File size: 3,886 Bytes
67aee1d | 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 | import OAI.Analysis.FreeEntropy.EntropyIntegral
namespace OAI
universe u1
noncomputable section
open scoped ENNReal NNReal Topology
open MeasureTheory Filter Set
namespace FiniteEntropySeparation
theorem lintegral_ofReal_finite {α : Type u1} [MeasurableSpace α] {μ : Measure α}
{f : α → ℝ} (hf : Integrable f μ) : (∫⁻ x, ENNReal.ofReal (f x) ∂μ) < ∞ :=
(lintegral_ofReal_le_lintegral_enorm _).trans_lt hf.hasFiniteIntegral
theorem reciprocal_posPart_finite {n a : ℝ} (ha : 0 < a) :
(∫⁻ s in Ioi (0 : ℝ), ENNReal.ofReal (n/(1+s)) - ENNReal.ofReal (n/(a+s))) < ∞ := by
have hh := lintegral_ofReal_finite ((reciprocalDifference_integrable ha).const_mul n)
apply lt_of_le_of_lt _ hh
apply setLIntegral_mono' measurableSet_Ioi
intro s hs
have hs : 0 < s := hs
by_cases hn : 0 ≤ n
· rw [← ENNReal.ofReal_sub _ (div_nonneg hn (by linarith : 0 ≤ a+s))]
apply le_of_eq
congr 1
simp [reciprocalDifference, mul_sub, div_eq_mul_inv]
· have hz : n / (1+s) ≤ 0 := div_nonpos_of_nonpos_of_nonneg (le_of_not_ge hn) (by linarith)
rw [ENNReal.ofReal_of_nonpos hz, zero_tsub]
exact bot_le
theorem reciprocal_negPart_finite {n a : ℝ} (ha : 0 < a) :
(∫⁻ s in Ioi (0 : ℝ), ENNReal.ofReal (n/(a+s)) - ENNReal.ofReal (n/(1+s))) < ∞ := by
have hh := lintegral_ofReal_finite (((reciprocalDifference_integrable ha).const_mul n).neg)
apply lt_of_le_of_lt _ hh
apply setLIntegral_mono' measurableSet_Ioi
intro s hs
have hs : 0 < s := hs
by_cases hn : 0 ≤ n
· rw [← ENNReal.ofReal_sub _ (div_nonneg hn (by linarith : 0 ≤ 1+s))]
apply le_of_eq
congr 1
simp [reciprocalDifference, mul_sub, div_eq_mul_inv]
· have hz : n / (a+s) ≤ 0 := div_nonpos_of_nonpos_of_nonneg (le_of_not_ge hn) (by linarith)
rw [ENNReal.ofReal_of_nonpos hz, zero_tsub]
exact bot_le
theorem entropy_parts_finite {f : ℝ → ℝ≥0∞} {n a b : ℝ}
(ha : 0 < a) (hb : 0 < b)
(hl : ∀ s, 0 < s → ENNReal.ofReal (n/(b+s)) ≤ f s)
(hu : ∀ s, 0 < s → f s ≤ ENNReal.ofReal (n/(a+s))) :
(∫⁻ s in Ioi (0 : ℝ), ENNReal.ofReal (n/(1+s)) - f s) < ∞ ∧
(∫⁻ s in Ioi (0 : ℝ), f s - ENNReal.ofReal (n/(1+s))) < ∞ := by
constructor
· apply lt_of_le_of_lt _ (reciprocal_posPart_finite (n := n) hb)
exact setLIntegral_mono' measurableSet_Ioi (fun s hs => tsub_le_tsub_left (hl s hs) _)
· apply lt_of_le_of_lt _ (reciprocal_negPart_finite (n := n) ha)
exact setLIntegral_mono' measurableSet_Ioi (fun s hs => tsub_le_tsub_right (hu s hs) _)
theorem finite_entropy_expression (c : ℝ) {P N : ℝ≥0∞} (hP : P < ∞) (hN : N < ∞) :
∃ r : ℝ, (c : EReal) + (1/2 : EReal) * P.toEReal - (1/2 : EReal) * N.toEReal = r := by
refine ⟨c + (1/2 : ℝ) * P.toReal - (1/2 : ℝ) * N.toReal, ?_⟩
rw [← EReal.coe_ennreal_toReal hP.ne, ← EReal.coe_ennreal_toReal hN.ne]
norm_cast
namespace Model
theorem chiStar_real (p : ℕ) [NeZero p] (hp : 2 ≤ p) (h : ℝ) (hh : 0 < h) :
∃ r : ℝ, chiStar (τ p) (X p h) (auxiliaryNoise p) = r := by
have hb : 0 < h+4 := by linarith
have hf := entropy_parts_finite (f := fun s => freeFisher (τ p) (doublyNoised p h s))
(n := (p+1 : ℕ)) hh hb
(fun s hs => by simpa [add_assoc, add_comm, add_left_comm] using
doublyNoised_fisher_lower p hp h s hh.le hs.le (by linarith))
(fun s hs => doublyNoised_fisher_le p h s hh.le hs.le (by linarith))
exact finite_entropy_expression _ hf.1 hf.2
theorem chiStar_finite (p : ℕ) [NeZero p] (hp : 2 ≤ p) (h : ℝ) (hh : 0 < h) :
(⊥ : EReal) < chiStar (τ p) (X p h) (auxiliaryNoise p) ∧
chiStar (τ p) (X p h) (auxiliaryNoise p) < ⊤ := by
obtain ⟨r,hr⟩ := chiStar_real p hp h hh
rw [hr]
exact ⟨EReal.bot_lt_coe r, EReal.coe_lt_top r⟩
end Model
end FiniteEntropySeparation
end
end OAI
|