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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 | import OAI.Probability.FreeIsing.Critical.Comparison
namespace OAI
noncomputable section
open Set MeasureTheory
namespace Problem367
theorem eq_zero_of_ae_criticalTimeCoefficient {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)
(hbound : ∀ᵐ t ∂volume.restrict (Icc 0 T), 0 < V t →
deriv V t ≤ criticalTimeCoefficient cutoff C t * V t *
(1 + Real.log (K / V t))) :
∀ t ∈ Icc 0 T, V t = 0 := by
exact eq_zero_of_ae_osgood hV hzero hnonneg hupper hK
(criticalTimeCoefficient_intervalIntegrable C hT hc) hbound
theorem eq_zero_of_ae_small_critical_large_linear
{V : ℝ → ℝ} {T K cutoff C : ℝ}
(hT : 0 ≤ T) (hc : cutoff < 1) (hC : 0 ≤ C)
(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) :
∀ 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, ae_restrict_mem measurableSet_Ioc] with t hs hl ht
intro hpos
by_cases htc : t ≤ cutoff
· simpa only [criticalTimeCoefficient, ite_eq_left htc, mul_assoc, mul_left_comm,
mul_comm] using hs htc hpos
· have hlog : 0 ≤ Real.log (K / V t) :=
Real.log_nonneg ((le_div_iff₀ hpos).2 (by
simpa using hupper t ⟨ht.1.le, ht.2⟩))
have hmul : C * V t ≤ C * V t * (1 + Real.log (K / V t)) := by
nlinarith [mul_nonneg (mul_nonneg hC hpos.le) hlog]
simpa only [criticalTimeCoefficient, ite_eq_right htc, mul_one] using
(hl (lt_of_not_ge htc) hpos).trans hmul
end Problem367
end
end OAI
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