| import OAI.Probability.FreeIsing.Critical.Reduction |
| import OAI.Probability.FreeIsing.Critical.TimeUniqueness |
| import OAI.Probability.FreeIsing.Estimates.LogInterpolation |
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|
| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory |
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| namespace Problem367 |
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| structure CriticalEndpointBounds (D Q L Z S V C A B c K t : ℝ) : Prop where |
| noise_nonneg : 0 ≤ S |
| noise_le : S ≤ C * V |
| small_loss : S ≤ A * t ^ ((1 : ℝ) / 4) * V + |
| min (A * V) (B * Real.exp (-c * (Real.log (1 / t)) ^ 2)) |
| quadratic : Q ≤ C * (S * (1 / t + (Real.log (1 + Real.log (K / V))) ^ 2) + V ^ 2) |
| linear : L ≤ C * (V * (t ^ (-(3 : ℝ) / 4) + Real.log (1 + Real.log (K / V))) + V ^ 2) |
| square_root : Z ≤ C * (Real.sqrt S * (t ^ (-(3 : ℝ) / 4) + |
| (Real.log (1 + Real.log (K / V))) ^ ((3 : ℝ) / 2)) + V) |
| drift : D ≤ C * (V + Q + L + Real.sqrt V * Z) |
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| def criticalEndgameConstant (C H A B c late : ℝ) : ℝ := |
| max (C ^ 2 * (9 * C + 3 * H + 3) * (A + 2 * (A + B) / c + 1)) late |
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| theorem CriticalEndpointBounds.differential_bound |
| {D Q L Z S V C H A B c K t : ℝ} |
| (h : CriticalEndpointBounds D Q L Z S V C A B c K t) |
| (hC : 1 ≤ C) (hH : 1 ≤ H) (hA : 0 ≤ A) (hB : 0 ≤ B) |
| (hc : 0 < c) (hK : 1 ≤ K) (hV : 0 < V) (hVK : V ≤ K) |
| (hVH : V ≤ H) (ht : 0 < t) (ht1 : t < 1) : |
| D ≤ (C ^ 2 * (9 * C + 3 * H + 3) * (A + 2 * (A + B) / c + 1)) * |
| V * (1 + Real.log (K / V)) * |
| (1 + t ^ (-(3 : ℝ) / 4) + 1 / (t * (Real.log (1 / t)) ^ 2)) := by |
| have hI : 1 ≤ 1 + Real.log (K / V) := by |
| have := Real.log_nonneg ((one_le_div hV).2 hVK) |
| linarith |
| have hJ : 0 < (Real.log (1 / t)) ^ 2 := |
| sq_pos_of_pos (Real.log_pos ((one_lt_div ht).2 ht1)) |
| exact critical_differential_bound_of_endpoint_sums hC hH hI |
| (by positivity) h.noise_nonneg hV.le hVH h.noise_le hJ ht |
| (small_time_log_loss_of_minimum hA hB hc hK hV hVK ht ht1 h.small_loss) |
| h.quadratic h.linear h.square_root h.drift |
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| theorem eq_zero_of_ae_critical_endpoint_bounds |
| {V Q L Z S : ℝ → ℝ} {T cutoff C H A B c K late : ℝ} |
| (hT : 0 ≤ T) (hcutoff : cutoff < 1) |
| (hC : 1 ≤ C) (hH : 1 ≤ H) (hA : 0 ≤ A) (hB : 0 ≤ B) |
| (hc : 0 < c) (hK : 1 ≤ K) |
| (hV : AbsolutelyContinuousOnInterval V 0 T) (hzero : V 0 = 0) |
| (hnonneg : ∀ t ∈ Icc 0 T, 0 ≤ V t) |
| (hupperK : ∀ t ∈ Icc 0 T, V t ≤ K) |
| (hupperH : ∀ t ∈ Icc 0 T, V t ≤ H) |
| (hendpoint : ∀ᵐ t ∂volume.restrict (Ioc 0 T), t ≤ cutoff → 0 < V t → |
| CriticalEndpointBounds (deriv V t) (Q t) (L t) (Z t) (S t) (V t) C A B c K t) |
| (hlate : ∀ᵐ t ∂volume.restrict (Ioc 0 T), cutoff < t → 0 < V t → |
| deriv V t ≤ late * V t) : |
| ∀ t ∈ Icc 0 T, V t = 0 := by |
| let M := criticalEndgameConstant C H A B c late |
| have hbase : 0 ≤ C ^ 2 * (9 * C + 3 * H + 3) * (A + 2 * (A + B) / c + 1) := by |
| positivity |
| have hM : 0 ≤ M := hbase.trans (le_max_left _ _) |
| apply eq_zero_of_ae_small_critical_large_linear hT hcutoff hM hV hzero |
| hnonneg hupperK (lt_of_lt_of_le zero_lt_one hK) |
| · filter_upwards [hendpoint, ae_restrict_mem measurableSet_Ioc] with t he ht |
| intro htc hpos |
| have hmem : t ∈ Icc 0 T := ⟨ht.1.le, ht.2⟩ |
| have htpos : 0 < t := ht.1 |
| have hb := (he htc hpos).differential_bound hC hH hA hB hc hK hpos |
| (hupperK t hmem) (hupperH t hmem) ht.1 (htc.trans_lt hcutoff) |
| have hlog : 0 ≤ Real.log (K / V t) := |
| Real.log_nonneg ((one_le_div hpos).2 (hupperK t hmem)) |
| apply hb.trans |
| apply mul_le_mul_of_nonneg_right _ (by positivity) |
| apply mul_le_mul_of_nonneg_right _ (by positivity) |
| exact mul_le_mul_of_nonneg_right (le_max_left _ _) hpos.le |
| · filter_upwards [hlate] with t hl |
| intro ht hpos |
| exact (hl ht hpos).trans (mul_le_mul_of_nonneg_right (le_max_right _ _) hpos.le) |
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| end Problem367 |
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| end |
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| end OAI |
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