openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphRetainedDirect.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.AngularMoment | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFiniteDirectEstimate | |
| 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 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 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] | |
| 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 GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| 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 | |
| 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 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] | |
| theorem graphFiniteDirectEstimateND (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) | |
| (hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, | |
| Integrable (fun p : ℝ × (V × V) => (Ψ p.1:ℂ)*(η k p.2:ℂ)* | |
| ((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))* | |
| conj (U b.1 (p.1,p.2.1)*W b.2 (p.1,p.2.2)))) (volume.prod (μ.prod μ))) | |
| {ε : ℝ} (hε : 0 ≤ ε) | |
| (hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, ¬G k a b → | |
| ‖∫ p : ℝ × (V × V), (Ψ p.1:ℂ)*(η k p.2:ℂ)* | |
| ((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))* | |
| conj (U b.1 (p.1,p.2.1)*W b.2 (p.1,p.2.2))) ∂volume.prod (μ.prod μ)‖ ≤ ε) : | |
| (∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r* | |
| sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r) ≤ | |
| D*∑ i ∈ A, ∑ j ∈ B, (∫ r, Ψ r*sourceAngularMomentND μ (U i) r* | |
| sourceAngularMomentND μ (W j) r) + (K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε := by | |
| let F := fun (k : κ) (a b : α × β) (p : ℝ × (V × V)) => (Ψ p.1:ℂ)*(η k p.2:ℂ)* | |
| ((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))*conj (U b.1 (p.1,p.2.1)*W b.2 (p.1,p.2.2))) | |
| have hiF : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, | |
| Integrable (F k a b) (volume.prod (μ.prod μ)) := hi | |
| have hiR : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, | |
| Integrable (fun p => if G k a b then F k a b p else 0) (volume.prod (μ.prod μ)) := by | |
| intro k hk a ha b hb | |
| by_cases h : G k a b | |
| · simpa only [h,ite_true] using hiF k hk a ha b hb | |
| · simp only [h,ite_false]; exact integrable_zero _ _ _ | |
| have heF : (fun p => ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, F k a b p) = | |
| fun p => (Ψ p.1:ℂ)*graphRetainedDirectND A B K (fun _ _ _ => True) η U W p.1 p.2 := by | |
| funext p | |
| simp only [F,graphRetainedDirectND,ite_true,Finset.mul_sum,mul_assoc] | |
| have heR : (fun p => ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, | |
| if G k a b then F k a b p else 0) = | |
| fun p => (Ψ p.1:ℂ)*graphRetainedDirectND A B K G η U W p.1 p.2 := by | |
| funext p | |
| simp only [F,graphRetainedDirectND,Finset.mul_sum,mul_ite,mul_zero,mul_assoc] | |
| have hintF := integrable_finsetSum K (fun k hk => integrable_finsetSum (A.product B) | |
| (fun a ha => integrable_finsetSum (A.product B) (fun b hb => hiF k hk a ha b hb))) | |
| have hintR := integrable_finsetSum K (fun k hk => integrable_finsetSum (A.product B) | |
| (fun a ha => integrable_finsetSum (A.product B) (fun b hb => hiR k hk a ha b hb))) | |
| rw [heF] at hintF | |
| rw [heR] at hintR | |
| have hl := graph_integral_localizationND (volume.prod (μ.prod μ)) A B K G F hiF hε hne | |
| rw [heF,heR,integral_prod _ hintF,integral_prod _ hintR] at hl | |
| simp_rw [integral_const_mul] at hl | |
| have he (r : ℝ) := graphExpandedDirectND_integral μ A B K η U W r | |
| (hpart.mono (fun _ h => h.2)) | |
| simp_rw [he] at hl | |
| simp only [← Complex.ofReal_mul] at hl | |
| rw [integral_complex_ofReal] at hl | |
| have hb := graphRetainedDirectND_integral_bound μ A B K G η hpart U W hU hW C E hbU hbW | |
| D hrow hcol hΨ hΨpos | |
| have hnon : 0 ≤ ∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r* | |
| sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r := integral_nonneg (fun r => | |
| mul_nonneg (mul_nonneg (hΨpos r) (sourceAngularMomentND_nonneg _ _ _)) | |
| (sourceAngularMomentND_nonneg _ _ _)) | |
| have htt : ‖((∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r* | |
| sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r : ℝ):ℂ)‖ ≤ ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ)‖ + ‖((∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r* | |
| sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ)‖ := by | |
| calc | |
| _ = ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ)+(((∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r* | |
| sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ))‖ := by congr 1; abel | |
| _ ≤ _ := norm_add_le _ _ | |
| simp only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hnon] at htt | |
| exact htt.trans (add_le_add hb (by simpa only [mul_assoc] using hl)) | |
| end Attempt4Near | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |