openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /BlockedProjectionWindow.lean
| import OAI.Analysis.IrrationalRotation.Analysis.BlockedSourceEvent | |
| import OAI.Analysis.IrrationalRotation.Product.ProductScaleCocycleWindow | |
| import OAI.Analysis.IrrationalRotation.Rotation.ProjectionCocycleWindow | |
| import OAI.Analysis.IrrationalRotation.Analysis.FiniteWindowIntegralLower | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology BigOperators ComplexConjugate lp | |
| namespace Invariant233.BlockedProductCoupling | |
| theorem measurable_sourceEvent (p : ProductScaleData) | |
| (e : ℕ ≃ Σ s, Fin (p.m s)) {r : ProductModel.Base → ℝ} | |
| (hr : Measurable r) (j : ℕ) : MeasurableSet (sourceEvent p e r j) := by | |
| have hx : Measurable (circleLeftEndpoint e) := by | |
| unfold circleLeftEndpoint toCircle leftEndpoint flatten OrderedProductCoupling.leftEndpoint | |
| fun_prop | |
| have hy : Measurable (circleRightEndpoint e) := by | |
| unfold circleRightEndpoint toCircle rightEndpoint flatten OrderedProductCoupling.rightEndpoint | |
| fun_prop | |
| have hc : MeasurableSet (currentEvent p.m j) := by | |
| change MeasurableSet ({z : Sample p.m | ∃ i, (z j i).1 = some (4 : Fin 5)} ∩ | |
| {z | ∀ i, (z j i).2.1 < 1 - 2 / (p.m j : ℝ) ∧ | |
| 2 / (p.m j : ℝ) < (z j i).2.2}) | |
| apply MeasurableSet.inter | |
| · simp only [ofPred_exists] | |
| exact MeasurableSet.iUnion fun i => (by fun_prop : | |
| Measurable (fun z : Sample p.m => (z j i).1)) (measurableSet_singleton _) | |
| · simp only [ofPred_forall] | |
| apply MeasurableSet.iInter | |
| intro i | |
| have hxi : Measurable (fun z : Sample p.m => (z j i).2.1) := by fun_prop | |
| have hyi : Measurable (fun z : Sample p.m => (z j i).2.2) := by fun_prop | |
| exact (measurableSet_lt hxi measurable_const).inter | |
| (measurableSet_lt measurable_const hyi) | |
| have hf : MeasurableSet (futureBad p.m (fun s => (p.K s : ℝ)) j) := by | |
| apply MeasurableSet.iUnion | |
| intro n | |
| apply MeasurableSet.iUnion | |
| intro i | |
| have hxi : Measurable (fun z : Sample p.m => (z (j + 1 + n) i).2.1) := by fun_prop | |
| have hyi : Measurable (fun z : Sample p.m => (z (j + 1 + n) i).2.2) := by fun_prop | |
| exact (measurableSet_le measurable_const hxi).union | |
| (measurableSet_le hyi measurable_const) | |
| exact hc.diff (hf.union | |
| ((measurableSet_lt (hr.comp hx) measurable_const).union | |
| (measurableSet_lt (measurable_const.sub (hr.comp hy)) measurable_const))) | |
| theorem ae_sourceEvent_coefficient_properties {p : ProductScaleData} | |
| (hp : ProductScaleLaws p) (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ) | |
| (hphase : ∀ i, θ i ∈ Icc (1 / p.N (e i).1) (2 / p.N (e i).1)) | |
| (P : ProjectionField) (hP : ∀ᵐ x ∂ProductModel.baseMeasure, IsStarProjection (P x)) (j : ℕ) : | |
| ∀ᵐ z ∂sourceLaw p j, | |
| z ∈ sourceEvent p e (projectionDiagonal P) j → | |
| (∀ v : ℤ, |v| ≤ (p.K j : ℤ) → | |
| ‖sourceCoefficientA p e θ P z v‖ ≤ p.coefficientBound j) ∧ | |
| (∀ v : ℤ, |v| ≤ (p.K j : ℤ) → | |
| ‖sourceCoefficientB p e θ P z v‖ ≤ p.coefficientBound j) ∧ | |
| 1 / ((j + 1 : ℕ) : ℝ) ≤ ‖sourceCoefficientA p e θ P z 0‖ ∧ | |
| 1 / ((j + 1 : ℕ) : ℝ) ≤ ‖sourceCoefficientB p e θ P z 0‖ := by | |
| have hrep := ae_circle_representatives hp.m_ge_100 source_kappa_nonneg source_kappa_le | |
| (fun s _ => (hp.a_bounds s).1) (fun s _ => (hp.a_bounds s).2) | |
| (fun s _ => (hp.b_bounds s).1) (fun s _ => (hp.b_bounds s).2) e j | |
| filter_upwards [hrep, (sourceLaw_quasiMeasurePreserving_left hp e j).ae hP, | |
| (sourceLaw_quasiMeasurePreserving_right hp e j).ae hP] with z hz hPx hPy | |
| intro he | |
| obtain ⟨_, hcurrent, hfuture, hdiagx, hdiagy⟩ := mem_sourceRetainedEvent_iff.mp he | |
| have hcur : ∀ i, (e i).1 = j → | |
| ProductCocycle.representative (circleLeftEndpoint e z i) < 1 - 2 / (p.m j : ℝ) ∧ | |
| 2 / (p.m j : ℝ) < ProductCocycle.representative (circleRightEndpoint e z i) := by | |
| intro i hi | |
| change circleRepresentative _ < _ ∧ _ < circleRepresentative _ | |
| rw [(hz i).1, (hz i).2] | |
| change (z (e i).1 (e i).2).2.1 < _ ∧ _ < (z (e i).1 (e i).2).2.2 | |
| subst j | |
| exact hcurrent (e i).2 | |
| have hfut : ∀ i, j < (e i).1 → | |
| ProductCocycle.representative (circleLeftEndpoint e z i) < | |
| 1 - 2 * (p.K j : ℝ) / p.N (e i).1 ∧ | |
| 2 * (p.K j : ℝ) / p.N (e i).1 < | |
| ProductCocycle.representative (circleRightEndpoint e z i) := by | |
| intro i hi | |
| change circleRepresentative _ < _ ∧ _ < circleRepresentative _ | |
| rw [(hz i).1, (hz i).2] | |
| exact hfuture (e i).1 hi (e i).2 | |
| obtain ⟨hA, hB⟩ := hp.product_finite_window_coefficients e θ hphase j | |
| (circleLeftEndpoint e z) (circleRightEndpoint e z) hcur hfut | |
| (P (circleLeftEndpoint e z) sourceBasisZero) | |
| ((1 - P (circleRightEndpoint e z)) sourceBasisZero) | |
| (ProjectionCocycleWindow.projection_vector_norm_le _ hPx) | |
| (ProjectionCocycleWindow.complement_vector_norm_le _ hPy) | |
| obtain ⟨hA0, hB0⟩ := ProjectionCocycleWindow.central_lower_bounds | |
| (P (circleLeftEndpoint e z)) (P (circleRightEndpoint e z)) | |
| (1 / ((j + 1 : ℕ) : ℝ)) hdiagx (by | |
| change 1 / ((j + 1 : ℕ) : ℝ) ≤ | |
| 1 - ((P (circleRightEndpoint e z) ProjectionCocycleWindow.basisZero) 0).re at hdiagy | |
| linarith) | |
| exact ⟨hA, hB, | |
| by simpa [sourceCoefficientA, sourceBasisZero, ProjectionCocycleWindow.basisZero] using hA0, | |
| by simpa [sourceCoefficientB, sourceBasisZero, ProjectionCocycleWindow.basisZero] using hB0⟩ | |
| theorem measurable_sourceCoefficientA (p : ProductScaleData) | |
| (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ) (P : ProjectionField) | |
| (hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v)) (v : ℤ) : | |
| Measurable (fun z => sourceCoefficientA p e θ P z v) := by | |
| have hx : Measurable (circleLeftEndpoint e) := by | |
| unfold circleLeftEndpoint toCircle leftEndpoint flatten OrderedProductCoupling.leftEndpoint | |
| fun_prop | |
| exact ((hPm v).comp hx).div (Complex.measurable_ofReal.comp | |
| ((ProductCocycle.measurable_cocycleVersion θ (fun i => p.C (e i).1) v).comp hx)) | |
| theorem measurable_sourceCoefficientB (p : ProductScaleData) | |
| (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ) (P : ProjectionField) | |
| (hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v)) (v : ℤ) : | |
| Measurable (fun z => sourceCoefficientB p e θ P z v) := by | |
| have hy : Measurable (circleRightEndpoint e) := by | |
| unfold circleRightEndpoint toCircle rightEndpoint flatten OrderedProductCoupling.rightEndpoint | |
| fun_prop | |
| have hvec : Measurable (fun z : Sample p.m => | |
| sourceBasisZero (-v) - (P (circleRightEndpoint e z) sourceBasisZero) (-v)) := | |
| measurable_const.sub ((hPm (-v)).comp hy) | |
| have hvec' : Measurable (fun z : Sample p.m => | |
| ((1 - P (circleRightEndpoint e z)) sourceBasisZero) (-v)) := by | |
| simpa only [sub_apply, one_apply_eq_self, lp.coeFn_sub, Pi.sub_apply] using hvec | |
| exact (Complex.measurable_ofReal.comp | |
| ((ProductCocycle.measurable_cocycleVersion θ (fun i => p.C (e i).1) (-v)).comp hy)).mul | |
| (Complex.continuous_conj.measurable.comp hvec') | |
| theorem source_polynomial_sum_lower {p : ProductScaleData} (hp : ProductScaleLaws p) | |
| (j : ℕ) (A B : ℤ → ℂ) | |
| (hA : ∀ v, |v| ≤ (p.K j : ℤ) → ‖A v‖ ≤ p.coefficientBound j) | |
| (hB : ∀ v, |v| ≤ (p.K j : ℤ) → ‖B v‖ ≤ p.coefficientBound j) | |
| (hA0 : 1 / ((j + 1 : ℕ) : ℝ) ≤ ‖A 0‖) | |
| (hB0 : 1 / ((j + 1 : ℕ) : ℝ) ≤ ‖B 0‖) : | |
| p.ε j ≤ ∑ n ∈ Finset.range (2 * p.K j + 1), | |
| ‖FiniteWindow.convolution (p.K j) A B n‖ := by | |
| obtain ⟨n, hn, hε⟩ := hp.polynomial_lower j A B hA hB hA0 hB0 | |
| exact hε.trans (Finset.single_le_sum (fun k _ => norm_nonneg _) | |
| (Finset.mem_range.mpr (by omega))) | |
| theorem sourceEvent_integral_lower {p : ProductScaleData} | |
| (hp : ProductScaleLaws p) (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ) | |
| (hphase : ∀ i, θ i ∈ Icc (1 / p.N (e i).1) (2 / p.N (e i).1)) | |
| (P : ProjectionField) (hP : ∀ᵐ x ∂ProductModel.baseMeasure, IsStarProjection (P x)) | |
| (hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v)) (j : ℕ) : | |
| p.ε j * (sourceLaw p j).real (sourceEvent p e (projectionDiagonal P) j) ≤ | |
| ∫ z in sourceEvent p e (projectionDiagonal P) j, | |
| ∑ n ∈ Finset.range (2 * p.K j + 1), | |
| ‖FiniteWindow.convolution (p.K j) | |
| (sourceCoefficientA p e θ P z) (sourceCoefficientB p e θ P z) n‖ | |
| ∂sourceLaw p j := by | |
| have := sourceLaw_isProbabilityMeasure hp j | |
| have hr : Measurable (projectionDiagonal P) := Complex.measurable_re.comp (hPm 0) | |
| apply FiniteWindow.integral_lower_of_sum_bound (sourceLaw p j) | |
| (p.K j) (p.coefficientBound j) (1 / ((j + 1 : ℕ) : ℝ)) (p.ε j) | |
| (source_polynomial_sum_lower hp j) | |
| (sourceEvent p e (projectionDiagonal P) j) | |
| (sourceCoefficientA p e θ P) (sourceCoefficientB p e θ P) | |
| (measurable_sourceCoefficientA p e θ P hPm) | |
| (measurable_sourceCoefficientB p e θ P hPm) | |
| filter_upwards [ae_restrict_of_ae (ae_sourceEvent_coefficient_properties hp e θ hphase P hP j), | |
| ae_restrict_mem (measurable_sourceEvent p e hr j)] with z hz he | |
| exact hz he | |
| theorem eventually_sourceEvent_and_integral_lower {p : ProductScaleData} | |
| (hp : ProductScaleLaws p) (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ) | |
| (hphase : ∀ i, θ i ∈ Icc (1 / p.N (e i).1) (2 / p.N (e i).1)) | |
| (P : ProjectionField) (hP : ∀ᵐ x ∂ProductModel.baseMeasure, IsStarProjection (P x)) | |
| (hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v)) | |
| (hi : ∀ᵐ x ∂ProductModel.baseMeasure, | |
| 0 < projectionDiagonal P x ∧ projectionDiagonal P x < 1) : | |
| ∀ᶠ j in atTop, | |
| Real.exp (-82) / 2 ≤ (sourceLaw p j).real (sourceEvent p e (projectionDiagonal P) j) ∧ | |
| (Real.exp (-82) / 2) * p.ε j ≤ | |
| ∫ z in sourceEvent p e (projectionDiagonal P) j, | |
| ∑ n ∈ Finset.range (2 * p.K j + 1), | |
| ‖FiniteWindow.convolution (p.K j) | |
| (sourceCoefficientA p e θ P z) (sourceCoefficientB p e θ P z) n‖ | |
| ∂sourceLaw p j := by | |
| have hr : Measurable (projectionDiagonal P) := Complex.measurable_re.comp (hPm 0) | |
| filter_upwards [eventually_sourceEvent_mass hp e hr hi] with j hj | |
| refine ⟨hj, ?_⟩ | |
| calc | |
| (Real.exp (-82) / 2) * p.ε j ≤ | |
| p.ε j * (sourceLaw p j).real (sourceEvent p e (projectionDiagonal P) j) := by | |
| simpa only [mul_comm] using mul_le_mul_of_nonneg_right hj (hp.epsilon_pos j).le | |
| _ ≤ _ := sourceEvent_integral_lower hp e θ hphase P hP hPm j | |
| end Invariant233.BlockedProductCoupling | |
| end | |
| end OAI | |