openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /CircleConditionalConvolutionData.lean
| import OAI.Analysis.IrrationalRotation.Rotation.CircleTruncationIntegral | |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleTailExceptionalFamilies | |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleOrderedTransfer | |
| import OAI.Analysis.IrrationalRotation.Analysis.ParameterConvolutionIdentity | |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleObstruction | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped lp ComplexConjugate | |
| namespace Invariant233.CircleCoupling | |
| open InvariantProjection.ConditionalConvolution | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem conditionalData_of_truncation_mass | |
| (j : ℕ) (hj : 1 ≤ j) (hr : ∀ i, 0 < MixedRadix.radices p.q i) | |
| {P : Rotation.Hilbert →L[ℂ] Rotation.Hilbert} | |
| (hmem : P ∈ Rotation.algebra θ) (hP : IsStarProjection P) | |
| (hT : (1 - P) * p.measurableOperator θ hp * P = 0) | |
| (F : Circle → Rotation.Fiber →L[ℂ] Rotation.Fiber) | |
| (hF : ∀ z, IsStarProjection (F z)) | |
| (hFm : ∀ v : Rotation.Fiber, StronglyMeasurable (fun z => F z v)) | |
| (hrep : ∀ ξ : Rotation.Hilbert, P ξ =ᵐ[circleMeasure] fun z => F z (ξ z)) | |
| (c : ℝ) (hmass : c ≤ (law p j).real (truncationEvent p j hr | |
| (fun z => ((F z ProjectionCocycleWindow.basisZero) 0).re))) : | |
| Nonempty (Data Sample (law p j) (p.K j) (p.coefficientBound j) | |
| (1 / (j : ℝ)) c (Real.exp (-p.L j)) (Real.exp (-p.L j / 400))) := by | |
| classical | |
| let := law_isProbabilityMeasure hp j | |
| let r : Circle → ℝ := fun z => ((F z ProjectionCocycleWindow.basisZero) 0).re | |
| let E := truncationEvent p j hr r | |
| let ξ : Sample → Rotation.Fiber := fun ω => | |
| F (point p hr false ω) ProjectionCocycleWindow.basisZero | |
| let γ : Sample → Rotation.Fiber := fun ω => | |
| (1 - F (point p hr true ω)) ProjectionCocycleWindow.basisZero | |
| let Fx : Sample → ℤ → ℝ := fun ω k => p.floorCocycle θ k (point p hr false ω) | |
| let Fy : Sample → ℤ → ℝ := fun ω k => p.floorCocycle θ k (point p hr true ω) | |
| have hrm : Measurable r := Complex.continuous_re.measurable.comp | |
| (((lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 0).continuous.comp_stronglyMeasurable | |
| (hFm ProjectionCocycleWindow.basisZero)).measurable) | |
| have hEm : MeasurableSet E := measurableSet_truncationEvent hp j hr hrm | |
| obtain ⟨H₁, H₂, hm₁, hm₂, hc₁, hc₂, hatt⟩ := | |
| tail_exceptional_families_of_subset hp j hj hr E hEm | |
| (truncationEvent_subset_tagEvent p j hr r) | |
| have hordered := orderedCouplingIdentities_of_rational hp hr F hF | |
| (fun k => (hFm (lp.single 2 k 1)).measurable) | |
| (hp.rational_graphs_of_representation hmem hP hT F hrep) j | |
| have hbound := hordered.mono (fun _ h k => (h.2 k).2) | |
| have hzero := hp.ae_parameter_convolution_absolute_and_zero (law p j) | |
| (point p hr false) (point p hr true) (measurable_point hp hr true).aemeasurable | |
| (endpoint_absolutelyContinuous hp j hr true) F hF hrep hT | |
| (hordered.mono (fun _ h => h.1)) hbound | |
| have hratio := hp.ae_parameter_convolution_ratio_bounds (law p j) | |
| (point p hr false) (point p hr true) (measurable_point hp hr false).aemeasurable | |
| (measurable_point hp hr true).aemeasurable | |
| (endpoint_absolutelyContinuous hp j hr false) | |
| (endpoint_absolutelyContinuous hp j hr true) hbound | |
| have hγm : StronglyMeasurable (fun z => | |
| (1 - F z) ProjectionCocycleWindow.basisZero) := by | |
| have h : StronglyMeasurable | |
| ((fun _ : Circle => ProjectionCocycleWindow.basisZero) - | |
| fun z => F z ProjectionCocycleWindow.basisZero) := | |
| stronglyMeasurable_const.sub (hFm ProjectionCocycleWindow.basisZero) | |
| convert h using 1 | |
| funext z | |
| simp only [Pi.sub_apply, sub_apply, one_apply_eq_self] | |
| refine ⟨{ | |
| mX := endpointSigma p hr false | |
| mY := endpointSigma p hr true | |
| mX_le := (measurable_point hp hr false).comap_le | |
| mY_le := (measurable_point hp hr true).comap_le | |
| ξ := ξ | |
| γ := γ | |
| ξ_measurable := (hFm ProjectionCocycleWindow.basisZero).comp_measurable | |
| (measurable_iff_comap_le.mpr le_rfl) | |
| γ_measurable := hγm.comp_measurable (measurable_iff_comap_le.mpr le_rfl) | |
| ξ_unit := Eventually.of_forall (fun ω => | |
| ProjectionCocycleWindow.projection_vector_norm_le _ (hF _)) | |
| γ_unit := Eventually.of_forall (fun ω => | |
| ProjectionCocycleWindow.complement_vector_norm_le _ (hF _)) | |
| Fx := Fx | |
| Fy := Fy | |
| Fx_measurable := fun k => (hp.measurable_parameter_cocycle k).comp | |
| (measurable_point hp hr false) | |
| Fy_measurable := fun k => (hp.measurable_parameter_cocycle k).comp | |
| (measurable_point hp hr true) | |
| Fx_pos := Eventually.of_forall (fun ω k => p.floorCocycle_pos θ k _) | |
| Fy_pos := Eventually.of_forall (fun ω k => p.floorCocycle_pos θ k _) | |
| Fx_zero := Eventually.of_forall (fun ω => p.floorCocycle_zero θ _) | |
| Fy_zero := Eventually.of_forall (fun ω => p.floorCocycle_zero θ _) | |
| zero_convolution := ?_ | |
| ratio_bounds := fun n _ => hratio.mono (fun _ h => h n) | |
| E := E | |
| E_measurable := hEm | |
| E_mass := hmass | |
| coefficient_bounds := ?_ | |
| H₁ := H₁ | |
| H₂ := H₂ | |
| H₁_measurable := hm₁ | |
| H₂_measurable := hm₂ | |
| attenuation := hatt | |
| conditional₁ := hc₁ | |
| conditional₂ := hc₂ }⟩ | |
| · intro n _ | |
| filter_upwards [hzero] with ω hω | |
| exact (hω n ProjectionCocycleWindow.basisZero ProjectionCocycleWindow.basisZero).2 | |
| · filter_upwards [ae_restrict_mem hEm, | |
| ae_restrict_of_ae (ae_truncationEvent_exclusions hp j hj hr r)] with ω hω he | |
| have he' := he hω | |
| have hdiag := hω.2.2.2.2 | |
| have hright : ((F (point p hr true ω) ProjectionCocycleWindow.basisZero) 0).re ≤ | |
| 1 - 1 / (j : ℝ) := by | |
| change 1 / (j : ℝ) ≤ 1 - ((F (point p hr true ω) | |
| ProjectionCocycleWindow.basisZero) 0).re at hdiag | |
| linarith | |
| exact hp.canonical_projection_coefficient_properties j hj _ _ | |
| (fun s hs i hi => (he' s hs i hi).1) (fun s hs i hi => (he' s hs i hi).2) | |
| _ _ (hF _) (hF _) (1 / (j : ℝ)) hω.2.2.2.1 hright | |
| end Invariant233.CircleCoupling | |
| end | |
| end OAI | |