openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphFiniteDirectEstimate.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphChildSquare | |
| 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] | |
| 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 | |
| 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] | |
| theorem graphFiniteCrossedEstimateND (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.2.2)*W b.2 (p.1,p.2.1)))) (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.2.2)*W b.2 (p.1,p.2.1))) ∂volume.prod (μ.prod μ)‖ ≤ ε) : | |
| (∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2) ≤ | |
| 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.2.2)*W b.2 (p.1,p.2.1))) | |
| 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)*graphRetainedCrossedND A B K (fun _ _ _ => True) η U W p.1 p.2 := by | |
| funext p | |
| simp only [F,graphRetainedCrossedND,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)*graphRetainedCrossedND A B K G η U W p.1 p.2 := by | |
| funext p | |
| simp only [F,graphRetainedCrossedND,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) := graphExpandedCrossedND_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 := graphRetainedCrossedND_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*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 := integral_nonneg (fun r => | |
| mul_nonneg (hΨpos r) (sq_nonneg _)) | |
| have htt : ‖((∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 : ℝ):ℂ)‖ ≤ ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ)‖ + ‖((∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ)‖ := by | |
| calc | |
| _ = ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ)+(((∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND 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 | |
| end | |
| end OAI | |