| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFMassRemainder |
| import OAI.Probability.SKBarriers.Parisi.CDFMassSegment |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem scalarCDFParisi_mass_quotient_bound (β : ℝ) (α γ : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hα1 : α 1=1) |
| (hg : ∀ z, γ z∈Icc (0:ℝ) 1) (hγ1 : γ 1=1) |
| {θ h : ℝ} (hθ : θ∈Ioc (0:ℝ) 1) (hh : 0<h) : |
| |(scalarCDFParisi β (cdfMassSegment α γ θ)-scalarCDFParisi β α)/θ- |
| (β^2/2)*∫ s in Icc (0:ℝ) 1, (γ s-α s)*(scalarCDFOverlap β α s-s)|≤ |
| (β^2/2)*cdfDistance γ α*scalarOverlapModulus β (θ*cdfDistance γ α) h := by |
| let η := cdfMassSegmentStieltjes α γ θ ⟨hθ.1.le,hθ.2⟩ |
| have H := scalarCDFParisi_mass_remainder β α η ha hα1 |
| (cdfMassSegment_bounds ha hg ⟨hθ.1.le,hθ.2⟩) (cdfMassSegment_one hα1 hγ1 θ) hh |
| change |scalarCDFParisi β (cdfMassSegment α γ θ)-scalarCDFParisi β α- |
| (β^2/2)*∫ s in Icc (0:ℝ) 1, (cdfMassSegment α γ θ s-α s)*(scalarCDFOverlap β α s-s)|≤ |
| (β^2/2)*cdfDistance (cdfMassSegment α γ θ) α* |
| scalarOverlapModulus β (cdfDistance (cdfMassSegment α γ θ) α) h at H |
| rw [cdfMassSegment_distance α γ hθ.1.le,cdfMassSegment_integral] at H |
| have he (a b L : ℝ) : |(a-b)/θ-L|=|a-b-θ*L|/θ := by |
| have E : (a-b)/θ-L=(a-b-θ*L)/θ := by field_simp [ne_of_gt hθ.1] |
| rw [E,abs_div,abs_of_pos hθ.1] |
| rw [he] |
| have HE : (β^2/2)*(θ*(∫ s in Icc (0:ℝ) 1, (γ s-α s)*(scalarCDFOverlap β α s-s)))= |
| θ*((β^2/2)*(∫ s in Icc (0:ℝ) 1, (γ s-α s)*(scalarCDFOverlap β α s-s))) := by ring |
| rw [HE] at H |
| apply (div_le_div_of_nonneg_right H hθ.1.le).trans_eq |
| field_simp [ne_of_gt hθ.1] |
|
|
| theorem scalarCDFParisi_mass_quotient_tendsto (β : ℝ) (α γ : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hα1 : α 1=1) |
| (hg : ∀ z, γ z∈Icc (0:ℝ) 1) (hγ1 : γ 1=1) : |
| Tendsto (fun θ : ℝ => (scalarCDFParisi β (cdfMassSegment α γ θ)-scalarCDFParisi β α)/θ) |
| (𝓝[>] (0:ℝ)) (𝓝 ((β^2/2)*∫ s in Icc (0:ℝ) 1, |
| (γ s-α s)*(scalarCDFOverlap β α s-s))) := by |
| let K := (β^2/2)*cdfDistance γ α |
| have hK : 0≤K := mul_nonneg (by positivity) (cdfDistance_nonneg _ _) |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| let h := ε/(24*(K+1)) |
| have hh : 0<h := div_pos hε (by positivity) |
| have he : K*(12*h)<ε := by |
| have HE : (K+1)*(12*h)=ε/2 := by dsimp [h]; field_simp; ring |
| have HH := mul_le_mul_of_nonneg_right (show K≤K+1 by linarith) (by positivity : 0≤12*h) |
| rw [HE] at HH |
| linarith |
| have HC : Continuous (fun θ : ℝ => K*scalarOverlapModulus β (θ*cdfDistance γ α) h) := |
| continuous_const.mul ((scalarOverlapModulus_continuous β h).comp (continuous_id.mul continuous_const)) |
| have HT : Tendsto (fun θ : ℝ => K*scalarOverlapModulus β (θ*cdfDistance γ α) h) |
| (𝓝[>] (0:ℝ)) (𝓝 (K*(12*h))) := by |
| simpa only [scalarOverlapModulus,zero_mul,mul_zero,zero_div,Real.exp_zero,sub_self,add_zero,zero_add] |
| using (HC.tendsto (0:ℝ)).mono_left nhdsWithin_le_nhds |
| have Hε := (tendsto_order.mp HT).2 ε he |
| have H₁ : ∀ᶠ θ : ℝ in 𝓝[>] (0:ℝ), θ<1 := (eventually_lt_nhds (by norm_num : (0:ℝ)<1)).filter_mono nhdsWithin_le_nhds |
| filter_upwards [Hε,H₁,self_mem_nhdsWithin] with θ hθε hθ1 hθ0 |
| rw [Real.dist_eq] |
| exact (scalarCDFParisi_mass_quotient_bound β α γ ha hα1 hg hγ1 ⟨hθ0,hθ1.le⟩ hh).trans_lt hθε |
|
|
| theorem scalarCDFParisi_hasDerivWithinAt_mass (β : ℝ) (α γ : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hα1 : α 1=1) |
| (hg : ∀ z, γ z∈Icc (0:ℝ) 1) (hγ1 : γ 1=1) : |
| HasDerivWithinAt (fun θ : ℝ => scalarCDFParisi β (cdfMassSegment α γ θ)) |
| ((β^2/2)*∫ s in Icc (0:ℝ) 1, (γ s-α s)*(scalarCDFOverlap β α s-s)) (Ici (0:ℝ)) 0 := by |
| have he : Ici (0:ℝ)\{0}=Ioi (0:ℝ) := by |
| ext x |
| simp only [Set.mem_sdiff,mem_Ici,mem_singleton_iff,mem_Ioi] |
| exact ⟨fun H => lt_of_le_of_ne H.1 (Ne.symm H.2),fun H => ⟨H.le,ne_of_gt H⟩⟩ |
| rw [hasDerivWithinAt_iff_tendsto_slope,he] |
| change Tendsto (fun θ => slope (fun θ : ℝ => scalarCDFParisi β (cdfMassSegment α γ θ)) 0 θ) _ _ |
| simpa only [slope_def_field,sub_zero,cdfMassSegment_zero] using |
| scalarCDFParisi_mass_quotient_tendsto β α γ ha hα1 hg hγ1 |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
