openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphFiniteCutoffEstimate.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFiniteDirectEstimate | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.AngularMomentLe | |
| 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 graphFiniteCutoffEstimateND (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} (hD : 0 ≤ D) | |
| (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) | |
| {ψ : V → ℂ} (hmψ : Measurable ψ) (hψ : ∀ᵐ w ∂μ, ‖ψ w‖ ≤ 1) | |
| (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)* | |
| (((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))* | |
| conj ((ψ p.2.2*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)* | |
| (((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))* | |
| conj ((ψ p.2.2*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, ψ 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 | |
| have hUb (i) (hi : i ∈ A) (r : ℝ) : ∀ᵐ w ∂μ, ‖ψ w*U i (r,w)‖ ≤ C i := by | |
| filter_upwards [hψ,hbU i hi r] with w hw hu | |
| rw [norm_mul] | |
| exact (mul_le_mul_of_nonneg_right hw (norm_nonneg _)).trans (by simpa using hu) | |
| have h := graphFiniteCrossedEstimateND μ A B K G η hpart (fun i z => ψ z.2*U i z) | |
| W (fun i hi => (hmψ.comp measurable_snd).mul (hU i hi)) hW C E hUb hbW D | |
| hrow hcol hΨ hΨpos hi hε hne | |
| simp_rw [← Finset.mul_sum] at h | |
| apply h.trans | |
| apply add_le_add _ le_rfl | |
| apply mul_le_mul_of_nonneg_left _ hD | |
| apply Finset.sum_le_sum | |
| intro i hi | |
| apply Finset.sum_le_sum | |
| intro j hj | |
| apply integral_mono_of_nonneg (Filter.Eventually.of_forall (fun r => mul_nonneg | |
| (mul_nonneg (hΨpos r) (sourceAngularMomentND_nonneg μ _ r)) (sourceAngularMomentND_nonneg μ _ r))) | |
| (graph_weighted_moments_integrableND μ (hU i hi) (hW j hj) (hbU i hi) (hbW j hj) hΨ) | |
| filter_upwards with r | |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left | |
| (sourceAngularMomentND_cutoff_le μ (hU i hi) (hbU i hi) ψ hψ r) (hΨpos r)) | |
| (sourceAngularMomentND_nonneg μ _ r) | |
| 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 graphFiniteHND_bound (d : ℕ) (a p v lam₁ lam₂ : ℝ) | |
| (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 (fun r => Ψ (r/(2:ℝ)^v))) (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/(2:ℝ)^v):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/(2:ℝ)^v):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) : | |
| sourceAngularHND d μ a v Ψ (fun z => lam₁ • ∑ i ∈ A, U i z) | |
| (fun z => lam₂ • ∑ j ∈ B, W j z) ≤ | |
| lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D* | |
| (∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ p v Ψ (U i) (W j)) + | |
| lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹* | |
| ((K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε) := by | |
| have h := graphFiniteDirectEstimateND μ A B K G η hpart U W hU hW C E hbU hbW | |
| D hrow hcol hΨ (fun r => hΨpos (r/(2:ℝ)^v)) hi hε hne | |
| have he : (2:ℝ)^((d:ℝ)*(p-a))*(∑ i ∈ A, ∑ j ∈ B, | |
| sourceAngularHND d μ p v Ψ (U i) (W j)) = | |
| ∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ a v Ψ (U i) (W j) := by | |
| simp_rw [Finset.mul_sum,← sourceAngularHND_depth d μ a p v Ψ] | |
| have hm := mul_le_mul_of_nonneg_left h | |
| (show 0 ≤ lam₁^2*lam₂^2*((2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹) by positivity) | |
| rw [sourceAngularHND_smul] | |
| have hr : lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D* | |
| (∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ p v Ψ (U i) (W j)) = | |
| lam₁^2*lam₂^2*D*(∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ a v Ψ (U i) (W j)) := by | |
| rw [← he] | |
| ring | |
| rw [hr] | |
| simp only [sourceAngularHND,← Finset.mul_sum] at * | |
| nlinarith only [hm] | |
| end Attempt4Near | |
| end | |
| end | |
| end OAI | |