| import OAI.Probability.MatroidProphet.Residual.Nominal |
| import OAI.Probability.MatroidProphet.Residual.Transfer |
| import OAI.Probability.MatroidProphet.Reverse.Paths |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Set Finset Pivots |
| variable {α : Type*} [Fintype α] |
|
|
| noncomputable def nominalLayerSet (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (U : Set α) (ε : Fin 2) |
| (b : ParityWindow (activation h) ε) : Set α := |
| {e | e ∈ U ∧ activation h + 2 ≤ b.val.val ∧ b.val.val ≤ 0 ∧ |
| e ∈ nominalPath M hE κ D C h b.val.val ∧ e ∉ nominalPath M hE κ D C h (b.val.val - 1)} |
|
|
| noncomputable def nominalRankStatistic (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U O : Set α) (ε : Fin 2) : ℕ := |
| ∑ b : ParityWindow (activation h) ε, |
| conditionalRank M (nominalLayerSet M hE κ D C h U ε b \ O) |
| (nominalPath M hE κ D C h (b.val.val - 2) ∪ lowerCompetition M hE κ D C T b.val.val h) |
|
|
| noncomputable def safeLayerSet (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U I : Set α) (ε : Fin 2) |
| (b : ParityWindow (activation h) ε) : Set α := |
| (I ∩ nominalLayerSet M hE κ D C h U ε b) \ |
| densityExpansion M hE κ (D h) |
| (guardedPath M hE κ D C h (b.val.val - 2) ∪ lowerCompetition M hE κ D C T b.val.val h) |
|
|
| noncomputable def safeLayerStatistic (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U I : Set α) (ε : Fin 2) : ℕ := |
| ∑ b : ParityWindow (activation h) ε, (safeLayerSet M hE κ D C T h U I ε b).ncard |
|
|
| def boolParity (parity : Bool) : Fin 2 := if parity then 1 else 0 |
|
|
| lemma boolParity_val (parity : Bool) : ((boolParity parity).val : ℤ) = (if parity then 1 else 0) := by |
| cases parity <;> rfl |
|
|
| lemma nominalLayerSet_subset (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (U : Set α) (ε : Fin 2) |
| (b : ParityWindow (activation h) ε) : nominalLayerSet M hE κ D C h U ε b ⊆ U := |
| fun _ he => he.1 |
|
|
| lemma nominalLayerSet_disjoint (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (U : Set α) (ε : Fin 2) : |
| Pairwise (fun b c => Disjoint (nominalLayerSet M hE κ D C h U ε b) |
| (nominalLayerSet M hE κ D C h U ε c)) := by |
| intro b c hbc |
| apply Set.disjoint_left.mpr |
| intro e hb hc |
| have hval : b.val.val ≠ c.val.val := by |
| intro heq |
| exact hbc (Subtype.ext (Subtype.ext heq)) |
| rcases lt_or_gt_of_ne hval with hlt | hlt |
| · exact hc.2.2.2.2 (nominalPath_mono M hE κ D C h (by omega) hb.2.2.2.1) |
| · exact hb.2.2.2.2 (nominalPath_mono M hE κ D C h (by omega) hc.2.2.2.1) |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
