openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ProductBiasedIdentityFromTriangular.lean
| import OAI.Analysis.IrrationalRotation.Sampling.PrefixReferenceCompletion | |
| import OAI.Analysis.IrrationalRotation.Couplings.ProductBiasedPrefixTransfer | |
| import OAI.Analysis.IrrationalRotation.Couplings.BlockProductCoupling | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped BigOperators | |
| namespace Invariant233.OrderedProductCoupling | |
| theorem circlePairLaw_pairIdentity_of_triangular | |
| (P : Parameters) (hsum : Summable (energyBound P)) | |
| (θ c : ℕ → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ) | |
| (hc : ∀ i, 0 ≤ c i) | |
| (F : ProductModel.Base → ProductModel.Fiber →L[ℂ] ProductModel.Fiber) | |
| (hF : ∀ x, IsStarProjection (F x)) | |
| (hFm : ∀ i : ℤ, Measurable (fun x => F x (lp.single 2 i 1))) | |
| (htri : ∀ I : Finset ℕ, ∀ᵐ xy ∂ProductModel.triangularLaw I, | |
| ProductModel.PairIdentity θ c F xy) : | |
| ∀ᵐ xy ∂circlePairLaw P, ProductModel.PairIdentity θ c F xy := | |
| circlePairLaw_pairIdentity_of_triangular_domination P hsum θ c hθ0 hθ hc | |
| (circle_prefix_absolutelyContinuous_triangular_sum P hsum) F hF hFm htri | |
| theorem circlePairLaw_pairIdentity_of_covariantField | |
| (P : Parameters) (hsum : Summable (energyBound P)) | |
| (θ c : ℕ → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ) | |
| (hc : ∀ i, 0 ≤ c i) | |
| (A : ProductModel.Hilbert →L[ℂ] ProductModel.Hilbert) | |
| (F : ProductModel.Base → ProductModel.Fiber →L[ℂ] ProductModel.Fiber) | |
| (hrep : ProductModel.IsCovariantFieldRepresentation θ A F 1) | |
| (hF : ∀ x, IsStarProjection (F x)) | |
| (htri : ∀ I : Finset ℕ, ∀ᵐ xy ∂ProductModel.triangularLaw I, | |
| ProductModel.PairIdentity θ c F xy) : | |
| ∀ᵐ xy ∂circlePairLaw P, ProductModel.PairIdentity θ c F xy := | |
| circlePairLaw_pairIdentity_of_triangular P hsum θ c hθ0 hθ hc F hF | |
| (fun i => (hrep.2.2.1 (lp.single 2 i 1)).measurable) htri | |
| theorem source_circlePairLaw_pairIdentity_of_triangular | |
| (m : ℕ → ℕ) (a b : ℕ → ℝ) (hm : ∀ s, 100 ≤ m s) | |
| (ha : ∀ s, 1 / (m s : ℝ) ≤ a s) (ha' : ∀ s, a s ≤ 1 / 4) | |
| (hb : ∀ s, 1 / (m s : ℝ) ≤ b s) (hb' : ∀ s, b s ≤ 1 / 4) | |
| (e : ℕ ≃ Σ s, Fin (m s)) | |
| (θ c : ℕ → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ) | |
| (hc : ∀ i, 0 ≤ c i) | |
| (F : ProductModel.Base → ProductModel.Fiber →L[ℂ] ProductModel.Fiber) | |
| (hF : ∀ x, IsStarProjection (F x)) | |
| (hFm : ∀ i : ℤ, Measurable (fun x => F x (lp.single 2 i 1))) | |
| (htri : ∀ I : Finset ℕ, ∀ᵐ xy ∂ProductModel.triangularLaw I, | |
| ProductModel.PairIdentity θ c F xy) : | |
| ∀ tag : ℕ, ∀ᵐ xy ∂circlePairLaw | |
| (ofBlocks m sourceKappa a b sourceKappa_nonneg sourceKappa_le hm ha ha' hb hb' e tag), | |
| ProductModel.PairIdentity θ c F xy := by | |
| intro tag | |
| exact circlePairLaw_pairIdentity_of_triangular _ | |
| (ofBlocks_energy_sum m sourceKappa a b sourceKappa_nonneg sourceKappa_le hm | |
| ha ha' hb hb' e tag summable_sourceKappa_sq).1 θ c hθ0 hθ hc F hF hFm htri | |
| end Invariant233.OrderedProductCoupling | |
| end | |
| end OAI | |