openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /FiniteHistoryAdditive.lean
| import OAI.Combinatorics.BalancedRyser.Probability.BoundedDrift | |
| import OAI.Combinatorics.BalancedRyser.Probability.FiniteHistory | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Preorder | |
| open scoped MeasureTheory ProbabilityTheory BigOperators | |
| namespace Balanced.FiniteHistory | |
| variable {S : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] | |
| theorem stronglyMeasurable_prefix (n : ℕ) (f : Prefix S n → ℝ) : | |
| StronglyMeasurable[filtration n] (fun ω : Path S => f (frestrictLe n ω)) := by | |
| apply Measurable.stronglyMeasurable | |
| change Measurable[Filtration.piLE n] _ | |
| rw [Filtration.piLE_eq_comap_frestrictLe] | |
| exact (measurable_of_countable f).comp (comap_measurable _) | |
| theorem stronglyMeasurable_stepObservable (n : ℕ) (f : Prefix S n → S → ℝ) : | |
| StronglyMeasurable[filtration (n + 1)] (stepObservable n f) := by | |
| let g : Prefix S (n + 1) → ℝ := fun h => | |
| f (fun i => h ⟨i.1, Finset.mem_Iic.mpr (by have := Finset.mem_Iic.mp i.2; omega)⟩) | |
| (h ⟨n + 1, Finset.mem_Iic.mpr le_rfl⟩) | |
| exact stronglyMeasurable_prefix (n + 1) g | |
| def additiveTrack (x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ) | |
| (t : ℕ) (ω : Path S) : ℝ := | |
| x₀ + ∑ j ∈ Finset.range t, stepObservable j (f j) ω | |
| omit [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] in | |
| @[simp] theorem additiveTrack_zero (x₀ : ℝ) | |
| (f : (n : ℕ) → Prefix S n → S → ℝ) (ω : Path S) : | |
| additiveTrack x₀ f 0 ω = x₀ := by simp [additiveTrack] | |
| omit [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] in | |
| @[simp] theorem additiveTrack_increment (x₀ : ℝ) | |
| (f : (n : ℕ) → Prefix S n → S → ℝ) (t : ℕ) (ω : Path S) : | |
| additiveTrack x₀ f (t + 1) ω - additiveTrack x₀ f t ω = | |
| stepObservable t (f t) ω := by | |
| simp only [additiveTrack, Finset.sum_range_succ] | |
| ring | |
| theorem stronglyMeasurable_additiveTrack (x₀ : ℝ) | |
| (f : (n : ℕ) → Prefix S n → S → ℝ) (t : ℕ) : | |
| StronglyMeasurable[filtration t] (additiveTrack x₀ f t) := by | |
| apply StronglyMeasurable.add stronglyMeasurable_const | |
| have hs := Finset.stronglyMeasurable_sum (Finset.range t) (fun j hj => | |
| (stronglyMeasurable_stepObservable j (f j)).mono | |
| (filtration.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hj)))) | |
| convert hs using 1 | |
| funext ω | |
| simp only [Finset.sum_apply] | |
| theorem condExp_additiveTrack_increment (step : ℕ → S → PMF S) (s₀ : S) | |
| (x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ) (t : ℕ) : | |
| (pathLaw step s₀)[fun ω => additiveTrack x₀ f (t + 1) ω - | |
| additiveTrack x₀ f t ω | filtration t] =ᵐ[pathLaw step s₀] | |
| fun ω => ∑ s, (step t (ω t) s).toReal * f t (frestrictLe t ω) s := by | |
| simpa only [additiveTrack_increment] using condExp_stepObservable step s₀ t (f t) | |
| theorem additiveTrack_drift_nonnegative (step : ℕ → S → PMF S) (s₀ : S) | |
| (x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ) (T : ℕ) | |
| (hmean : ∀ t < T, ∀ h : Prefix S t, | |
| 0 ≤ ∑ s, (step t (h ⟨t, Finset.mem_Iic.mpr le_rfl⟩) s).toReal * f t h s) : | |
| ∀ t ≤ T, ∀ᵐ ω ∂pathLaw step s₀, | |
| 0 ≤ ∑ j ∈ Finset.range t, | |
| (pathLaw step s₀)[fun ω => additiveTrack x₀ f (j + 1) ω - | |
| additiveTrack x₀ f j ω | filtration j] ω := by | |
| intro t ht | |
| have hall := ae_all_iff.2 (fun j => condExp_additiveTrack_increment step s₀ x₀ f j) | |
| filter_upwards [hall] with ω hω | |
| apply Finset.sum_nonneg | |
| intro j hj | |
| rw [hω j] | |
| exact hmean j (lt_of_lt_of_le (Finset.mem_range.mp hj) ht) (frestrictLe j ω) | |
| theorem additiveTrack_lower_tail (step : ℕ → S → PMF S) (s₀ : S) | |
| (x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ) (T : ℕ) (L : ℝ) | |
| (hT : 1 ≤ T) (hL : 0 < L) | |
| (hbound : ∀ j < T, ∀ h s, |f j h s| ≤ L) | |
| (hmean : ∀ j < T, ∀ h : Prefix S j, | |
| 0 ≤ ∑ s, (step j (h ⟨j, Finset.mem_Iic.mpr le_rfl⟩) s).toReal * f j h s) | |
| (u : ℝ) (hu : 0 < u) (t : ℕ) (ht : t ≤ T) : | |
| (pathLaw step s₀).real {ω | additiveTrack x₀ f t ω ≤ x₀ - u} ≤ | |
| 2 * Real.exp (-u ^ 2 / (8 * (T : ℝ) * L ^ 2)) := by | |
| have hc := boundedDriftConcentration_proof (pathLaw step s₀) filtration | |
| (additiveTrack x₀ f) T L hT hL | |
| (fun j _ => stronglyMeasurable_additiveTrack x₀ f j) | |
| (fun j hj => Filter.Eventually.of_forall (fun ω => by | |
| simpa only [additiveTrack_increment, stepObservable] using hbound j hj (frestrictLe j ω) (ω (j + 1)))) | |
| u hu t ht | |
| refine le_trans ?_ hc | |
| apply ENNReal.toReal_mono (measure_ne_top _ _) | |
| apply measure_mono_ae | |
| filter_upwards [additiveTrack_drift_nonnegative step s₀ x₀ f T hmean t ht] | |
| with ω hω hbad | |
| change additiveTrack x₀ f t ω ≤ x₀ - u at hbad | |
| simp only [additiveTrack_zero] | |
| have hneg : u ≤ -(additiveTrack x₀ f t ω - x₀ - | |
| ∑ j ∈ Finset.range t, (pathLaw step s₀)[fun ω => | |
| additiveTrack x₀ f (j + 1) ω - additiveTrack x₀ f j ω | filtration j] ω) := by | |
| linarith | |
| exact hneg.trans (neg_le_abs _) | |
| end Balanced.FiniteHistory | |
| end | |
| end OAI | |