openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Rotation /CircleOrderedSharpness.lean
| import OAI.Analysis.IrrationalRotation.Rotation.CircleOrderedHaarRigidity | |
| import OAI.Analysis.IrrationalRotation.Analysis.RationalGraphTrimming | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| namespace Invariant233.CircleOrderedRigidity | |
| def rationalPeriodGraph (p : CircleParameterData) : Measure (Circle × Circle) := | |
| circleMeasure.map (fun x : Circle => | |
| (x, x + (((1 : ℝ) / p.q 1 : ℝ) : Circle))) | |
| theorem measurable_rationalPeriodPair (p : CircleParameterData) : | |
| Measurable (fun x : Circle => | |
| (x, x + (((1 : ℝ) / p.q 1 : ℝ) : Circle))) := by | |
| fun_prop | |
| theorem rationalPeriodGraph_fst (p : CircleParameterData) : | |
| (rationalPeriodGraph p).map Prod.fst = circleMeasure := by | |
| rw [rationalPeriodGraph, Measure.map_map measurable_fst | |
| (measurable_rationalPeriodPair p)] | |
| exact Measure.map_id | |
| theorem rationalPeriodGraph_snd (p : CircleParameterData) : | |
| (rationalPeriodGraph p).map Prod.snd = circleMeasure := by | |
| rw [rationalPeriodGraph, Measure.map_map measurable_snd | |
| (measurable_rationalPeriodPair p)] | |
| unfold circleMeasure | |
| exact map_add_right_eq_self volume _ | |
| theorem rationalPeriodGraph_probability (p : CircleParameterData) : | |
| IsProbabilityMeasure (rationalPeriodGraph p) := by | |
| exact (Measure.isProbabilityMeasure_map_iff | |
| (measurable_rationalPeriodPair p).aemeasurable).2 circleMeasure_probability | |
| theorem first_frequency_gt_one {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) : 1 < p.q 1 := by | |
| simpa only [hp.frequency_zero] using hp.frequency_increasing (by omega : 0 < 1) | |
| theorem rationalPeriod_ne_zero {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) : | |
| (((1 : ℝ) / p.q 1 : ℝ) : Circle) ≠ 0 := by | |
| have hq : (1 : ℝ) < p.q 1 := by exact_mod_cast first_frequency_gt_one hp | |
| have ht : (1 : ℝ) / p.q 1 ∈ Set.Ico (0 : ℝ) 1 := by | |
| constructor | |
| · positivity | |
| · exact (div_lt_one (by positivity)).2 hq | |
| intro he | |
| have hz := (AddCircle.coe_eq_zero_iff_of_mem_Ico ht).1 he | |
| have hpos : (0 : ℝ) < 1 / p.q 1 := by positivity | |
| exact hpos.ne' hz | |
| theorem rationalPeriodGraph_ae_coordinate_eq {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) : | |
| ∀ᵐ xy ∂rationalPeriodGraph p, ∀ i : ℕ, 1 ≤ i → | |
| p.coordinate i xy.1 = p.coordinate i xy.2 := by | |
| apply ae_all_iff.2 | |
| intro i | |
| by_cases hi : 1 ≤ i | |
| · have hmeas : MeasurableSet {xy : Circle × Circle | | |
| p.coordinate i xy.1 = p.coordinate i xy.2} := | |
| measurableSet_eq_fun ((p.measurable_coordinate i).comp measurable_fst) | |
| ((p.measurable_coordinate i).comp measurable_snd) | |
| have he : ∀ᵐ xy ∂rationalPeriodGraph p, | |
| p.coordinate i xy.1 = p.coordinate i xy.2 := by | |
| apply (ae_map_iff (measurable_rationalPeriodPair p).aemeasurable hmeas).2 | |
| exact Filter.Eventually.of_forall fun x => by | |
| simpa using (hp.coordinate_rational_translation hi (1 : ℤ) x).symm | |
| exact he.mono fun _ h _ => h | |
| · exact Filter.Eventually.of_forall fun _ h => (hi h).elim | |
| theorem rationalPeriodGraph_ae_ne {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) : | |
| ∀ᵐ xy ∂rationalPeriodGraph p, xy.1 ≠ xy.2 := by | |
| apply (ae_map_iff (measurable_rationalPeriodPair p).aemeasurable | |
| (measurableSet_eq_fun measurable_fst measurable_snd).compl).2 | |
| exact Filter.Eventually.of_forall fun x h => by | |
| have he : (((1 : ℝ) / p.q 1 : ℝ) : Circle) = 0 := by | |
| exact (add_left_cancel (show x + (((1 : ℝ) / p.q 1 : ℝ) : Circle) = x + 0 by | |
| simpa using h.symm)) | |
| exact rationalPeriod_ne_zero hp he | |
| theorem exists_ordered_haar_law_off_diagonal {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) : | |
| ∃ η : Measure (Circle × Circle), IsProbabilityMeasure η ∧ | |
| η.map Prod.fst = circleMeasure ∧ η.map Prod.snd = circleMeasure ∧ | |
| (∀ᵐ xy ∂η, ∀ i : ℕ, 1 ≤ i → | |
| p.coordinate i xy.1 ≤ p.coordinate i xy.2) ∧ | |
| (∀ᵐ xy ∂η, xy.1 ≠ xy.2) := by | |
| refine ⟨rationalPeriodGraph p, rationalPeriodGraph_probability p, | |
| rationalPeriodGraph_fst p, rationalPeriodGraph_snd p, ?_, | |
| rationalPeriodGraph_ae_ne hp⟩ | |
| exact (rationalPeriodGraph_ae_coordinate_eq hp).mono fun _ h i hi => (h i hi).le | |
| end Invariant233.CircleOrderedRigidity | |
| end | |
| end OAI | |