File size: 6,927 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 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 | import OAI.Probability.FreeIsing.Estimates.OsgoodIntegration
namespace OAI
noncomputable section
open Set Real MeasureTheory
namespace Problem367
theorem exists_last_level_crossing {V : ℝ → ℝ} {a t z : ℝ}
(hat : a ≤ t) (hV : ContinuousOn V (Icc a t))
(ha : V a < z) (ht : z < V t) :
∃ u ∈ Icc a t, V u = z ∧ ∀ s ∈ Icc u t, z ≤ V s := by
let S : Set ℝ := Icc a t ∩ V ⁻¹' {z}
have hSclosed : IsClosed S :=
hV.preimage_isClosed_of_isClosed isClosed_Icc isClosed_singleton
have hScompact : IsCompact S :=
isCompact_Icc.of_isClosed_subset hSclosed inter_subset_left
have hSne : S.Nonempty := by
obtain ⟨u, hu, hVu⟩ := intermediate_value_Icc hat hV ⟨ha.le, ht.le⟩
exact ⟨u, hu, hVu⟩
obtain ⟨u, hu, hmax⟩ := hScompact.exists_isMaxOn hSne continuousOn_id
have huat : u ∈ Icc a t := hu.1
have hVu : V u = z := hu.2
refine ⟨u, huat, hVu, ?_⟩
intro s hs
by_contra hsz
have hsz' : V s < z := lt_of_not_ge hsz
have hus : u < s := by
apply lt_of_le_of_ne hs.1
intro h
subst s
linarith
have hst : s ≤ t := hs.2
have hsub : Icc s t ⊆ Icc a t := by
intro x hx
exact ⟨huat.1.trans (hs.1.trans hx.1), hx.2⟩
obtain ⟨v, hv, hVv⟩ :=
intermediate_value_Icc hst (hV.mono hsub) ⟨hsz'.le, ht.le⟩
have hvS : v ∈ S := ⟨hsub hv, hVv⟩
have hvu : v ≤ u := hmax hvS
linarith [hv.1]
theorem eq_zero_of_positive_interval_lower_bound {V : ℝ → ℝ} {a b : ℝ}
(hV : ContinuousOn V (Icc a b)) (ha : V a = 0)
(hnonneg : ∀ t ∈ Icc a b, 0 ≤ V t)
(hbound : ∀ t ∈ Icc a b, 0 < V t →
∃ c > 0, ∀ u ∈ Icc a t,
(∀ s ∈ Icc u t, 0 < V s) → c ≤ V u) :
∀ t ∈ Icc a b, V t = 0 := by
intro t ht
by_contra hzero
have hpos : 0 < V t := lt_of_le_of_ne (hnonneg t ht) (Ne.symm hzero)
obtain ⟨c, hc, hcbound⟩ := hbound t ht hpos
let z : ℝ := min c (V t) / 2
have hz : 0 < z := by dsimp [z]; positivity
have hzc : z < c := by
dsimp [z]
linarith [min_le_left c (V t)]
have hzt : z < V t := by
dsimp [z]
linarith [min_le_right c (V t)]
have hsub : Icc a t ⊆ Icc a b := by
intro x hx
exact ⟨hx.1, hx.2.trans ht.2⟩
obtain ⟨u, hu, hVu, hafter⟩ :=
exists_last_level_crossing ht.1 (hV.mono hsub) (by simpa [ha] using hz) hzt
have hcu : c ≤ V u := hcbound u hu (fun s hs => hz.trans_le (hafter s hs))
linarith
theorem lower_bound_of_log_log_le {K z B : ℝ} (hK : 0 < K) (hz : 0 < z)
(hbound : Real.log (1 + Real.log (K / z)) ≤ B) :
min K (K / Real.exp (Real.exp B - 1)) ≤ z := by
by_cases hKz : K ≤ z
· exact (min_le_left _ _).trans hKz
have hzK : z < K := lt_of_not_ge hKz
have hratio : 1 ≤ K / z := (le_div_iff₀ hz).2 (by simpa using hzK.le)
have hinner : 0 < 1 + Real.log (K / z) := by
have := Real.log_nonneg hratio
linarith
have hlog : Real.log (K / z) ≤ Real.exp B - 1 := by
have := (Real.log_le_iff_le_exp hinner).1 hbound
linarith
have hratio_bound : K / z ≤ Real.exp (Real.exp B - 1) :=
(Real.log_le_iff_le_exp (div_pos hK hz)).1 hlog
have hlast : K / Real.exp (Real.exp B - 1) ≤ z := by
apply (div_le_iff₀ (Real.exp_pos _)).2
have := (div_le_iff₀ hz).1 hratio_bound
nlinarith
exact (min_le_right _ _).trans hlast
theorem eq_zero_of_log_log_comparison {V : ℝ → ℝ} {a b K C : ℝ}
(hV : ContinuousOn V (Icc a b)) (ha : V a = 0)
(hnonneg : ∀ t ∈ Icc a b, 0 ≤ V t) (hK : 0 < K)
(hcompare : ∀ t ∈ Icc a b, ∀ u ∈ Icc a t,
(∀ s ∈ Icc u t, 0 < V s) →
Real.log (1 + Real.log (K / V u)) ≤
Real.log (1 + Real.log (K / V t)) + C) :
∀ t ∈ Icc a b, V t = 0 := by
apply eq_zero_of_positive_interval_lower_bound hV ha hnonneg
intro t ht hpos
let B := Real.log (1 + Real.log (K / V t)) + C
refine ⟨min K (K / Real.exp (Real.exp B - 1)), ?_, ?_⟩
· exact lt_min hK (div_pos hK (Real.exp_pos _))
· intro u hu hpositive
apply lower_bound_of_log_log_le hK
(hpositive u (left_mem_Icc.2 hu.2))
exact hcompare t ht u hu hpositive
theorem eq_zero_of_ae_deriv_le_mul_self {V : ℝ → ℝ} {a b C : ℝ}
(hV : AbsolutelyContinuousOnInterval V a b) (ha : V a = 0)
(hnonneg : ∀ t ∈ Icc a b, 0 ≤ V t)
(hderiv : ∀ᵐ t : ℝ, t ∈ Icc a b → deriv V t ≤ C * V t) :
∀ t ∈ Icc a b, V t = 0 := by
intro t ht
have hab : a ≤ b := ht.1.trans ht.2
let E : ℝ → ℝ := fun x => Real.exp (-C * x)
let G : ℝ → ℝ := fun x => E x * V x
have hEsmooth : ContDiff ℝ 1 E := by dsimp [E]; fun_prop
have hEac : AbsolutelyContinuousOnInterval E a b :=
hEsmooth.contDiffOn.absolutelyContinuousOnInterval
have hGac : AbsolutelyContinuousOnInterval G a b := hEac.fun_mul hV
have hGderiv : ∀ᵐ x : ℝ, x ∈ Icc a b → deriv G x ≤ 0 := by
filter_upwards [hV.ae_differentiableAt, hderiv] with x hdiff hle hx
have hVdiff : DifferentiableAt ℝ V x := hdiff (by simpa [uIcc_of_le hab] using hx)
have hEdiff : HasDerivAt E (Real.exp (-C * x) * (-C)) x := by
simpa [E, mul_comm] using ((hasDerivAt_id x).const_mul (-C)).exp
have hGdiff := hEdiff.mul hVdiff.hasDerivAt
change deriv (E * V) x ≤ 0
rw [hGdiff.deriv]
have hb := hle hx
dsimp [E]
nlinarith [Real.exp_pos (-C * x)]
have hsub : uIcc a t ⊆ uIcc a b := by
simp only [uIcc_of_le ht.1, uIcc_of_le hab]
intro x hx
exact ⟨hx.1, hx.2.trans ht.2⟩
have hGat := hGac.mono hsub
have hint : (∫ x in a..t, deriv G x) ≤ 0 := by
calc
_ ≤ ∫ x in a..t, (0 : ℝ) := by
apply intervalIntegral.integral_mono_ae_restrict ht.1
hGat.intervalIntegrable_deriv intervalIntegrable_const
apply (ae_restrict_iff' measurableSet_Icc).2
filter_upwards [hGderiv] with x hx hxat
exact hx ⟨hxat.1, hxat.2.trans ht.2⟩
_ = 0 := by simp
rw [hGat.integral_deriv_eq_sub] at hint
have hprod : Real.exp (-C * t) * V t ≤ 0 := by
simpa [G, E, ha] using hint
exact le_antisymm ((mul_le_mul_iff_right₀ (Real.exp_pos (-C * t))).1 (by simpa using hprod))
(hnonneg t ht)
theorem eq_zero_of_ae_osgood {V k : ℝ → ℝ} {a b K : ℝ}
(hV : AbsolutelyContinuousOnInterval V a b) (ha : V a = 0)
(hnonneg : ∀ t ∈ Icc a b, 0 ≤ V t)
(hupper : ∀ t ∈ Icc a b, V t ≤ K) (hK : 0 < K)
(hk : IntervalIntegrable k volume a b)
(hbound : ∀ᵐ t ∂volume.restrict (Icc a b), 0 < V t →
deriv V t ≤ k t * V t * (1 + Real.log (K / V t))) :
∀ t ∈ Icc a b, V t = 0 := by
apply eq_zero_of_log_log_comparison
(C := ∫ t in a..b, |k t|)
(hV.continuousOn.mono Icc_subset_uIcc) ha hnonneg hK
intro t ht u hu hpositive
simpa only [osgoodTransform] using
osgoodTransform_le_add_integral_abs hu.1 hu.2 ht.2 hV hpositive hupper hk hbound
end Problem367
end
end OAI
|