| import Mathlib | |
| import OAI.Probability.SKBarriers.Parisi.CDFSupportContact | |
| import OAI.Probability.SKBarriers.Parisi.CDFHeatContact | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped NNReal Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| namespace SK.Analytic | |
| theorem cdf_zero_below_least_support (μ : ProbabilityMeasure ℝ) {q z : ℝ} | |
| (hq : IsLeast (μ : Measure ℝ).support q) (hz : z<q) : | |
| cdf (μ : Measure ℝ) z=0 := by | |
| have H : (μ : Measure ℝ) (Iic z)=0 := by | |
| apply measure_mono_null (t:=(μ : Measure ℝ).supportᶜ) _ (μ : Measure ℝ).measure_compl_support | |
| intro x hx hs | |
| exact (not_le_of_gt hz) (le_trans (hq.2 hs) hx) | |
| rw [cdf_eq_real,measureReal_def,H,ENNReal.toReal_zero] | |
| theorem scalarCDFParisi_zero_mem_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) : | |
| (0:ℝ)∈(μ : Measure ℝ).support := by | |
| have HS := Measure.support_subset_of_isClosed (μ:=(μ : Measure ℝ)) | |
| isClosed_Icc (supported_probability_ae μ hμ) | |
| have HC : IsCompact (μ : Measure ℝ).support := | |
| isCompact_Icc.of_isClosed_subset (μ : Measure ℝ).isClosed_support HS | |
| obtain ⟨q,hq⟩ := HC.exists_isLeast ((μ : Measure ℝ).nonempty_support (by | |
| intro H | |
| have H1 := measure_univ (μ:=(μ : Measure ℝ)) | |
| simp only [H,Measure.coe_zero,Pi.zero_apply,zero_ne_one] at H1)) | |
| have hbound := HS hq.1 | |
| have he : q=0 := by | |
| by_contra H | |
| have hq0 : 0<q := lt_of_le_of_ne hbound.1 (Ne.symm H) | |
| have hm := (scalarCDFParisi_support_minima hβ μ hμ hmin q hq.1).2.2 | |
| exact scalarCDFPotential_not_min_heat_prefix β (cdf (μ : Measure ℝ)) | |
| (fun z => ⟨cdf_nonneg _ z,cdf_le_one _ z⟩) (supported_probability_cdf_one μ hμ) | |
| hq0 hbound.2 (fun z hz => cdf_zero_below_least_support μ hq hz.2) | |
| (scalarCDFParisi_overlap_on_support hβ μ hμ hmin hq.1) hm | |
| simpa only [he] using hq.1 | |
| theorem cdf_pos_of_zero_mem_support (μ : ProbabilityMeasure ℝ) | |
| (h0 : (0:ℝ)∈(μ : Measure ℝ).support) {s : ℝ} (hs : 0<s) : | |
| 0<cdf (μ : Measure ℝ) s := by | |
| have H : 0<(μ : Measure ℝ) (Iic s) := | |
| (Measure.mem_support_iff_forall (μ:=(μ : Measure ℝ)) 0).mp h0 _ (Iic_mem_nhds hs) | |
| rw [cdf_eq_real,measureReal_def] | |
| exact ENNReal.toReal_pos H.ne' (measure_ne_top _ _) | |
| theorem cdf_area_strict (μ : ProbabilityMeasure ℝ) | |
| (h0 : (0:ℝ)∈(μ : Measure ℝ).support) {h : ℝ} (hh : 0<h) : | |
| (∫ s in h..1, cdf (μ : Measure ℝ) s)<(∫ s in (0:ℝ)..1, cdf (μ : Measure ℝ) s) := by | |
| have HI (a b : ℝ) : IntervalIntegrable (cdf (μ : Measure ℝ)) volume a b := | |
| (cdf (μ : Measure ℝ)).mono.intervalIntegrable | |
| have HP := intervalIntegral.intervalIntegral_pos_of_pos_on (HI 0 h) | |
| (fun s hs => cdf_pos_of_zero_mem_support μ h0 hs.1) hh | |
| have HE := intervalIntegral.integral_add_adjacent_intervals (HI 0 h) (HI h 1) | |
| linarith | |
| end SK.Analytic | |
| end | |
| end | |
| end OAI | |