| 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 | |