File size: 3,820 Bytes
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 | import OAI.Probability.FreeIsing.Critical.Reduction
import OAI.Probability.FreeIsing.Critical.TimeUniqueness
import OAI.Probability.FreeIsing.Estimates.LogInterpolation
namespace OAI
noncomputable section
open Set MeasureTheory
namespace Problem367
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)
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
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
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)
end Problem367
end
end OAI
|