| import OAI.Probability.MatroidProphet.Algorithm.Selection |
| import OAI.Probability.MatroidProphet.Groups |
| import OAI.Probability.MatroidProphet.PathRankCost |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Set Finset |
| variable {α : Type*} [Fintype α] |
|
|
| lemma mem_nominal_difference_iff (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) (he : e ∉ M.closure ∅) (b : ℤ) : |
| e ∈ nominalPath M hE κ D C h b \ nominalPath M hE κ D C h (b-1) ↔ |
| nominalBirth M hE κ D C h e = b := by |
| simp only [Set.mem_sdiff, mem_nominal_iff_birth_le M hE κ D C h e he] |
| omega |
|
|
| lemma lowerCompetition_member (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (b : ℤ) {j h : ℕ} (hj : j < h) |
| {e : α} (he : e ∈ T j ∩ (nominalPath M hE κ D C j b \ |
| nominalPath M hE κ D C j (b-1))) : |
| e ∈ lowerCompetition M hE κ D C T b h := by |
| induction h with |
| | zero => omega |
| | succ h ih => |
| change e ∈ lowerCompetition M hE κ D C T b h ∪ |
| (T h ∩ (nominalPath M hE κ D C h b \ nominalPath M hE κ D C h (b-1))) |
| by_cases heq : j = h |
| · exact Or.inr (heq ▸ he) |
| · exact Or.inl (ih (by omega)) |
|
|
| namespace MainAlgorithm |
| variable {n : ℕ} |
|
|
| lemma candidate_nonloop (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (e : Fin n) (v : Option ℤ) |
| (he : candidate M d w e v) : e ∉ M.closure ∅ := |
| fun hx => he.2.2 (M.closure_subset_closure (Set.empty_subset _) hx) |
|
|
| lemma groups_index_of_get (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) {h : ℕ} {i : ℤ} (hi : (groups M d w)[h]? = some i) : |
| i ∈ groups M d w ∧ (groups M d w).idxOf i = h := by |
| obtain ⟨hh, rfl⟩ := List.getElem?_eq_some_iff.mp hi |
| exact ⟨List.getElem_mem hh, (groups_nodup M d w).idxOf_getElem h hh⟩ |
|
|
| noncomputable def groupBirthRegion (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) (b : ℤ) : Set (Fin n) := |
| unobservedBirthRegion M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C) |
| (groupMask M d w univ h) (d.H ∪ d.D ∪ d.C : Finset (Fin n)) h b |
|
|
| noncomputable def competition (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) (b : ℤ) : Set (Fin n) := |
| lowerCompetition M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C) |
| (groupMask M d w d.T) b h |
|
|
| lemma groupBirthRegion_eligible (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) (b : ℤ) |
| (hb : b % 2 = (if d.odd then 1 else 0)) {e : Fin n} |
| (he : e ∈ groupBirthRegion M hE d w h b ∩ (d.T : Set (Fin n))) |
| (hf : e ∉ finalPath M hE d w (b-2)) : |
| e ∈ assignedEligible (assign M hE d w) w ∧ |
| assignedLayer (assign M hE d w) w e = b := by |
| classical |
| have ht := he.2 |
| have hr := he.1 |
| unfold groupBirthRegion unobservedBirthRegion at hr |
| split_ifs at hr with ha |
| · obtain ⟨⟨⟨_, hc, hg⟩, hX⟩, hO⟩ := hr |
| obtain ⟨i, hwi⟩ : ∃ i, w e = some i := Option.ne_none_iff_exists'.mp hc.1 |
| have hget : (groups M d w)[h]? = some i := hg.symm.trans hwi |
| obtain ⟨hi, hidx⟩ := groups_index_of_get M d w hget |
| have hbirth : birth M hE d w e i = b := by |
| unfold birth |
| rw [hidx] |
| exact (mem_nominal_difference_iff M hE (2^100) _ _ h e |
| (candidate_nonloop M d w e _ hc) b).mp hX |
| have hel : eligible M hE d w e i := by |
| refine ⟨hO, ?_, hi, ht, ?_, ?_, ?_⟩ |
| · rwa [← hwi] |
| · rwa [hidx, hbirth] |
| · rwa [hbirth] |
| · rwa [hbirth] |
| have has : assign M hE d w e (w e) = some b := |
| (assign_spec M hE d w e _ b).mpr ⟨i, hwi, hel, hbirth⟩ |
| constructor |
| · simp [assignedEligible, has] |
| · simp [assignedLayer, has] |
| · exact False.elim hr |
|
|
| lemma selection_low_mem_competition (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) |
| (h : ℕ) (b : ℤ) {e : Fin n} (he : e ∈ selection M hE d w π) |
| (hb : assignedLayer (assign M hE d w) w e = b) |
| (hlow : ¬ ∃ i, w e = some i ∧ h ≤ (groups M d w).idxOf i) : |
| e ∈ competition M hE d w h b := by |
| have hel := layerGreedy_subset M _ _ _ _ he |
| obtain ⟨i, hwi, hi, hbi⟩ := (assign_spec M hE d w e _ _).mp |
| (assigned_some_of_eligible (assign M hE d w) w e hel) |
| have hidx : (groups M d w).idxOf i < h := by |
| by_contra hn |
| exact hlow ⟨i, hwi, by omega⟩ |
| apply lowerCompetition_member M hE (2^100) _ _ _ b hidx |
| constructor |
| · exact ⟨hi.2.2.2.1, by simpa only [hwi] using hi.2.1, hwi.trans (groups_get?_idxOf M d w i hi.2.2.1).symm⟩ |
| · apply (mem_nominal_difference_iff M hE (2^100) _ _ _ e |
| (candidate_nonloop M d w e _ hi.2.1) b).mpr |
| exact hbi.trans hb |
|
|
| theorem selection_cumulative_rank (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) |
| (h : ℕ) (L : Finset ℤ) (hL : ∀ b ∈ L, b % 2 = (if d.odd then 1 else 0)) : |
| (∑ b ∈ L, conditionalRank M |
| (groupBirthRegion M hE d w h b ∩ (d.T : Set (Fin n))) |
| (finalPath M hE d w (b-2) ∪ competition M hE d w h b)) ≤ |
| ((selection M hE d w π).filter (fun e => |
| ∃ i, w e = some i ∧ h ≤ (groups M d w).idxOf i)).card := by |
| classical |
| exact layerGreedy_sum_rank_lower M hE _ _ _ _ L _ _ _ |
| (fun b hb e he => groupBirthRegion_eligible M hE d w h b (hL b hb) he.1 he.2) |
| (fun b _ e he hb hn => selection_low_mem_competition M hE d w π h b he hb hn) |
|
|
| end MainAlgorithm |
| end MatroidProphet |
|
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| end OAI |
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