File size: 3,058 Bytes
9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | 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
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