α) (B : Finset β) |
| (U₁ U₂ : α → ℂ) (W₁ W₂ : β → ℂ) : |
| (∑ a ∈ A.product B, ∑ b ∈ A.product B, |
| (U₁ a.1*W₁ a.2)*conj (U₂ b.1*W₂ b.2)) = |
| ((∑ i ∈ A, U₁ i)*(∑ j ∈ B, W₁ j))*conj ((∑ i ∈ A, U₂ i)*(∑ j ∈ B, W₂ j)) := by |
| rw [← Finset.sum_mul_sum,← map_sum] |
| simp_rw [Finset.product_eq_sprod,Finset.sum_product,← Finset.sum_mul_sum] |
|
|
| omit [MeasurableSpace V] in |
| lemma graphRetainedDirectND_complete (A : Finset α) (B : Finset β) (K : Finset κ) |
| (η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (r : ℝ) (z : V × V) (hpart : ∑ k ∈ K, η k z=1) : |
| graphRetainedDirectND A B K (fun _ _ _ => True) η U W r z = |
| ((‖∑ i ∈ A, U i (r,z.1)‖^2*‖∑ j ∈ B, W j (r,z.2)‖^2:ℝ):ℂ) := by |
| simp only [graphRetainedDirectND,ite_true] |
| rw [← Finset.sum_mul,← Complex.ofReal_sum,hpart,Complex.ofReal_one,one_mul, |
| graph_four_sumND A B (fun i => U i (r,z.1)) (fun i => U i (r,z.1)) |
| (fun j => W j (r,z.2)) (fun j => W j (r,z.2)), |
| Complex.mul_conj,Complex.normSq_eq_norm_sq,norm_mul,mul_pow] |
|
|
| omit [MeasurableSpace V] in |
| lemma graphRetainedCrossedND_complete (A : Finset α) (B : Finset β) (K : Finset κ) |
| (η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (r : ℝ) (z : V × V) (hpart : ∑ k ∈ K, η k z=1) : |
| graphRetainedCrossedND A B K (fun _ _ _ => True) η U W r z = |
| ((∑ i ∈ A, U i (r,z.1))*conj (∑ j ∈ B, W j (r,z.1)))* |
| conj ((∑ i ∈ A, U i (r,z.2))*conj (∑ j ∈ B, W j (r,z.2))) := by |
| simp only [graphRetainedCrossedND,ite_true] |
| rw [← Finset.sum_mul,← Complex.ofReal_sum,hpart,Complex.ofReal_one,one_mul] |
| rw [graph_four_sumND A B (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))] |
| simp only [map_mul,starRingEnd_self_apply] |
| ring |
|
|
| lemma graphExpandedDirectND_integral (μ : Measure V) [SFinite μ] |
| (A : Finset α) (B : Finset β) (K : Finset κ) |
| (η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (r : ℝ) (hpart : ∀ᵐ z ∂μ.prod μ, ∑ k ∈ K, η k z=1) : |
| (∫ z, graphRetainedDirectND A B K (fun _ _ _ => True) η U W r z ∂μ.prod μ) = |
| ((sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r* |
| sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r:ℝ):ℂ) := by |
| calc |
| _ = ∫ z : V × V, ((‖∑ i ∈ A, U i (r,z.1)‖^2*‖∑ j ∈ B, W j (r,z.2)‖^2:ℝ):ℂ) |
| ∂μ.prod μ := integral_congr_ae (by |
| filter_upwards [hpart] with z hz |
| exact graphRetainedDirectND_complete A B K η U W r z hz) |
| _ = _ := by |
| rw [integral_complex_ofReal,integral_prod_mul |
| (fun w => ‖∑ i ∈ A, U i (r,w)‖^2) (fun w => ‖∑ j ∈ B, W j (r,w)‖^2)] |
| rfl |
|
|
| lemma graphExpandedCrossedND_integral (μ : Measure V) [SFinite μ] |
| (A : Finset α) (B : Finset β) (K : Finset κ) |
| (η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) |
| (r : ℝ) (hpart : ∀ᵐ z ∂μ.prod μ, ∑ k ∈ K, η k z=1) : |
| (∫ z, graphRetainedCrossedND A B K (fun _ _ _ => True) η U W r z ∂μ.prod μ) = |
| ((‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2:ℝ):ℂ) := by |
| calc |
| _ = ∫ z : V × V, ((∑ i ∈ A, U i (r,z.1))*conj (∑ j ∈ B, W j (r,z.1)))* |
| conj ((∑ i ∈ A, U i (r,z.2))*conj (∑ j ∈ B, W j (r,z.2))) ∂μ.prod μ := |
| integral_congr_ae (by |
| filter_upwards [hpart] with z hz |
| exact graphRetainedCrossedND_complete A B K η U W r z hz) |
| _ = _ := by |
| rw [integral_prod_mul |
| (fun w => (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w))) |
| (fun w => conj ((∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)))), |
| integral_conj,Complex.mul_conj,Complex.normSq_eq_norm_sq] |
|
|
| end Attempt4Near |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace Attempt4Near |
| open MeasureTheory |
| open scoped BigOperators |
| variable {Z α β κ : Type*} [MeasurableSpace Z] |
|
|
| lemma graph_integral_sum_truncateND (μ : Measure Z) (S : Finset α) |
| (G : α → Prop) [DecidablePred G] (F : α → Z → ℂ) |
| (hi : ∀ i ∈ S, Integrable (F i) μ) {ε : ℝ} (hε : 0 ≤ ε) |
| (hne : ∀ i ∈ S, ¬G i → ‖∫ z, F i z ∂μ‖ ≤ ε) : |
| ‖(∫ z, ∑ i ∈ S, F i z ∂μ)-(∫ z, ∑ i ∈ S, if G i then F i z else 0 ∂μ)‖ ≤ |
| S.card*ε := by |
| have hit (i : α) (hiS : i ∈ S) : Integrable (fun z => if G i then F i z else 0) μ := by |
| by_cases h : G i |
| · simpa only [h,ite_true] using hi i hiS |
| · simp only [h,ite_false]; exact integrable_zero _ _ _ |
| rw [integral_finsetSum S hi,integral_finsetSum S hit] |
| have he : (∑ i ∈ S, ∫ z, if G i then F i z else 0 ∂μ) = |
| ∑ i ∈ S.filter G, ∫ z, F i z ∂μ := by |
| rw [Finset.sum_filter] |
| apply Finset.sum_congr rfl |
| intro i _ |
| by_cases h : G i <;> simp [h] |
| rw [he] |
| exact graph_discarded_sumND S G (fun i => ∫ z, F i z ∂μ) hε hne |
|
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| lemma graph_integral_localizationND (μ : Measure Z) (A : Finset α) (B : Finset β) (K : Finset κ) |
| (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] |
| (F : κ → (α × β) → (α × β) → Z → ℂ) |
| (hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, Integrable (F k a b) μ) |
| {ε : ℝ} (hε : 0 ≤ ε) |
| (hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, |
| ¬G k a b → ‖∫ z, F k a b z ∂μ‖ ≤ ε) : |
| ‖(∫ z, ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, F k a b z ∂μ)- |
| (∫ z, ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, |
| if G k a b then F k a b z else 0 ∂μ)‖ ≤ |
| (K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε := by |
| let S := K.product ((A.product B).product (A.product B)) |
| have h := graph_integral_sum_truncateND μ S |
| (fun z => G z.1 z.2.1 z.2.2) (fun z => F z.1 z.2.1 z.2.2) |
| (fun z hz => hi z.1 (Finset.mem_product.mp hz).1 |
| z.2.1 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).1 |
| z.2.2 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).2) hε |
| (fun z hz => hne z.1 (Finset.mem_product.mp hz).1 |
| z.2.1 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).1 |
| z.2.2 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).2) |
| simpa only [S,Finset.product_eq_sprod,Finset.sum_product,Finset.card_product, |
| Nat.cast_mul,pow_two,mul_assoc] using h |
|
|
| end Attempt4Near |
| end |
| end |
|
|
| end OAI |
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|
