openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /ContinuousProjectionTransport.lean
| import OAI.Analysis.IrrationalRotation.Rotation.ContinuousWeightOperatorBridge | |
| import OAI.Analysis.IrrationalRotation.Rotation.RotationProjectionTransport | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Invariant233.CircleParameterLaws | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem continuousWeight_hasOnlyTrivialInvariantProjections | |
| (d : BackgroundBumpData θ) | |
| (hstar : HasOnlyTrivialInvariantProjections (Rotation.algebra θ) | |
| (Rotation.U * Rotation.measurableRepresentation θ | |
| (fun x => (p.measurableWeight θ x : ℂ)) 1 zero_le_one | |
| hp.measurable_complexParameterWeight hp.norm_complexParameterWeight_le_one)) : | |
| HasOnlyTrivialInvariantProjections (Rotation.algebra θ) | |
| (Rotation.weightedReal θ | |
| ⟨p.continuousWeight θ d, (hp.isContinuousCircleWeight_continuousWeight d).1⟩) := by | |
| let f : C(Circle, ℝ) := | |
| ⟨p.continuousWeight θ d, (hp.isContinuousCircleWeight_continuousWeight d).1⟩ | |
| let H := Rotation.exponentialTransfer θ (p.continuousTransfer θ d) | |
| let Tstar := Rotation.U * Rotation.measurableRepresentation θ | |
| (fun x => (p.measurableWeight θ x : ℂ)) 1 zero_le_one | |
| hp.measurable_complexParameterWeight hp.norm_complexParameterWeight_le_one | |
| have h := hp.continuousWeight_natural_intertwining d | |
| let hD : ∀ ξ : H.domain, Rotation.weightedReal θ f ξ.val ∈ H.domain := | |
| fun ξ => (h ξ).choose | |
| have hI : ∀ ξ : H.domain, H ⟨Rotation.weightedReal θ f ξ.val, hD ξ⟩ = | |
| (Real.exp (-1) : ℂ) • Tstar (H ξ) := fun ξ => (h ξ).choose_spec | |
| exact Rotation.hasOnlyTrivialInvariantProjections_of_scaled_exponentialTransfer θ | |
| (measurable_continuousTransfer d) (Rotation.weightedReal θ f) Tstar | |
| (Real.exp (-1) : ℂ) (Complex.ofReal_ne_zero.mpr (Real.exp_ne_zero _)) hD hI hstar | |
| end Invariant233.CircleParameterLaws | |
| end | |
| end OAI | |