| import OAI.Analysis.FreeEntropy.ModelFlowLimit |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology ENNReal |
| namespace FiniteEntropySeparation |
| def chosenStart (h : ℝ) : ℝ := h/(4*((chosenP+1:ℕ):ℝ)) |
| theorem chosenStart_pos {h : ℝ} (hh : 0<h) : 0<chosenStart h := by unfold chosenStart; positivity |
| theorem chosenStart_le_time {h : ℝ} (hh : h<1) : chosenStart h≤chosenTime := by |
| unfold chosenStart |
| apply (div_le_iff₀ (by positivity : (0:ℝ)<4*((chosenP+1:ℕ):ℝ))).mpr |
| norm_num [chosenP,chosenTime] at * |
| linarith |
| theorem chosen_short_bound {h : ℝ} (hh : 0<h) : |
| (1/2:ℝ)*(∫u in h..(h+chosenStart h),Model.heatFisherReal chosenP u)≤(1:ℝ)/8 := by |
| have hb := Model.heatFisher_short_bound chosenP chosenP_ge_two hh (chosenStart_pos hh).le |
| have he : ((chosenP+1:ℕ):ℝ)*chosenStart h/(2*h)=(1:ℝ)/8 := by |
| unfold chosenStart |
| field_simp |
| ; ring |
| rwa [he] at hb |
| theorem flow_to_chiStar_gap {h β : ℝ} (hh0 : 0<h) (hh1 : h<1) |
| (hflow : β+(1/2:ℝ)*(∫u in (h+chosenStart h)..(h+chosenTime),Model.heatFisherReal chosenP u)≤ |
| Model.gaussianEntropy chosenP (h+chosenTime)-1) : |
| (β:EReal)≤chiStar (Model.τ chosenP) (Model.X chosenP h) (Model.auxiliaryNoise chosenP)-(1/2:EReal) := by |
| have ht : 0<h+chosenTime := by linarith [chosenTime_pos] |
| have hstart := chosenStart_pos hh0 |
| have hst := chosenStart_le_time hh1 |
| have ha : h≤h+chosenStart h := by linarith |
| have hab : h+chosenStart h≤h+chosenTime := by linarith |
| have hi1 := Model.heatFisherReal_intervalIntegrable chosenP chosenP_ge_two h (h+chosenStart h) hh0 ha |
| have hi2 := Model.heatFisherReal_intervalIntegrable chosenP chosenP_ge_two (h+chosenStart h) (h+chosenTime) (by linarith) hab |
| have he := intervalIntegral.integral_add_adjacent_intervals hi1 hi2 |
| have hf := Model.chiStar_flow chosenP chosenP_ge_two h (h+chosenTime) hh0 (by linarith [chosenTime_pos]) |
| rw [Model.chiStar_tail_formula chosenP chosenP_ge_two h hh0, |
| Model.chiStar_tail_formula chosenP chosenP_ge_two (h+chosenTime) ht,← EReal.coe_sub] at hf |
| have hfR := EReal.coe_injective hf |
| have htail := Model.entropy_deficit_le chosenP chosenP_ge_two (h+chosenTime) ht (chosen_large_noise hh0.le) |
| have hshort := chosen_short_bound hh0 |
| rw [Model.chiStar_tail_formula chosenP chosenP_ge_two h hh0] |
| change (β:EReal)≤((Model.gaussianEntropy chosenP h-(1/2:ℝ)*(∫u in Ioi h,Model.fisherDeficit chosenP u):ℝ):EReal)-(((1/2:ℝ):ℝ):EReal) |
| rw [← EReal.coe_sub,EReal.coe_le_coe_iff] |
| linarith |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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