| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarOverlapContinuity |
| import OAI.Probability.SKBarriers.Replicas.TwoPartitionOverlap |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem scalarCDFOverlap_continuousOn_partition (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hαm : Monotone α) |
| (n : ℕ) (q : Fin (n+1) → ℝ) (hq : Monotone q) |
| (hq0 : q 0=0) (hq1 : q (Fin.last n)=1) |
| (m : Fin n → ℝ) (hm : ∀ i, m i∈Icc (0:ℝ) 1) |
| (hmodel : ∀ i : Fin n, ∀ z∈Ico (q i.castSucc) (q i.succ), α z=m i) : |
| ContinuousOn (scalarCDFOverlap β α) (Icc (0:ℝ) 1) := by |
| let v : ℝ → Fin (n+n) → ℝ := fun r => |
| Fin.append (timeGridCoefficients n β (fun i => min (q i) r)) |
| (timeGridCoefficients n β (fun i => max (q i) r)) |
| have hc : Continuous v := by |
| apply continuous_pi |
| intro i |
| refine Fin.addCases ?_ ?_ i |
| · intro j |
| simp only [v,Fin.append_left,timeGridCoefficients] |
| fun_prop |
| · intro j |
| simp only [v,Fin.append_right,timeGridCoefficients] |
| fun_prop |
| have H := (scalarMomentSquare_contDiff_coeff (n+n) (Fin.append m m) ⟨n,by omega⟩).continuous.comp hc |
| apply H.continuousOn.congr |
| intro r hr |
| have hpre (i : Fin n) (z : ℝ) (hz : z∈Ico (min (q i.castSucc) r) (min (q i.succ) r)) : |
| α z=m i := by |
| apply hmodel i z |
| constructor |
| · rcases min_le_iff.mp hz.1 with h|h |
| · exact h |
| · exact False.elim (not_lt_of_ge h (lt_min_iff.mp hz.2).2) |
| · exact (lt_min_iff.mp hz.2).1 |
| have hpost (i : Fin n) (z : ℝ) (hz : z∈Ico (max (q i.castSucc) r) (max (q i.succ) r)) : |
| α z=m i := by |
| apply hmodel i z |
| constructor |
| · exact (le_max_left _ _).trans hz.1 |
| · rcases lt_max_iff.mp hz.2 with h|h |
| · exact h |
| · exact False.elim (not_lt_of_ge ((le_max_right _ _).trans hz.1) h) |
| have H' := scalarCDFOverlap_eq_two_partitions β hα hαm n n |
| (fun i => min (q i) r) (fun i => max (q i) r) |
| (by intro i j hij; exact min_le_min (hq hij) le_rfl) |
| (by intro i j hij; exact max_le_max (hq hij) le_rfl) |
| (by simp only [hq0,min_eq_left hr.1]) |
| (by simp only [hq1,max_eq_left hr.2]) |
| (by simp only [hq1,hq0,min_eq_right hr.2,max_eq_right hr.1]) |
| m m hm hm hpre hpost |
| simpa only [hq0,max_eq_right hr.1,Function.comp_def,v] using H' |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
