| 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 | |