| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.QuantileTransport |
| import OAI.Probability.SKBarriers.Parisi.CDFOverlapConvergence |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem scalarOverlapModulus_continuous (β h : ℝ) : |
| Continuous (fun D : ℝ => scalarOverlapModulus β D h) := by |
| unfold scalarOverlapModulus |
| fun_prop |
|
|
| theorem scalarOverlapModulus_mono (β : ℝ) {h : ℝ} (hh : 0≤h) : |
| Monotone (fun D : ℝ => scalarOverlapModulus β D h) := by |
| intro D E hDE |
| unfold scalarOverlapModulus |
| have hc : 0≤4*scalarTimeMassConstant β+2*scalarTimeMassConstantK β 3 := |
| add_nonneg (mul_nonneg (by norm_num) (scalarTimeMassConstant_nonneg β)) |
| (mul_nonneg (by norm_num) (scalarTimeMassConstantK_nonneg β 3)) |
| apply add_le_add |
| · exact add_le_add (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hDE hc) hh) le_rfl |
| · apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| apply sub_le_sub_right |
| apply Real.exp_le_exp.mpr |
| exact mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hDE (scalarTimeMassConstant_nonneg β)) (by norm_num) |
| |
| theorem scalarOverlapModulus_nonneg (β : ℝ) {D h : ℝ} (hD : 0≤D) (hh : 0≤h) : |
| 0 ≤ scalarOverlapModulus β D h := by |
| have H := scalarOverlapModulus_mono β hh hD |
| simp only [scalarOverlapModulus,mul_zero,zero_div,Real.exp_zero,sub_self,add_zero,zero_add] at H |
| exact (mul_nonneg (by norm_num : (0:ℝ)≤12) hh).trans H |
|
|
| theorem quantileCDF_segment_distance {k : ℕ} (A B : Fin (k+1) → ℝ) |
| (hA : A∈admissibleQuantiles k) (hB : B∈admissibleQuantiles k) |
| {t : ℝ} (ht : t∈Icc (0:ℝ) 1) |
| (hQ : (1-t) • A+t • B∈admissibleQuantiles k) : |
| cdfDistance (quantileCDF k ((1-t) • A+t • B)) (quantileCDF k A)= |
| t*cdfDistance (quantileCDF k B) (quantileCDF k A) := by |
| rw [cdfDistance,cdfDistance,quantileCDF_L1_eq _ _ hQ hA,quantileCDF_L1_eq _ _ hB hA,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| have he : ((1-t) • A+t • B) j-A j=t*(B j-A j) := by |
| simp only [Pi.add_apply,Pi.smul_apply,smul_eq_mul]; ring |
| rw [he,abs_mul,abs_of_nonneg ht.1] |
| ring |
|
|
| def cdfGradientTest (β : ℝ) (α : ℝ → ℝ) (s : ℝ) : ℝ := |
| s-scalarCDFOverlap β α (projIcc 0 1 (by norm_num) s) |
|
|
| theorem cdfGradientTest_continuous {β : ℝ} {α : ℝ → ℝ} |
| (hΓ : ContinuousOn (scalarCDFOverlap β α) (Icc (0:ℝ) 1)) : |
| Continuous (cdfGradientTest β α) := |
| continuous_id.sub ((continuousOn_iff_continuous_domRestrict.mp hΓ).comp continuous_projIcc) |
|
|
| theorem cdfGradientTest_eq {β : ℝ} {α : ℝ → ℝ} {s : ℝ} (hs : s∈Icc (0:ℝ) 1) : |
| cdfGradientTest β α s=s-scalarCDFOverlap β α s := by |
| simp only [cdfGradientTest,projIcc_of_mem _ hs] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
