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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | 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
/-- The source coordinates, retaining the block and the position in that block. -/
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
/-- Actual augmented law on the block-indexed coordinates. -/
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
/-- The density belongs to the actual law of the selected endpoint map, rather
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
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