openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Rotation /CircleRationalGraphFields.lean
| import OAI.Analysis.IrrationalRotation.Rotation.CircleRationalGraphApproximants | |
| import OAI.Analysis.IrrationalRotation.Operators.ProjectionFieldRepresentative | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| namespace Invariant233.CircleParameterData | |
| theorem rationalGraphIdentities_congr_ae (p : CircleParameterData) (θ : ℝ) | |
| {P Q : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ} | |
| (hPQ : P =ᵐ[circleMeasure] Q) : | |
| p.RationalGraphIdentities θ P ↔ p.RationalGraphIdentities θ Q := by | |
| have htrans : ∀ᵐ x ∂circleMeasure, ∀ (A : ℕ) (r : ℤ), | |
| P (x + p.rationalTranslation A r) = Q (x + p.rationalTranslation A r) := by | |
| apply ae_all_iff.mpr | |
| intro A | |
| apply ae_all_iff.mpr | |
| intro r | |
| have hmp : MeasurePreserving (fun x : Circle => x + p.rationalTranslation A r) | |
| circleMeasure circleMeasure := by | |
| change MeasurePreserving _ volume volume | |
| exact measurePreserving_add_right volume _ | |
| exact hmp.quasiMeasurePreserving.ae (p := fun x => P x = Q x) hPQ | |
| constructor | |
| · intro h | |
| filter_upwards [h, hPQ, htrans] with x hx he hs | |
| intro A hA r | |
| rw [← he, ← hs A r] | |
| exact hx A hA r | |
| · intro h | |
| filter_upwards [h, hPQ, htrans] with x hx he hs | |
| intro A hA r | |
| rw [he, hs A r] | |
| exact hx A hA r | |
| end Invariant233.CircleParameterData | |
| namespace Invariant233.CircleParameterLaws | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem rationalGraphIdentities_of_ae_positive_orbits | |
| (P : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ) | |
| (hP : ∀ᵐ x ∂circleMeasure, IsStarProjection (P x)) | |
| (hPm : ∀ k : ℤ, Measurable (fun x => P x (lp.single 2 k 1))) | |
| (horbit : ∀ n : ℕ, 0 < n → ∀ᵐ x ∂circleMeasure, | |
| (1 - P (CircleWeights.rotate θ (n : ℤ) x)).comp | |
| ((ratioDiagonal (p.floorCocycle θ) | |
| (x, CircleWeights.rotate θ (n : ℤ) x)).comp (P x)) = 0) : | |
| p.RationalGraphIdentities θ P := by | |
| obtain ⟨Q, hQ, hQm, hQP⟩ := Fields.exists_everywhere_projection_field | |
| circleMeasure P hP hPm | |
| apply (p.rationalGraphIdentities_congr_ae θ hQP).mp | |
| apply hp.rationalGraphIdentities_of_positive_orbits Q hQ hQm | |
| intro n hn | |
| have hmp : MeasurePreserving (CircleWeights.rotate θ (n : ℤ)) | |
| circleMeasure circleMeasure := by | |
| change MeasurePreserving (fun x : Circle => x + (n : ℤ) • (θ : Circle)) volume volume | |
| exact measurePreserving_add_right volume _ | |
| have hrot := hmp.quasiMeasurePreserving.ae (p := fun x => Q x = P x) hQP | |
| filter_upwards [hQP, hrot, horbit n hn] with x he hr hx | |
| rw [he, hr] | |
| exact hx | |
| end Invariant233.CircleParameterLaws | |
| end | |
| end OAI | |