| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.PartitionOverlapIndex |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def appendTimeGrid (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) : |
| Fin (a+b+1) → ℝ := Fin.append (m:=a) (n:=b+1) (fun i : Fin a => q i.castSucc) p |
|
|
| theorem appendTimeGrid_right (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (j : Fin (b+1)) : appendTimeGrid a b q p (j.natAdd a)=p j := by |
| exact Fin.append_right _ _ _ |
|
|
| theorem appendTimeGrid_left (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (hr : q (Fin.last a)=p 0) (i : Fin (a+1)) : appendTimeGrid a b q p ⟨i,by omega⟩=q i := by |
| refine Fin.lastCases ?_ (fun j => ?_) i |
| · rw [show (⟨(Fin.last a).val,by omega⟩:Fin (a+b+1))=(0:Fin (b+1)).natAdd a by ext; simp, |
| appendTimeGrid_right,hr] |
| · change appendTimeGrid a b q p (j.castAdd (b+1))=q j.castSucc |
| exact Fin.append_left _ _ _ |
|
|
| theorem appendTimeGrid_left_start (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (hr : q (Fin.last a)=p 0) (i : Fin a) : |
| appendTimeGrid a b q p (i.castAdd b).castSucc=q i.castSucc := |
| appendTimeGrid_left a b q p hr i.castSucc |
|
|
| theorem appendTimeGrid_left_end (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (hr : q (Fin.last a)=p 0) (i : Fin a) : |
| appendTimeGrid a b q p (i.castAdd b).succ=q i.succ := |
| appendTimeGrid_left a b q p hr i.succ |
|
|
| theorem appendTimeGrid_right_start (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (j : Fin b) : appendTimeGrid a b q p (j.natAdd a).castSucc=p j.castSucc := |
| appendTimeGrid_right a b q p j.castSucc |
| |
| theorem appendTimeGrid_right_end (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (j : Fin b) : appendTimeGrid a b q p (j.natAdd a).succ=p j.succ := by |
| rw [show (j.natAdd a).succ=j.succ.natAdd a by ext; simp |
| exact appendTimeGrid_right a b q p j.succ |
| |
| theorem appendTimeGrid_monotone (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (hq : Monotone q) (hp : Monotone p) (hr : q (Fin.last a)=p 0) : |
| Monotone (appendTimeGrid a b q p) := by |
| apply Fin.monotone_iff_le_succ.mpr |
| intro i |
| refine Fin.addCases ?_ ?_ i |
| · intro i |
| rw [appendTimeGrid_left_start a b q p hr,appendTimeGrid_left_end a b q p hr] |
| exact hq (Fin.castSucc_le_succ i) |
| · intro j |
| rw [appendTimeGrid_right_start,appendTimeGrid_right_end] |
| exact hp (Fin.castSucc_le_succ j) |
| |
| theorem timeGridCoefficients_append (a b : ℕ) (β : ℝ) |
| (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) (hr : q (Fin.last a)=p 0) : |
| timeGridCoefficients (a+b) β (appendTimeGrid a b q p)= |
| Fin.append (timeGridCoefficients a β q) (timeGridCoefficients b β p) := by |
| funext i |
| refine Fin.addCases ?_ ?_ i |
| · intro i |
| simp only [timeGridCoefficients,Fin.append_left, |
| appendTimeGrid_left_start a b q p hr,appendTimeGrid_left_end a b q p hr] |
| · intro j |
| simp only [timeGridCoefficients,Fin.append_right,appendTimeGrid_right_start,appendTimeGrid_right_end] |
|
|
| theorem appendTimeGrid_models {α : ℝ → ℝ} (a b : ℕ) |
| (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) (hr : q (Fin.last a)=p 0) |
| (m : Fin a → ℝ) (l : Fin b → ℝ) |
| (hm : ∀ i : Fin a, ∀ z∈Ico (q i.castSucc) (q i.succ), α z=m i) |
| (hl : ∀ i : Fin b, ∀ z∈Ico (p i.castSucc) (p i.succ), α z=l i) : |
| ∀ i : Fin (a+b), ∀ z∈Ico (appendTimeGrid a b q p i.castSucc) |
| (appendTimeGrid a b q p i.succ), α z=Fin.append m l i := by |
| intro i |
| refine Fin.addCases ?_ ?_ i |
| · intro i z hz |
| rw [appendTimeGrid_left_start a b q p hr,appendTimeGrid_left_end a b q p hr] at hz |
| simpa only [Fin.append_left] using hm i z hz |
| · intro j z hz |
| rw [appendTimeGrid_right_start,appendTimeGrid_right_end] at hz |
| simpa only [Fin.append_right] using hl j z hz |
|
|
| theorem scalarCDFOverlap_eq_two_partitions_index (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hαm : Monotone α) |
| (a b : ℕ) (q : Fin (a+1) → ℝ) (p : Fin (b+1) → ℝ) |
| (hq : Monotone q) (hp : Monotone p) (hq0 : q 0=0) (hp1 : p (Fin.last b)=1) |
| (hr : q (Fin.last a)=p 0) (m : Fin a → ℝ) (l : Fin b → ℝ) |
| (hm : ∀ i, m i∈Icc (0:ℝ) 1) (hl : ∀ i, l i∈Icc (0:ℝ) 1) |
| (hmodelq : ∀ i : Fin a, ∀ z∈Ico (q i.castSucc) (q i.succ), α z=m i) |
| (hmodelp : ∀ i : Fin b, ∀ z∈Ico (p i.castSucc) (p i.succ), α z=l i) |
| (j : Fin (b+1)) : |
| scalarCDFOverlap β α (p j)= |
| scalarMomentSquare (a+b) (Fin.append m l) |
| (Fin.append (timeGridCoefficients a β q) (timeGridCoefficients b β p)) |
| scalarSpinTerminal scalarMagnetization (j.natAdd a) 0 := by |
| have h0 : appendTimeGrid a b q p 0=0 := by |
| have H := appendTimeGrid_left a b q p hr 0 |
| simpa only [Fin.val_zero,Fin.mk_zero,hq0] using H |
| have h1 : appendTimeGrid a b q p (Fin.last (a+b))=1 := by |
| have H := appendTimeGrid_right a b q p (Fin.last b) |
| rw [show (Fin.last b).natAdd a=Fin.last (a+b) by ext; simp] at H |
| exact H.trans hp1 |
| have hml (i : Fin (a+b)) : Fin.append m l i∈Icc (0:ℝ) 1 := by |
| refine Fin.addCases ?_ ?_ i |
| · intro i |
| · intro j; simpa only [Fin.append_right] using hl j |
| have H := scalarCDFOverlap_eq_partition_index β hα hαm (a+b) (appendTimeGrid a b q p) |
| (appendTimeGrid_monotone a b q p hq hp hr) h0 h1 (Fin.append m l) hml |
| (appendTimeGrid_models a b q p hr m l hmodelq hmodelp) (j.natAdd a) |
| rwa [appendTimeGrid_right,timeGridCoefficients_append a b β q p hr] at H |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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