| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarSplit |
| import OAI.Probability.SKBarriers.Parisi.CDFOverlap |
| import OAI.Probability.SKBarriers.Parisi.CDFPartitionRepresentation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def gridPrefix (a b : ℕ) (q : Fin (a+b+1) → ℝ) (i : Fin (a+1)) : ℝ := |
| q ⟨i.1, by omega⟩ |
| def gridSuffix (a b : ℕ) (q : Fin (a+b+1) → ℝ) (i : Fin (b+1)) : ℝ := |
| q ⟨a+i.1, by omega⟩ |
|
|
| theorem gridPrefix_monotone {a b : ℕ} {q : Fin (a+b+1) → ℝ} |
| (h : Monotone q) : Monotone (gridPrefix a b q) := by |
| intro i j hij |
| exact h (show (⟨i.1,by omega⟩:Fin (a+b+1)) ≤ ⟨j.1,by omega⟩ from hij) |
| theorem gridSuffix_monotone {a b : ℕ} {q : Fin (a+b+1) → ℝ} |
| (h : Monotone q) : Monotone (gridSuffix a b q) := by |
| intro i j hij |
| apply h |
| exact Nat.add_le_add_left hij a |
|
|
| theorem scalarCDFOverlap_eq_partition (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hαm : Monotone α) |
| (a b : ℕ) (q : Fin (a+b+1) → ℝ) (hq : Monotone q) |
| (hq0 : q 0=0) (hq1 : q (Fin.last (a+b))=1) |
| (m : Fin (a+b) → ℝ) (hm : ∀ i, m i∈Icc (0:ℝ) 1) |
| (hmodel : ∀ i : Fin (a+b), ∀ z∈Ico (q i.castSucc) (q i.succ), α z=m i) : |
| scalarCDFOverlap β α (q ⟨a,by omega⟩)= |
| scalarMomentSquare (a+b) m (timeGridCoefficients (a+b) β q) |
| scalarSpinTerminal scalarMagnetization ⟨a,by omega⟩ 0 := by |
| let r := q ⟨a,by omega⟩ |
| have hr : r∈Icc (0:ℝ) 1 := by |
| constructor |
| · rw [← hq0] |
| · rw [← hq1] |
| let t := Real.toNNReal r |
| let s := Real.toNNReal (1-r) |
| have ht : (t:ℝ)=r := Real.coe_toNNReal r hr.1 |
| have hs : (s:ℝ)=1-r := Real.coe_toNNReal (1-r) (sub_nonneg.mpr hr.2) |
| have ht1 : t≤1 := by rw [← NNReal.coe_le_coe,ht,NNReal.coe_one] |
| have hs1 : s≤1 := by rw [← NNReal.coe_le_coe,hs,NNReal.coe_one] |
| let v := timeGridCoefficients (a+b) β q |
| let f := scalarCDFValue β α r s |
| let u := scalarCDFGradient β α r s |
| have hsuffix : f=scalarHierarchy b (fun i => m (i.natAdd a)) |
| (fun i => v (i.natAdd a)) scalarSpinTerminal := by |
| funext x |
| have H := scalarCDFOperator_eq_partition scalarSpinTerminal_regular scalarSpinTerminal_lipschitz |
| β hα hαm b (gridSuffix a b q) (gridSuffix_monotone hq) |
| (fun i => m (i.natAdd a)) (fun i => hm (i.natAdd a)) |
| (fun i z hz => hmodel (i.natAdd a) z hz) s hs1 |
| (by change (s:ℝ)=q (Fin.last (a+b))-r; rw [hq1,hs]) x |
| exact H |
| have hu : u=scalarHierarchyAverage b (fun i => m (i.natAdd a)) |
| (fun i => v (i.natAdd a)) scalarSpinTerminal scalarMagnetization := by |
| funext x |
| have H := (scalarCDFValue_hasDerivAt β hα hαm r s hs1 x).deriv |
| change deriv f x=u x at H |
| rw [hsuffix,scalarHierarchy_gradient_average _ _ _ scalarSpinTerminal_regular |
| scalarSpinTerminal_hasDerivAt] at H |
| exact H.symm |
| have HG := scalarCDFGradient_sq_regular β hα hαm r s hs1 |
| have hp0 : gridPrefix a b q 0=0 := by simpa [gridPrefix] using hq0 |
| have H := scalarCDFAverage_eq_partition (scalarCDFValue_regular β hα hαm r s hs1) |
| HG.1 HG.2.1 (scalarCDFValue_lipschitz β hα hαm r s hs1) |
| (B:=1) (C:=2) HG.2.2.1 HG.2.2.2 β hα hαm a (gridPrefix a b q) |
| (gridPrefix_monotone hq) (fun i => m (i.castAdd b)) (fun i => hm (i.castAdd b)) |
| (fun i z hz => hmodel (i.castAdd b) z hz) t ht1 |
| (by change (t:ℝ)=r-gridPrefix a b q 0 |
| rw [hp0] at H |
| change scalarCDFAverage β α 0 t f (fun z => (u z)^2) 0=_ at H |
| change scalarCDFAverage β α 0 t f (fun z => (u z)^2) 0=_ |
| rw [H,scalarMomentSquare_split] |
| change scalarHierarchyAverage a (fun i => m (i.castAdd b)) (fun i => v (i.castAdd b)) |
| f (fun x => (u x)^2) 0=_ |
| rw [hsuffix,hu] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
