openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Analysis /BackwardRecurrenceProbability.lean
| import OAI.GroupTheory.PolycyclicRecognition.Algebra.BackwardRecurrenceSets | |
| import OAI.GroupTheory.PolycyclicRecognition.Analysis.BackwardSwitchProbability | |
| namespace OAI | |
| /-! # Positive probability of all prescribed returns -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal | |
| /-- Backward recurrence and independent sampling together. The thresholds | |
| are the actual normalized masses of the subsequent retained sets. -/ | |
| theorem exists_backward_recurrence_probability | |
| {G : Type*} [Group G] [MeasurableSpace G] [MeasurableMul₂ G] [MeasurableInv G] | |
| (μ : Measure G) [SFinite μ] [μ.IsMulRightInvariant] | |
| (σ : ℕ → Measure G) [∀ j, IsProbabilityMeasure (σ j)] | |
| (Q : Set G) (hQfin : μ Q ≠ ∞) (hQpos : 0 < (μ Q).toReal) | |
| (K R : ℕ → Set G) (hK : ∀ j, ∀ᵐ s ∂σ j, s ∈ K j) | |
| (hRfin : ∀ j, μ (R j) ≠ ∞) (hRpos : ∀ j, 0 < (μ (R j)).toReal) | |
| (hQR : ∀ j, ∀ a ∈ Q, ∀ s ∈ K j, a * s ∈ R j) | |
| (D : ℕ → ℝ) (hD : ∀ j, 0 < D j) | |
| (hvol : ∀ j, (μ (R j)).toReal ≤ D j * (μ Q).toReal) | |
| (h : ℕ) (A : Set G) (hA : MeasurableSet A) (hAQ : A ⊆ Q) | |
| (hApos : 0 < (μ A).toReal) : | |
| ∃ S : ℕ → Set G, S h = A ∧ | |
| (∀ j, j ≤ h → MeasurableSet (S j) ∧ S j ⊆ A ∧ 0 < (μ (S j)).toReal) ∧ | |
| (∀ j, j < h → S j ⊆ S (j + 1) ∧ | |
| (μ (S (j + 1))).toReal ^ 2 / (2 * D j * (μ Q).toReal) ≤ (μ (S j)).toReal) ∧ | |
| let τ := fun j => Real.toNNReal ((μ (S (j + 1))).toReal / (2 * D j * (μ Q).toReal)) | |
| 0 < switchThresholdProduct τ 0 h ∧ | |
| ∀ x ∈ S 0, switchThresholdProduct τ 0 h ≤ survivingSwitchProbability σ S 0 h x := by | |
| obtain ⟨S, hlast, hsets, hstep⟩ := exists_backward_recurrence_sets μ σ Q hQfin hQpos | |
| K R hK hRfin hRpos hQR D hD hvol h A hA hAQ hApos | |
| refine ⟨S, hlast, hsets, fun j hj => ⟨(hstep j hj).1, (hstep j hj).2.1⟩, ?_⟩ | |
| let τ := fun j => Real.toNNReal ((μ (S (j + 1))).toReal / (2 * D j * (μ Q).toReal)) | |
| have hpos (j : ℕ) (hj : j < h) : | |
| 0 < (μ (S (j + 1))).toReal / (2 * D j * (μ Q).toReal) := | |
| div_pos (hsets (j + 1) (by omega)).2.2 | |
| (mul_pos (mul_pos (by norm_num) (hD j)) hQpos) | |
| have hτ (j : ℕ) (hj : j < h) : 0 < τ j := by | |
| exact Real.toNNReal_pos.mpr (hpos j hj) | |
| refine ⟨switchThresholdProduct_pos τ 0 h (fun j _ hj => hτ j (by omega)), ?_⟩ | |
| intro x hx | |
| apply switchThresholdProduct_le_survival σ S τ 0 h _ hx | |
| intro j hj y hy | |
| have hjh : j < h := by omega | |
| have hr := (hstep j hjh).2.2 y hy | |
| have hfin : twoSampleReturn (σ j) (S (j + 1)) y ≠ ∞ := | |
| (twoSampleReturn_le_one (σ j) (S (j + 1)) y).trans_lt (by simp) |>.ne | |
| have hr' := ENNReal.ofReal_le_ofReal hr | |
| rw [ENNReal.ofReal_toReal hfin] at hr' | |
| exact hr' | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |