openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Rotation /CircleCurrentBlockEvent.lean
| import OAI.Analysis.IrrationalRotation.Model.CircleCoupling | |
| import OAI.Analysis.IrrationalRotation.Couplings.CircleCouplingParameters | |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleRawTagEvent | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal BigOperators | |
| namespace Invariant233.CircleCoupling | |
| open CircleDigitCoupling CircleParameterData OneCoordinateCoupling | |
| def rawCurrentBlockEvent (p : CircleParameterData) (j : ℕ) : Set Sample := | |
| {ω | (∃ i ∈ p.block j, label i ω = some (4 : Fin 5)) ∧ | |
| ∀ i ∈ p.block j, (ω i).2.1 < 1 - 3 / (p.m j : ℝ) ∧ | |
| 3 / (p.m j : ℝ) < (ω i).2.2} | |
| lemma measurableSet_rawCurrentBlockEvent (p : CircleParameterData) (j : ℕ) : | |
| MeasurableSet (rawCurrentBlockEvent p j) := by | |
| classical | |
| unfold rawCurrentBlockEvent label | |
| measurability | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| private lemma coordinate_probability_for_tag (j i : ℕ) : | |
| IsProbabilityMeasure (coordinateLaw p j i) := by | |
| by_cases hi : i = 0 | |
| · simp only [coordinateLaw, ite_eq_left hi, initialLaw] | |
| infer_instance | |
| have hs := p.blockIndex_pos hp.m_pos (show 1 ≤ i by omega) | |
| have hm : (100 : ℝ) ≤ p.m (p.blockIndex i) := by | |
| exact_mod_cast (show 100 ≤ p.m (p.blockIndex i) by have := hp.m_ge_200 _ hs; omega) | |
| have ht : 0 ≤ tag j (p.blockIndex i) ∧ tag j (p.blockIndex i) ≤ 1 := by | |
| unfold tag | |
| split_ifs <;> norm_num | |
| simpa only [coordinateLaw, ite_eq_right hi] using rawLaw_isProbabilityMeasure | |
| (CircleParameterLaws.coupling_kappa_bounds _ hs).1 | |
| (CircleParameterLaws.coupling_kappa_bounds _ hs).2 hm | |
| (hp.coupling_a_bounds _ hs).1 (hp.coupling_a_bounds _ hs).2 | |
| (hp.coupling_b_bounds _ hs).1 (hp.coupling_b_bounds _ hs).2 ht.1 ht.2 | |
| lemma coordinateLaw_currentBlock {j i : ℕ} (hi : i ∈ p.block j) : | |
| coordinateLaw p j i = rawLaw (kappa j) (p.m j) (p.a j) (p.b j) 1 := by | |
| have hi0 : i ≠ 0 := by | |
| have h := (p.mem_block_iff j i).mp hi | |
| omega | |
| simp only [coordinateLaw, ite_eq_right hi0, p.blockIndex_eq_of_mem hp.m_pos hi, | |
| tag, ite_true] | |
| theorem rawCurrentBlockEvent_mass_lower (j : ℕ) (hj : 1 ≤ j) : | |
| Real.exp (-90) ≤ (law p j).real (rawCurrentBlockEvent p j) := by | |
| classical | |
| let := coordinate_probability_for_tag hp j | |
| let μ₀ := rawLaw (kappa j) (p.m j) (p.a j) (p.b j) 1 | |
| have : IsProbabilityMeasure μ₀ := rawLaw_isProbabilityMeasure | |
| (CircleParameterLaws.coupling_kappa_bounds j hj).1 | |
| (CircleParameterLaws.coupling_kappa_bounds j hj).2 | |
| (by exact_mod_cast (show 100 ≤ p.m j by have := hp.m_ge_200 j hj; omega)) | |
| (hp.coupling_a_bounds j hj).1 (hp.coupling_a_bounds j hj).2 | |
| (hp.coupling_b_bounds j hj).1 (hp.coupling_b_bounds j hj).2 | |
| (by norm_num) (by norm_num) | |
| let e : Fin (p.m j) ≃ (p.block j) := Fintype.equivOfCardEq (by simp [p.card_block j hj]) | |
| let E := MeasurableEquiv.piCongrLeft (fun _ : Fin (p.m j) => CoordinateSample) e.symm | |
| let f : Sample → Fin (p.m j) → CoordinateSample := fun ω i => ω (e i) | |
| have hf : Measurable f := by unfold f; fun_prop | |
| have hcomp : f = E ∘ (p.block j).restrict := by | |
| funext ω i | |
| simp [E, f, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply] | |
| have hmap : (law p j).map f = Measure.pi (fun _ : Fin (p.m j) => μ₀) := by | |
| rw [hcomp, ← Measure.map_map E.measurable (Finset.measurable_restrict _)] | |
| change ((Measure.infinitePi (coordinateLaw p j)).map (p.block j).restrict).map E = _ | |
| rw [Measure.infinitePi_map_restrict] | |
| have hcoord : (fun i : p.block j => coordinateLaw p j i) = fun _ => μ₀ := by | |
| funext i | |
| exact coordinateLaw_currentBlock hp i.property | |
| rw [hcoord] | |
| exact Measure.pi_map_piCongrLeft e.symm (fun _ : Fin (p.m j) => μ₀) | |
| let A : Set (Fin (p.m j) → CoordinateSample) := | |
| {z | (∃ i, (z i).1 = some (4 : Fin 5)) ∧ | |
| ∀ i, (z i).2.1 < 1 - 3 / (p.m j : ℝ) ∧ 3 / (p.m j : ℝ) < (z i).2.2} | |
| have hA : MeasurableSet A := by | |
| dsimp [A] | |
| measurability | |
| have hpre : f ⁻¹' A = rawCurrentBlockEvent p j := by | |
| ext ω | |
| constructor | |
| · rintro ⟨⟨i, hi⟩, hω⟩ | |
| refine ⟨⟨e i, (e i).property, hi⟩, ?_⟩ | |
| intro k hk | |
| obtain ⟨i, he⟩ := e.surjective ⟨k, hk⟩ | |
| simpa only [f, he] using hω i | |
| · rintro ⟨⟨k, hk, htag⟩, hω⟩ | |
| obtain ⟨i, he⟩ := e.surjective ⟨k, hk⟩ | |
| refine ⟨⟨i, ?_⟩, ?_⟩ | |
| · simpa only [f, he, label] using htag | |
| · intro i | |
| exact hω (e i) (e i).property | |
| have hmass : (law p j).real (rawCurrentBlockEvent p j) = | |
| (Measure.pi (fun _ : Fin (p.m j) => μ₀)).real A := by | |
| rw [← hmap] | |
| simp only [measureReal_def, Measure.map_apply hf hA, hpre] | |
| rw [hmass] | |
| exact concrete_circle_currentBlock_mass (p.m j) (hp.m_ge_200 j hj) (kappa j) | |
| (CircleParameterLaws.coupling_kappa_bounds j hj).1 | |
| (CircleParameterLaws.coupling_kappa_bounds j hj).2 | |
| (fun _ => p.a j) (fun _ => p.b j) | |
| (fun _ => (hp.coupling_a_bounds j hj).1) (fun _ => (hp.coupling_a_bounds j hj).2) | |
| (fun _ => (hp.coupling_b_bounds j hj).1) (fun _ => (hp.coupling_b_bounds j hj).2) | |
| end Invariant233.CircleCoupling | |
| end | |
| end OAI | |