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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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
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