| import OAI.Analysis.FreeEntropy.MicrostatesLower |
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| namespace OAI |
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| 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 |
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| end |
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| end OAI |
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