| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.PartitionOverlap |
| import OAI.Probability.SKBarriers.Parisi.QuantileScalarOverlap |
|
|
| namespace OAI |
|
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| section |
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|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem scalarCDFOverlap_eq_partition_index (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) |
| (j : Fin (n+1)) : |
| scalarCDFOverlap β α (q j)= |
| scalarMomentSquare n m (timeGridCoefficients n β q) scalarSpinTerminal scalarMagnetization j 0 := by |
| let a := j.1 |
| let b := n-a |
| have hn : a+b=n := Nat.add_sub_of_le (Nat.le_of_lt_succ j.isLt) |
| let q' : Fin (a+b+1) → ℝ := fun i => q (Fin.cast (congrArg (·+1) hn) i) |
| let m' : Fin (a+b) → ℝ := fun i => m (Fin.cast hn i) |
| have hq' : Monotone q' := fun i l hil => hq hil |
| have h0 : q' 0=0 := by simpa [q'] using hq0 |
| have h1 : q' (Fin.last (a+b))=1 := by simpa [q'] using hq1 |
| have hm' (i : Fin (a+b)) : m' i∈Icc (0:ℝ) 1 := hm _ |
| have hmodel' (i : Fin (a+b)) (z : ℝ) (hz : z∈Ico (q' i.castSucc) (q' i.succ)) : α z=m' i := |
| hmodel (Fin.cast hn i) z hz |
| have H := scalarCDFOverlap_eq_partition β hα hαm a b q' hq' h0 h1 m' hm' hmodel' |
| have heq : q' ⟨a,by omega⟩=q j := by congr 1 |
| rw [heq] at H |
| have hv : timeGridCoefficients (a+b) β q'= |
| fun i => timeGridCoefficients n β q (Fin.cast hn i) := rfl |
| rw [hv] at H |
| have hj : (⟨a,by omega⟩:Fin (a+b+1))=Fin.cast (congrArg (·+1) hn.symm) j := by ext; rfl |
| rw [hj] at H |
| exact H.trans (scalarMomentSquare_cast hn m (timeGridCoefficients n β q) |
| scalarSpinTerminal scalarMagnetization j 0) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
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| end OAI |
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|
