| import OAI.Analysis.FiniteFactor.Coupling.Limit |
| import OAI.Analysis.FiniteFactor.Coupling.Infinite |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Topology ENNReal |
| noncomputable section |
| namespace CouplingParameters |
|
|
| / |
| abbrev BlockCoordinate (m : ℕ → ℕ) := Σ s : ℕ, Fin (m s) |
|
|
| def blockPrefix (m : ℕ → ℕ) (n : ℕ) : Finset (BlockCoordinate m) := |
| (Finset.range n).sigma (fun _ => Finset.univ) |
|
|
| theorem mem_blockPrefix (m : ℕ → ℕ) (n : ℕ) (i : BlockCoordinate m) : |
| i ∈ blockPrefix m n ↔ i.1 < n := by simp [blockPrefix] |
|
|
| theorem blockPrefix_mono (m : ℕ → ℕ) : Monotone (blockPrefix m) := by |
| intro n k hnk i hi |
| exact (mem_blockPrefix m k i).2 (lt_of_lt_of_le ((mem_blockPrefix m n i).1 hi) hnk) |
|
|
| theorem blockPrefix_cover (m : ℕ → ℕ) (J : Finset (BlockCoordinate m)) : |
| ∃ n, J ⊆ blockPrefix m n := by |
| refine ⟨J.sup (fun i => i.1) + 1, fun i hi => ?_⟩ |
| exact (mem_blockPrefix m _ i).2 (Nat.lt_succ_of_le (Finset.le_sup (f := fun i => i.1) hi)) |
|
|
| theorem blockPrefix_energy (p : ℕ → CouplingParameters) (m : ℕ → ℕ) |
| (hm : ∀ s, (p s).m = (m s : ℝ)) (j : ℕ) |
| (htag : ∀ s, (p s).tagged = decide (s=j)) |
| (hκ : Summable (fun s => (p s).kappa^2)) |
| (right : BlockCoordinate m → Bool) (n : ℕ) : |
| (∫ x, prefixDensity (fun i : BlockCoordinate m => p i.1) right (blockPrefix m n) x ^ 2 |
| ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ≤ |
| Real.exp (20*(2*(∑' s, (p s).kappa^2)+1)) := by |
| apply (prefixDensity_energy _ right _).trans |
| apply Real.exp_le_exp.mpr |
| have heq : (∑ i ∈ blockPrefix m n, 20*(2*(p i.1).kappa^2/(p i.1).m + (p i.1).tagMass)) = |
| 20 * ∑ s ∈ Finset.range n, (p s).m * (2*(p s).kappa^2/(p s).m + (p s).tagMass) := by |
| rw [blockPrefix, Finset.sum_sigma] |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro s _ |
| rw [hm s] |
| ring |
| rw [heq] |
| exact mul_le_mul_of_nonneg_left (uniform_block_budget p j htag hκ (Finset.range n)) (by norm_num) |
|
|
| theorem infinite_block_source_density (p : ℕ → CouplingParameters) (m : ℕ → ℕ) |
| (hm : ∀ s, (p s).m = (m s : ℝ)) (j : ℕ) |
| (htag : ∀ s, (p s).tagged = decide (s=j)) |
| (hκ : ∀ s, (p s).kappa = sourceKappa s) (right : BlockCoordinate m → Bool) : |
| ∃ g : (BlockCoordinate m → ℝ) → ℝ, |
| MemLp g 2 (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ∧ |
| (∀ᵐ x ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure), 0 ≤ g x) ∧ |
| (∫ x, g x ^ 2 ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ≤ |
| Real.exp (20*(2*(∑' s, sourceKappa s ^ 2)+1)) ∧ |
| Measure.infinitePi (fun i : BlockCoordinate m => (p i.1).selectedLaw true (right i)) = |
| (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)).withDensity |
| (fun x => ENNReal.ofReal (g x)) := by |
| apply infiniteEndpoint_density_of_prefix_bound _ right (blockPrefix m) |
| (blockPrefix_mono m) (blockPrefix_cover m) |
| intro n |
| have hs : Summable (fun s => (p s).kappa^2) := by simpa only [hκ] using sourceKappa_sq_summable |
| simpa only [hκ] using blockPrefix_energy p m hm j htag hs right n |
|
|
| / |
| def blockProductCoupling (p : ℕ → CouplingParameters) (m : ℕ → ℕ) : |
| Measure (BlockCoordinate m → Fin 6 × (ℝ × ℝ)) := |
| Measure.infinitePi (fun i => (p i.1).labelledCoupling) |
|
|
| instance blockProductCoupling_probability (p : ℕ → CouplingParameters) (m : ℕ → ℕ) : |
| IsProbabilityMeasure (blockProductCoupling p m) := by unfold blockProductCoupling; infer_instance |
|
|
| def blockProductEndpoint {m : ℕ → ℕ} (right : BlockCoordinate m → Bool) |
| (ω : BlockCoordinate m → Fin 6 × (ℝ × ℝ)) (i : BlockCoordinate m) : ℝ := |
| endpoint (right i) (ω i).2 |
|
|
| theorem blockProductEndpoint_law (p : ℕ → CouplingParameters) (m : ℕ → ℕ) |
| (right : BlockCoordinate m → Bool) : |
| (blockProductCoupling p m).map (blockProductEndpoint right) = |
| Measure.infinitePi (fun i : BlockCoordinate m => (p i.1).selectedLaw true (right i)) := by |
| unfold blockProductCoupling blockProductEndpoint |
| rw [Measure.infinitePi_map_pi _ (f := fun (i : BlockCoordinate m) (z : Fin 6 × (ℝ × ℝ)) => endpoint (right i) z.2) |
| (fun i => (endpoint_measurable (right i)).comp measurable_snd)] |
| simp_rw [labelled_endpoint_density] |
| rfl |
|
|
| / |
| than only to an abstract measure with matching finite moments. -/ |
| theorem blockProductEndpoint_L2_density (p : ℕ → CouplingParameters) (m : ℕ → ℕ) |
| (hm : ∀ s, (p s).m = (m s : ℝ)) (j : ℕ) |
| (htag : ∀ s, (p s).tagged = decide (s=j)) |
| (hκ : ∀ s, (p s).kappa = sourceKappa s) (right : BlockCoordinate m → Bool) : |
| ∃ g : (BlockCoordinate m → ℝ) → ℝ, |
| MemLp g 2 (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ∧ |
| (∀ᵐ x ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure), 0 ≤ g x) ∧ |
| (∫ x, g x ^ 2 ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ≤ |
| Real.exp (20*(2*(∑' s, sourceKappa s ^ 2)+1)) ∧ |
| (blockProductCoupling p m).map (blockProductEndpoint right) = |
| (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)).withDensity |
| (fun x => ENNReal.ofReal (g x)) := by |
| rw [blockProductEndpoint_law] |
| exact infinite_block_source_density p m hm j htag hκ right |
|
|
| end CouplingParameters |
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
