| import OAI.Probability.MatroidProphet.Algorithm.LayerRank |
| import OAI.Probability.MatroidProphet.PathStatistics |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.MainAlgorithm |
| open Set Finset Pivots |
| variable {n : ℕ} |
|
|
| noncomputable def pathGroups (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (h : ℕ) : Finset (Fin n) := by |
| classical |
| exact (groupMask M d w univ h).toFinset |
|
|
| lemma pathGroups_coe (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (h : ℕ) : |
| (pathGroups M d w h : Set (Fin n)) = groupMask M d w univ h := by |
| classical |
| exact Set.coe_toFinset _ |
|
|
| lemma groupMask_eq_inter (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (mask : Finset (Fin n)) (h : ℕ) : |
| groupMask M d w mask h = (mask : Set (Fin n)) ∩ (pathGroups M d w h : Set (Fin n)) := by |
| classical |
| rw [pathGroups_coe] |
| ext e |
| simp [groupMask] |
|
|
| lemma pathGroups_disjoint (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) : Pairwise (fun i j => Disjoint (pathGroups M d w i) (pathGroups M d w j)) := by |
| classical |
| intro i j hij |
| apply Finset.disjoint_left.mpr |
| intro e hei hej |
| have heI : e ∈ groupMask M d w univ i := by simpa only [pathGroups, Set.mem_toFinset] using hei |
| have heJ : e ∈ groupMask M d w univ j := by simpa only [pathGroups, Set.mem_toFinset] using hej |
| obtain ⟨k, hk⟩ := Option.ne_none_iff_exists'.mp heI.2.1.1 |
| have hi := (groups_index_of_get M d w (heI.2.2.symm.trans hk)).2 |
| have hj := (groups_index_of_get M d w (heJ.2.2.symm.trans hk)).2 |
| exact hij (hi.symm.trans hj) |
|
|
| noncomputable def listedLambda (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) : ℕ := |
| finalRankStatistic M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C) |
| (groupMask M d w d.T) h (groups M d w).length (groupMask M d w univ h) |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (d.T : Set (Fin n)) (boolParity d.odd) |
|
|
| noncomputable def listedZ (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) : ℕ := |
| nominalRankStatistic M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C) |
| (groupMask M d w d.T) h (groupMask M d w univ h) |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (boolParity d.odd) |
|
|
| noncomputable def trueLambda (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ) : ℕ := by |
| classical |
| exact if i ∈ groups M d w then listedLambda M hE d w ((groups M d w).idxOf i) else 0 |
|
|
| noncomputable def trueZ (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ) : ℕ := by |
| classical |
| exact if i ∈ groups M d w then listedZ M hE d w ((groups M d w).idxOf i) else 0 |
|
|
| theorem listedLambda_le_count (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) (h : ℕ) : |
| listedLambda M hE d w h ≤ |
| ((selection M hE d w π).filter (fun e => |
| ∃ i, w e = some i ∧ h ≤ (groups M d w).idxOf i)).card := by |
| classical |
| have hc := selection_cumulative_rank M hE d w π h |
| (parityPoints (activation h) (boolParity d.odd)) (by |
| intro b hb |
| have hm := ((mem_parityPoints _ _ _).mp hb).2.2 |
| rwa [boolParity_val] at hm) |
| rw [sum_parityPoints] at hc |
| have hreg (b : ParityWindow (activation h) (boolParity d.odd)) : |
| groupBirthRegion M hE d w h b.val.val = |
| nominalLayerSet M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C) |
| h (groupMask M d w univ h) (boolParity d.odd) b \ |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) := |
| unobservedBirthRegion_eq_nominalLayer M hE (2^100) _ _ h _ _ _ b |
| simp_rw [hreg] at hc |
| exact hc |
|
|
| theorem trueLambda_le_count (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) (i : ℤ) : |
| trueLambda M hE d w i ≤ |
| ((selection M hE d w π).filter (fun e => ∃ j, w e = some j ∧ i ≤ j)).card := by |
| classical |
| unfold trueLambda |
| split_ifs with hi |
| · apply (listedLambda_le_count M hE d w π _).trans |
| apply Finset.card_le_card |
| intro e he |
| obtain ⟨hea, j, hwj, hij⟩ := Finset.mem_filter.mp he |
| refine Finset.mem_filter.mpr ⟨hea, j, hwj, ?_⟩ |
| have hel := layerGreedy_subset M _ _ _ _ hea |
| obtain ⟨k, hwk, hek, _⟩ := (assign_spec M hE d w e _ _).mp |
| (assigned_some_of_eligible (assign M hE d w) w e hel) |
| have hjk : j = k := Option.some.inj (hwj.symm.trans hwk) |
| have hj : j ∈ groups M d w := hjk ▸ hek.2.2.1 |
| by_contra hn |
| have hlt := (groups_idxOf_lt_iff M d w j i hj hi).mpr (by omega) |
| omega |
| · exact Nat.zero_le _ |
|
|
| end MatroidProphet.MainAlgorithm |
|
|
| end OAI |
|
|
