openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Rotation /CircleTruncationIntegral.lean
| import OAI.Analysis.IrrationalRotation.Model.CircleTruncationEvent | |
| import OAI.Analysis.IrrationalRotation.Couplings.CircleCouplingForcing | |
| import OAI.Analysis.IrrationalRotation.Rotation.CanonicalCircleFiniteLower | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped BigOperators ComplexConjugate lp | |
| namespace Invariant233.CircleCoupling | |
| open CircleDigitCoupling CircleParameterData OneCoordinateCoupling | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem ae_truncationEvent_exclusions (j : ℕ) (hj : 1 ≤ j) | |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) (r : Circle → ℝ) : | |
| ∀ᵐ ω ∂law p j, ω ∈ truncationEvent p j hr r → | |
| ∀ s, j ≤ s → ∀ i ∈ p.block s, | |
| p.coordinate i (point p hr false ω) < 1 - 2 * (p.K j : ℝ) / p.N s ∧ | |
| 2 * (p.K j : ℝ) / p.N s < p.coordinate i (point p hr true ω) := by | |
| have hu (side : Bool) : ∀ᵐ ω ∂law p j, ∀ i, | |
| endpoint side (ω i) ∈ Ico (0 : ℝ) 1 := | |
| ae_all_iff.mpr (fun i => ae_endpoint_unit hp j i side) | |
| filter_upwards [hu false, hu true, ae_nonterminal_digits hp j hr false, | |
| ae_nonterminal_digits hp j hr true] with ω hx hy hdx hdy | |
| intro hE | |
| have hfloor (side : Bool) (hunit : ∀ i, endpoint side (ω i) ∈ Ico (0 : ℝ) 1) | |
| (i : ℕ) : | |
| ⌊(MixedRadix.radices p.q i : ℝ) * endpoint side (ω i)⌋₊ = | |
| (digits p hr side ω i : ℕ) := by | |
| have hq : (0 : ℝ) < MixedRadix.radices p.q i := by exact_mod_cast hr i | |
| have hf : ⌊(MixedRadix.radices p.q i : ℝ) * endpoint side (ω i)⌋₊ < | |
| MixedRadix.radices p.q i := | |
| (Nat.floor_lt (mul_nonneg hq.le (hunit i).1)).mpr (by nlinarith [(hunit i).2]) | |
| change _ = _ % MixedRadix.radices p.q i | |
| rw [Nat.mod_eq_of_lt hf] | |
| exact hp.source_good_event_exclusions j hj (digits p hr false ω) (digits p hr true ω) | |
| hdx hdy (fun i => (ω i).2.1) (fun i => (ω i).2.2) | |
| (fun i => (hx i).1) (fun i => (hy i).1) | |
| (hfloor false hx) (hfloor true hy) hE.1 hE.2.2.1 | |
| theorem truncationEvent_integral_lower (j : ℕ) (hj : 1 ≤ j) | |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) | |
| (F : Circle → ℓ²(ℤ, ℂ) →L[ℂ] ℓ²(ℤ, ℂ)) | |
| (hFm : StronglyMeasurable (fun x => F x ProjectionCocycleWindow.basisZero)) | |
| (hF : ∀ x, IsStarProjection (F x)) : | |
| p.ε j * (law p j).real (truncationEvent p j hr | |
| (fun x => ((F x ProjectionCocycleWindow.basisZero) 0).re)) ≤ | |
| ∫ ω in truncationEvent p j hr | |
| (fun x => ((F x ProjectionCocycleWindow.basisZero) 0).re), | |
| ∑ n ∈ Finset.range (2 * p.K j + 1), | |
| ‖InvariantProjection.ConditionalConvolution.truncated (p.K j) n | |
| (fun v => (F (point p hr false ω) ProjectionCocycleWindow.basisZero) v / | |
| (p.floorCocycle θ v (point p hr false ω) : ℂ)) | |
| (fun v => (p.floorCocycle θ (-v) (point p hr true ω) : ℂ) * | |
| conj (((1 - F (point p hr true ω)) ProjectionCocycleWindow.basisZero) (-v)))‖ | |
| ∂law p j := by | |
| let := law_isProbabilityMeasure hp j | |
| let r : Circle → ℝ := fun x => ((F x ProjectionCocycleWindow.basisZero) 0).re | |
| have hrm : Measurable r := | |
| Complex.continuous_re.measurable.comp | |
| (((lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 0).continuous.comp_stronglyMeasurable hFm).measurable) | |
| have htag : MeasurableSet (tagEvent p j) := by | |
| simp only [tagEvent, Set.ofPred_exists] | |
| apply MeasurableSet.iUnion | |
| intro i | |
| by_cases hi : i ∈ p.block j | |
| · simpa only [hi, true_and] using | |
| (measurableSet_eq_fun (measurable_label i) (measurable_const (a := (some 4 : Label)))) | |
| · simp only [hi, false_and, Set.ofPred_false, MeasurableSet.empty] | |
| have hEm : MeasurableSet (truncationEvent p j hr r) := by | |
| have hx (i : ℕ) : Measurable (fun ω : Sample => (ω i).2.1) := | |
| measurable_fst.comp (measurable_snd.comp (measurable_pi_apply i)) | |
| have hy (i : ℕ) : Measurable (fun ω : Sample => (ω i).2.2) := | |
| measurable_snd.comp (measurable_snd.comp (measurable_pi_apply i)) | |
| simp only [truncationEvent, Set.ofPred_and, Set.ofPred_forall] | |
| refine MeasurableSet.inter ?_ (htag.inter | |
| (MeasurableSet.inter ?_ (MeasurableSet.inter ?_ ?_))) | |
| · exact MeasurableSet.iInter fun i => MeasurableSet.iInter fun _ => | |
| (measurableSet_lt (hx i) measurable_const).inter | |
| (measurableSet_lt measurable_const (hy i)) | |
| · exact MeasurableSet.iInter fun s => MeasurableSet.iInter fun _ => | |
| MeasurableSet.iInter fun i => MeasurableSet.iInter fun _ => | |
| (measurableSet_lt (hx i) measurable_const).inter | |
| (measurableSet_lt measurable_const (hy i)) | |
| · exact measurableSet_le measurable_const (hrm.comp (measurable_point hp hr false)) | |
| · exact measurableSet_le measurable_const | |
| (measurable_const.sub (hrm.comp (measurable_point hp hr true))) | |
| apply hp.canonical_projection_integral_lower (law p j) j hj | |
| (truncationEvent p j hr r) (point p hr false) (point p hr true) | |
| (measurable_point hp hr false) (measurable_point hp hr true) | |
| (fun ω => F (point p hr false ω)) (fun ω => F (point p hr true ω)) | |
| (hFm.comp_measurable (measurable_point hp hr false)) | |
| (hFm.comp_measurable (measurable_point hp hr true)) | |
| (fun ω => hF _) (fun ω => hF _) | |
| filter_upwards [ae_restrict_mem hEm, | |
| ae_restrict_of_ae (ae_truncationEvent_exclusions hp j hj hr r)] with ω hω hexcl | |
| have hcuts := hexcl hω | |
| refine ⟨(fun s hs i hi => (hcuts s hs i hi).1), | |
| (fun s hs i hi => (hcuts s hs i hi).2), hω.2.2.2.1, ?_⟩ | |
| have hdiag := hω.2.2.2.2 | |
| change 1 / (j : ℝ) ≤ 1 - ((F (point p hr true ω) ProjectionCocycleWindow.basisZero) 0).re at hdiag | |
| linarith | |
| end Invariant233.CircleCoupling | |
| end | |
| end OAI | |