| import Mathlib | |
| import OAI.Probability.SKBarriers.Parisi.CDFEndpoint | |
| import OAI.Probability.SKBarriers.Parisi.CDFPotential | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped NNReal Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| namespace SK.Analytic | |
| theorem derivative_nonpos_at_right_minimum {f : ℝ → ℝ} {d : ℝ} | |
| (hd : HasDerivAt f d 1) (hm : ∀ r∈Icc (0:ℝ) 1, f 1≤f r) : d≤0 := by | |
| have H := (hasDerivAt_iff_tendsto_slope_left_right.mp hd).1 | |
| apply le_of_tendsto H | |
| have HP : ∀ᶠ r : ℝ in 𝓝[<] (1:ℝ), 0<r := | |
| (eventually_gt_nhds zero_lt_one).filter_mono nhdsWithin_le_nhds | |
| filter_upwards [HP,self_mem_nhdsWithin] with r hr0 hr1 | |
| rw [slope_def_field] | |
| exact div_nonpos_of_nonneg_of_nonpos (sub_nonneg.mpr (hm r ⟨hr0.le,hr1.le⟩)) | |
| (sub_nonpos.mpr hr1.le) | |
| theorem scalarCDFParisi_overlap_on_support {β : ℝ} (hβ : β≠0) | |
| (μ : ProbabilityMeasure ℝ) (hμ : (μ : Measure ℝ) (Icc (0:ℝ) 1)=1) | |
| (hmin : scalarCDFParisi β (cdf (μ : Measure ℝ))=finiteParisiInf β) | |
| {q : ℝ} (hqs : q∈(μ : Measure ℝ).support) : | |
| scalarCDFOverlap β (cdf (μ : Measure ℝ)) q=q := by | |
| have hs := scalarCDFParisi_support_minima hβ μ hμ hmin q hqs | |
| have ha (x : ℝ) : cdf (μ : Measure ℝ) x∈Icc (0:ℝ) 1 := ⟨cdf_nonneg _ _,cdf_le_one _ _⟩ | |
| by_cases hq0 : q=0 | |
| · subst q | |
| exact scalarCDFOverlap_zero β ha (cdf (μ : Measure ℝ)).mono | |
| by_cases hq1 : q=1 | |
| · subst q | |
| have HC := scalarCDFOverlap_continuousOn β (cdf (μ : Measure ℝ)) ha | |
| (supported_probability_cdf_one μ hμ) | |
| have HD := derivative_nonpos_at_right_minimum (scalarCDFPotential_hasDerivAt HC 1) hs.2.2 | |
| rw [cdfGradientTest_eq ⟨zero_le_one,le_rfl⟩] at HD | |
| have HB := scalarCDFOverlap_nonneg_le_one β ha (cdf (μ : Measure ℝ)).mono | |
| (show (1:ℝ)∈Icc (0:ℝ) 1 from ⟨zero_le_one,le_rfl⟩) | |
| linarith [HB.2] | |
| · exact scalarCDFParisi_overlap_on_interior_support hβ μ hμ hmin hqs | |
| ⟨lt_of_le_of_ne hs.1.1 (Ne.symm hq0),lt_of_le_of_ne hs.1.2 hq1⟩ | |
| theorem scalarCDFParisi_overlap_eq_ae {β : ℝ} (hβ : β≠0) | |
| (μ : ProbabilityMeasure ℝ) (hμ : (μ : Measure ℝ) (Icc (0:ℝ) 1)=1) | |
| (hmin : scalarCDFParisi β (cdf (μ : Measure ℝ))=finiteParisiInf β) : | |
| ∀ᵐ q ∂(μ : Measure ℝ), scalarCDFOverlap β (cdf (μ : Measure ℝ)) q=q := by | |
| have H : ∀ᵐ q ∂(μ : Measure ℝ), q∈(μ : Measure ℝ).support := (μ : Measure ℝ).support_mem_ae | |
| exact H.mono (fun _ hq => scalarCDFParisi_overlap_on_support hβ μ hμ hmin hq) | |
| theorem supported_cdf_area (μ : ProbabilityMeasure ℝ) | |
| (hμ : (μ : Measure ℝ) (Icc (0:ℝ) 1)=1) : | |
| (∫ s in Icc (0:ℝ) 1, cdf (μ : Measure ℝ) s)=1-∫ q, q ∂(μ : Measure ℝ) := by | |
| have H := supported_cdf_integral_duality μ hμ (g:=fun _ => (1:ℝ)) (integrable_const 1) | |
| simp only [mul_one,intervalIntegral.integral_const,smul_eq_mul] at H | |
| have HI : Integrable (fun q : ℝ => q) (μ : Measure ℝ) := | |
| supported_continuous_integrable μ hμ continuousOn_id | |
| rw [H,integral_sub (integrable_const 1) HI] | |
| simp only [integral_const,probReal_univ,one_smul] | |
| end SK.Analytic | |
| end | |
| end | |
| end OAI | |