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f61e547 | 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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | import OAI.Probability.FreeIsing.Critical.Endgame
import OAI.Probability.FreeIsing.Estimates.EndpointTimeScale
namespace OAI
noncomputable section
open Set MeasureTheory
namespace Problem367
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
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
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)
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)
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
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))
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
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
end Problem367
end
end OAI
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