File size: 3,196 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 | import OAI.Analysis.FreeEntropy.FreeEntropyFlow
namespace OAI
noncomputable section
open MeasureTheory Filter Finset Set
open scoped Topology
namespace FiniteEntropySeparation
namespace Model
variable (p : ℕ) [NeZero p]
theorem isotropic_gaussian_correction {a : ℝ} (ha : 0<a) :
((p+1:ℕ):ℝ)/2*Real.log (2*Real.pi*Real.exp 1*(1+a))≤gaussianEntropy p a+1/(2*a) := by
have hz : 0<2*Real.pi*Real.exp 1 := by positivity
have hv := variance_nonneg p (Fin.last p)
have hl : Real.log (1+a)-Real.log (a+variance p (Fin.last p))≤1/a := by
have he : Real.log (1+a)-Real.log a≤1/a := by
rw [← Real.log_div (by positivity) ha.ne']
have hb := Real.log_le_sub_one_of_pos (div_pos (by positivity : 0<1+a) ha)
have he : (1+a)/a-1=1/a := by field_simp; ring
rwa [he] at hb
have hm := Real.log_le_log ha (show a≤a+variance p (Fin.last p) by linarith)
linarith
rw [Real.log_mul hz.ne' (by positivity)]
unfold gaussianEntropy
rw [Fin.sum_univ_castSucc]
simp only [variance_castSucc,sum_const,card_univ,Fintype.card_fin,nsmul_eq_mul]
push_cast
have he : 1/(2*a)=(1/2:ℝ)*(1/a) := by ring
rw [he,add_comm a 1]
nlinarith
end Model
theorem chosen_cumulantConstant : Model.cumulantConstant chosenP=(2:ℝ)^23 := by
norm_num [Model.cumulantConstant,chosenP]
theorem chosen_large_noise {h : ℝ} (hh : 0≤h) :
2*(Model.cumulantConstant chosenP)^2≤h+chosenTime := by
rw [chosen_cumulantConstant]
norm_num [chosenTime] at *
linarith
theorem chosen_gaussian_correction {h : ℝ} (hh : 0<h) :
((chosenP+1:ℕ):ℝ)/2*Real.log (2*Real.pi*Real.exp 1*(1+h+chosenTime))≤
Model.gaussianEntropy chosenP (h+chosenTime)+(1:ℝ)/2 := by
have ht : 1≤h+chosenTime := by norm_num [chosenTime] at *; linarith
have hb := Model.isotropic_gaussian_correction chosenP (show 0<h+chosenTime by linarith)
have hc : 1/(2*(h+chosenTime))≤(1:ℝ)/2 := by
apply (div_le_iff₀ (by positivity : 0<2*(h+chosenTime))).mpr
linarith
rw [← add_assoc] at hb
linarith
theorem chosen_terminal_gaussian {D : ℕ → ℕ} (hD0 : ∀k,0<D k) (hD : Tendsto D atTop atTop)
{h R δ Δ : ℝ} (hh0 : 0<h) (hh1 : h<1) (hR : 0≤R)
(hδ : 0<δ) (hΔ : 0<Δ) (hδΔ : δ≤Δ/4)
(hsep : ∀i j : Fin chosenP,i≠j → Δ≤|equidiscreteAtom chosenP i-equidiscreteAtom chosenP j|)
(hq : ∀l : Fin chosenP,∀r : ℝ,|r-equidiscreteAtom chosenP l|≤δ → (3:ℝ)/4≤((spectralBasis chosenP l).eval r)^2)
(hh : spectralGood chosenP δ Δ (wordMoment (Model.τ chosenP) (Model.X chosenP h)))
(hn : ∀k,(diagonalMicrostates (Model.τ chosenP) (Model.X chosenP h) R k (D k)).Nonempty)
(μ : (k:ℕ) → Measure (MatrixCoordinates (chosenP+1) (D k))) [∀k,IsProbabilityMeasure (μ k)]
(hμ : ∀k,∀ᵐa ∂μ k,a∈diagonalMicrostates (Model.τ chosenP) (Model.X chosenP h) R k (D k)) :
∀ᶠk in atTop,normalizedMatrixEntropy (μ k) chosenTime≤Model.gaussianEntropy chosenP (h+chosenTime)-1 := by
filter_upwards [chosen_terminal_isotropic hD0 hD hh0 hh1 hR hδ hΔ hδΔ hsep hq hh hn μ hμ] with k hk
linarith [chosen_gaussian_correction hh0]
end FiniteEntropySeparation
end
end OAI
|