| import OAI.Probability.FreeIsing.Critical.Endgame |
| import OAI.Probability.FreeIsing.Estimates.EndpointTimeScale |
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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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| theorem eq_zero_of_ae_small_critical_large_osgood |
| {V : ℝ → ℝ} {T K cutoff C : ℝ} |
| (hT : 0 ≤ T) (hc : cutoff < 1) |
| (hV : AbsolutelyContinuousOnInterval V 0 T) (hzero : V 0 = 0) |
| (hnonneg : ∀ t ∈ Icc 0 T, 0 ≤ V t) |
| (hupper : ∀ t ∈ Icc 0 T, V t ≤ K) (hK : 0 < K) |
| (hsmall : ∀ᵐ t ∂volume.restrict (Ioc 0 T), t ≤ cutoff → 0 < V t → |
| deriv V t ≤ C * V t * (1 + Real.log (K / V t)) * |
| (1 + t ^ (-(3 : ℝ) / 4) + 1 / (t * (Real.log (1 / t)) ^ 2))) |
| (hlarge : ∀ᵐ t ∂volume.restrict (Ioc 0 T), cutoff < t → 0 < V t → |
| deriv V t ≤ C * V t * (1 + Real.log (K / V t))) : |
| ∀ t ∈ Icc 0 T, V t = 0 := by |
| apply eq_zero_of_ae_criticalTimeCoefficient hT hc hV hzero hnonneg hupper hK |
| rw [← restrict_Ioc_eq_restrict_Icc] |
| filter_upwards [hsmall, hlarge] with t hs hl |
| intro hpos |
| by_cases ht : t ≤ cutoff |
| · simpa only [criticalTimeCoefficient, ite_eq_left ht, mul_assoc, mul_left_comm, |
| mul_comm] using hs ht hpos |
| · simpa only [criticalTimeCoefficient, ite_eq_right ht, mul_one] using |
| hl (lt_of_not_ge ht) hpos |
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| theorem endpoint_sums_bounded_scale_reduction |
| {D Q L Z S V C H I R W Rmax Wmax : ℝ} |
| (hC : 1 ≤ C) (hH : 1 ≤ H) (hI : 1 ≤ I) |
| (hS : 0 ≤ S) (hV : 0 ≤ V) (hVH : V ≤ H) |
| (hSV : S ≤ C * V) (hR : 0 ≤ R) (hW : 0 ≤ W) |
| (hRmax : R ≤ Rmax) (hWmax : W ≤ Wmax) |
| (hQ : Q ≤ C * (S * (R + (Real.log I) ^ 2) + V ^ 2)) |
| (hL : L ≤ C * (V * (W + Real.log I) + V ^ 2)) |
| (hZ : Z ≤ C * (Real.sqrt S * (W + (Real.log I) ^ ((3 : ℝ) / 2)) + V)) |
| (hD : D ≤ C * (V + Q + L + Real.sqrt V * Z)) : |
| D ≤ (C ^ 2 * (9 * C + 3 * H + 3) * (C * Rmax + 1 + Wmax)) * V * I := by |
| have hC0 : 0 ≤ C := zero_le_one.trans hC |
| have hRmax0 : 0 ≤ Rmax := hR.trans hRmax |
| have hWmax0 : 0 ≤ Wmax := hW.trans hWmax |
| have hVI : V ≤ V * I := by nlinarith |
| have hnoise : S * R ≤ (C * Rmax) * (V * I) := by |
| calc |
| S * R ≤ (C * V) * Rmax := mul_le_mul hSV hRmax hR (by positivity) |
| _ = (C * Rmax) * V := by ring |
| _ ≤ (C * Rmax) * (V * I) := |
| mul_le_mul_of_nonneg_left hVI (by positivity) |
| have hlinear : V * (I + W) ≤ (1 + Wmax) * (V * I) := by |
| calc |
| V * (I + W) ≤ V * I + Wmax * V := by |
| nlinarith [mul_le_mul_of_nonneg_left hWmax hV] |
| _ ≤ V * I + Wmax * (V * I) := |
| add_le_add le_rfl (mul_le_mul_of_nonneg_left hVI hWmax0) |
| _ = (1 + Wmax) * (V * I) := by ring |
| have hscale := endpoint_sums_critical_reduction hC hH hI hS hV hVH hSV |
| hR hW hQ hL hZ hD |
| calc |
| D ≤ C ^ 2 * (9 * C + 3 * H + 3) * (S * R + V * (I + W)) := hscale |
| _ ≤ C ^ 2 * (9 * C + 3 * H + 3) * |
| ((C * Rmax) * (V * I) + (1 + Wmax) * (V * I)) := |
| mul_le_mul_of_nonneg_left (add_le_add hnoise hlinear) (by positivity) |
| _ = _ := by ring |
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| structure CriticalSpatialEndpointBounds (D Q L Z S V C K ω : ℝ) : Prop where |
| noise_nonneg : 0 ≤ S |
| noise_le : S ≤ C * V |
| quadratic : Q ≤ C * (S * ((ω⁻¹) ^ 2 + (Real.log (1 + Real.log (K / V))) ^ 2) + V ^ 2) |
| linear : L ≤ C * (V * (ω⁻¹ + Real.log (1 + Real.log (K / V))) + V ^ 2) |
| square_root : Z ≤ C * (Real.sqrt S * (ω⁻¹ * Real.sqrt (ω⁻¹) + |
| (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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| theorem inv_le_inv_mul_sqrt {ω : ℝ} (hω : 0 < ω) (hω1 : ω ≤ 1) : |
| ω⁻¹ ≤ ω⁻¹ * Real.sqrt (ω⁻¹) := by |
| have hi : 1 ≤ ω⁻¹ := (one_le_inv₀ hω).2 hω1 |
| have hs : 1 ≤ Real.sqrt (ω⁻¹) := by |
| exact (Real.one_le_sqrt).2 hi |
| simpa using mul_le_mul_of_nonneg_left hs (inv_nonneg.mpr hω.le) |
|
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| theorem CriticalSpatialEndpointBounds.small_time |
| {D Q L Z S V C A B c K t : ℝ} |
| (h : CriticalSpatialEndpointBounds D Q L Z S V C K (Real.sqrt t)) |
| (hC : 0 ≤ C) (hV : 0 ≤ V) (ht : 0 < t) (ht1 : t ≤ 1) |
| (hsmall : S ≤ A * t ^ ((1 : ℝ) / 4) * V + |
| min (A * V) (B * Real.exp (-c * (Real.log (1 / t)) ^ 2))) : |
| CriticalEndpointBounds D Q L Z S V C A B c K t := by |
| refine ⟨h.noise_nonneg, h.noise_le, hsmall, ?_, ?_, ?_, h.drift⟩ |
| · simpa only [inv_sqrt_sq_time ht.le] using h.quadratic |
| · apply h.linear.trans |
| apply mul_le_mul_of_nonneg_left _ hC |
| apply add_le_add _ le_rfl |
| apply mul_le_mul_of_nonneg_left _ hV |
| exact add_le_add (inv_sqrt_le_time_three_quarters ht ht1) le_rfl |
| · simpa only [inv_sqrt_mul_sqrt_time ht] using h.square_root |
|
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| def criticalLateCoefficient (C H ε : ℝ) : ℝ := |
| C ^ 2 * (9 * C + 3 * H + 3) * |
| (C * (ε⁻¹) ^ 2 + 1 + ε⁻¹ * Real.sqrt (ε⁻¹)) |
|
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| theorem CriticalSpatialEndpointBounds.late_time |
| {D Q L Z S V C H K ω ε : ℝ} |
| (h : CriticalSpatialEndpointBounds D Q L Z S V C K ω) |
| (hC : 1 ≤ C) (hH : 1 ≤ H) (hV : 0 < V) (hVK : V ≤ K) (hVH : V ≤ H) |
| (hε : 0 < ε) (hεω : ε ≤ ω) (hω1 : ω ≤ 1) : |
| D ≤ criticalLateCoefficient C H ε * V * (1 + Real.log (K / V)) := by |
| have hω : 0 < ω := hε.trans_le hεω |
| have hωinv : 0 ≤ ω⁻¹ := inv_nonneg.mpr hω.le |
| have hεinv : 0 ≤ ε⁻¹ := inv_nonneg.mpr hε.le |
| have hinv : ω⁻¹ ≤ ε⁻¹ := (inv_le_inv₀ hω hε).2 hεω |
| have hI : 1 ≤ 1 + Real.log (K / V) := by |
| have := Real.log_nonneg ((one_le_div hV).2 hVK) |
| linarith |
| have hlinear : L ≤ C * (V * |
| (ω⁻¹ * Real.sqrt (ω⁻¹) + Real.log (1 + Real.log (K / V))) + V ^ 2) := by |
| apply h.linear.trans |
| apply mul_le_mul_of_nonneg_left _ (zero_le_one.trans hC) |
| apply add_le_add _ le_rfl |
| apply mul_le_mul_of_nonneg_left _ hV.le |
| exact add_le_add (inv_le_inv_mul_sqrt hω hω1) le_rfl |
| exact endpoint_sums_bounded_scale_reduction hC hH hI h.noise_nonneg hV.le hVH |
| h.noise_le (sq_nonneg _) (mul_nonneg hωinv (Real.sqrt_nonneg _)) |
| ((sq_le_sq₀ hωinv hεinv).2 hinv) |
| (mul_le_mul hinv (Real.sqrt_le_sqrt hinv) (Real.sqrt_nonneg _) hεinv) |
| h.quadratic hlinear h.square_root h.drift |
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| theorem eq_zero_of_ae_critical_spatial_endpoint_bounds |
| {V Q L Z S : ℝ → ℝ} {T cutoff C H A B c K : ℝ} |
| (hT : 0 ≤ T) (hcutoff0 : 0 < cutoff) (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), 0 < V t → |
| CriticalSpatialEndpointBounds (deriv V t) (Q t) (L t) (Z t) (S t) (V t) C K |
| (min (Real.sqrt t) 1)) |
| (hsmall : ∀ᵐ t ∂volume.restrict (Ioc 0 T), t ≤ cutoff → 0 < V t → |
| S t ≤ A * t ^ ((1 : ℝ) / 4) * V t + |
| min (A * V t) (B * Real.exp (-c * (Real.log (1 / t)) ^ 2))) : |
| ∀ t ∈ Icc 0 T, V t = 0 := by |
| let early := C ^ 2 * (9 * C + 3 * H + 3) * (A + 2 * (A + B) / c + 1) |
| let late := criticalLateCoefficient C H (Real.sqrt cutoff) |
| let M := max early late |
| apply eq_zero_of_ae_small_critical_large_osgood (C := M) hT hcutoff hV hzero |
| hnonneg hupperK (zero_lt_one.trans_le hK) |
| · filter_upwards [hendpoint, hsmall, ae_restrict_mem measurableSet_Ioc] with t he hs ht |
| intro htc hpos |
| have hmem : t ∈ Icc 0 T := ⟨ht.1.le, ht.2⟩ |
| have ht1 : t < 1 := htc.trans_lt hcutoff |
| have htpos : 0 < t := ht.1 |
| have hscale : min (Real.sqrt t) 1 = Real.sqrt t := |
| min_eq_left (Real.sqrt_le_one.2 ht1.le) |
| have he' := he hpos |
| rw [hscale] at he' |
| have hb := (he'.small_time (zero_le_one.trans hC) hpos.le ht.1 ht1.le |
| (hs htc hpos)).differential_bound hC hH hA hB hc hK hpos |
| (hupperK t hmem) (hupperH t hmem) ht.1 ht1 |
| 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 early late) hpos.le |
| · 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 hscale : Real.sqrt cutoff ≤ min (Real.sqrt t) 1 := |
| le_min (Real.sqrt_le_sqrt htc.le) (Real.sqrt_le_one.2 hcutoff.le) |
| have hb := (he hpos).late_time hC hH hpos (hupperK t hmem) (hupperH t hmem) |
| (Real.sqrt_pos.2 hcutoff0) hscale (min_le_right _ _) |
| 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) |
| exact mul_le_mul_of_nonneg_right (le_max_right early late) hpos.le |
|
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| end Problem367 |
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| end |
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| end OAI |
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