File size: 4,418 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 | import OAI.Analysis.FreeEntropy.MicrostatesLower
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology ENNReal
namespace FiniteEntropySeparation
theorem chosen_cutoff_real_bound {h R δ Δ β : 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)))
(hβ : (β:EReal)<chiCutoff (Model.τ chosenP) (Model.X chosenP h) R) :
(β:EReal)≤chiStar (Model.τ chosenP) (Model.X chosenP h) (Model.auxiliaryNoise chosenP)-(1/2:EReal) := by
obtain ⟨D,hDm,hD0,hvol⟩ := exists_microstate_dimensions (Model.τ chosenP) (Model.X chosenP h) hβ
let S := fun k => diagonalMicrostates (Model.τ chosenP) (Model.X chosenP h) R k (D k)
have hs k : MeasurableSet (S k) := diagonalMicrostates_measurable _ _ _ _ _
have hv0 k : volume (S k)≠0 := matrixVolumeEntropy_ne_bot_volume (hD0 k) _
(ne_bot_of_gt ((EReal.bot_lt_coe β).trans (hvol k)))
have hvt k : volume (S k)≠⊤ := ne_of_lt (microstates_volume_lt_top (Model.τ chosenP) (Model.X chosenP h) R hR k (D k) _)
let μ := fun k => cond volume (S k)
have hprob : ∀k,IsProbabilityMeasure (μ k) := fun k => cond_isProbabilityMeasure_of_finite (hv0 k) (hvt k)
let : ∀k,IsProbabilityMeasure (μ k) := hprob
have hμ k : ∀ᵐa ∂μ k,a∈S k := ae_cond_mem (hs k)
have hn k : (S k).Nonempty := by
apply Set.nonempty_iff_ne_empty.mpr
intro he
apply hv0 k
rw [he,measure_empty]
have hinit (k : ℕ) : β≤normalizedMatrixEntropy (μ k) (chosenStart h) := by
have hnorm : ∀ᵐa ∂μ k,‖a‖≤Real.sqrt (((chosenP+1:ℕ):ℝ)*D k)*R :=
FreeTensor.diagonalMicrostates_norm_ae (Model.X chosenP h) hR μ hμ k
have hb := normalizedMatrixEntropy_cond_lower (hD0 k) (S k) (hv0 k) (hvt k)
(show 0≤Real.sqrt (((chosenP+1:ℕ):ℝ)*D k)*R by positivity) hnorm (chosenStart_pos hh0)
have he : β<((D k:ℝ)^2)⁻¹*Real.log (volume (S k)).toReal+((chosenP+1:ℕ):ℝ)/2*Real.log (D k) := by
apply EReal.coe_lt_coe_iff.mp
have hbvol : (β:EReal)< matrixVolumeEntropy (S k) := hvol k
rwa [matrixVolumeEntropy_coe (S k) (hv0 k) (hvt k)] at hbvol
exact he.le.trans (by simpa only [zpow_neg,zpow_ofNat] using hb)
have hterm := chosen_terminal_gaussian hD0 hDm.tendsto_atTop hh0 hh1 hR hδ hΔ hδΔ hsep hq hh hn μ hμ
have hflow := Model.matrixFlow_limit chosenP chosenP_ge_two hD0 hDm.tendsto_atTop hh0 hR hn μ hμ
(chosenStart_pos hh0) (chosenStart_le_time hh1) hinit hterm
exact flow_to_chiStar_gap hh0 hh1 hflow
theorem chosen_cutoff_bound {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))) :
chiCutoff (Model.τ chosenP) (Model.X chosenP h) R≤
chiStar (Model.τ chosenP) (Model.X chosenP h) (Model.auxiliaryNoise chosenP)-(1/2:EReal) := by
by_contra he
obtain ⟨β,hb,hb'⟩ := EReal.exists_between_coe_real (lt_of_not_ge he)
exact (not_lt_of_ge (chosen_cutoff_real_bound hh0 hh1 hR hδ hΔ hδΔ hsep hq hh hb')) hb
theorem chosen_chi_bound {h δ Δ : ℝ} (hh0 : 0<h) (hh1 : h<1)
(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))) :
chi (Model.τ chosenP) (Model.X chosenP h)≤
chiStar (Model.τ chosenP) (Model.X chosenP h) (Model.auxiliaryNoise chosenP)-(1/2:EReal) := by
apply iSup_le
intro R
exact chosen_cutoff_bound hh0 hh1 R.property.le hδ hΔ hδΔ hsep hq hh
end FiniteEntropySeparation
end
end OAI
|