| import OAI.Probability.MatroidProphet.Algorithm.Expectation |
| import OAI.Probability.MatroidProphet.CandidateFilter |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.MainAlgorithm |
| open Set Finset |
| variable {n : ℕ} |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma sum_groupMask_card_le {ι : Type*} (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (J : Finset ι) (idx : ι → ℕ) (hinj : Set.InjOn idx (J : Set ι)) |
| (mask target : Finset (Fin n)) |
| (hsub : ∀ j ∈ J, groupMask M d w mask (idx j) ⊆ (target : Set (Fin n))) : |
| (∑ j ∈ J, (groupMask M d w mask (idx j)).ncard) ≤ target.card := by |
| classical |
| let f : ι → Finset (Fin n) := fun j => (groupMask M d w mask (idx j)).toFinset |
| have hf (j : ι) : f j ⊆ pathGroups M d w (idx j) := by |
| intro e he |
| have hx : e ∈ groupMask M d w mask (idx j) := by simpa [f] using he |
| have hg : e ∈ groupMask M d w univ (idx j) := ⟨mem_univ _, hx.2⟩ |
| simpa [pathGroups] using hg |
| have hd : (J : Set ι).PairwiseDisjoint f := by |
| intro i hi j hj hij |
| have hij' : idx i ≠ idx j := fun heq => hij (hinj hi hj heq) |
| exact (pathGroups_disjoint M d w hij').mono (hf i) (hf j) |
| have hs : J.biUnion f ⊆ target := by |
| intro e he |
| obtain ⟨j, hj, hej⟩ := mem_biUnion.mp he |
| exact hsub j hj (by simpa [f] using hej) |
| have hcard := Finset.card_le_card hs |
| rw [card_biUnion hd] at hcard |
| simpa only [f, ← Set.ncard_eq_toFinset_card'] using hcard |
|
|
| lemma groupMask_eq_focal (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (mask : Finset (Fin n)) {h : ℕ} {i : ℤ} |
| (hi : (groups M d w)[h]? = some i) : |
| (groupMask M d w mask h).toFinset = |
| mask ∩ ((candidateLabels M w d.H).filter (fun e => w e = some i)) := by |
| classical |
| ext e |
| simp only [Set.mem_toFinset, Finset.mem_inter, mem_filter, candidateLabels, candidateSet, Finset.mem_univ, true_and] |
| change (e ∈ mask ∧ candidate M d w e (w e) ∧ w e = (groups M d w)[h]?) ↔ |
| (e ∈ mask ∧ candidate M d w e (w e) ∧ w e = some i) |
| simp only [hi] |
|
|
| lemma groupMask_later_subset (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (mask : Finset (Fin n)) {h j : ℕ} {i : ℤ} |
| (hi : (groups M d w)[h]? = some i) (hj : j < (groups M d w).length) (hhj : h < j) : |
| groupMask M d w mask j ⊆ |
| (mask ∩ ((candidateLabels M w d.H).filter (fun e => ∃ k, w e = some k ∧ i < k)) : |
| Finset (Fin n)) := by |
| intro e he |
| have hjget : (groups M d w)[j]? = some ((groups M d w)[j]) := List.getElem?_eq_getElem hj |
| obtain ⟨k, hk, hkj⟩ := groups_earlier_key M d w hjget hhj |
| have hki : k = i := Option.some.inj (hk.symm.trans hi) |
| subst k |
| exact Finset.mem_inter.mpr ⟨he.1, mem_filter.mpr ⟨mem_filter.mpr ⟨mem_univ _, he.2.1⟩, |
| _, he.2.2.trans hjget, hkj⟩⟩ |
|
|
| theorem listed_cost_le_true_counts (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) {h : ℕ} {i : ℤ} (hi : (groups M d w)[h]? = some i) (κ : ℕ) : |
| κ * (groupMask M d w d.C h).ncard + |
| ∑ j ∈ range ((groups M d w).length-(h+1)), |
| (κ * (groupMask M d w d.C (h+1+j)).ncard + |
| (groupMask M d w d.D (h+1+j)).ncard) ≤ |
| κ * (d.C ∩ ((candidateLabels M w d.H).filter (fun e => w e = some i))).card + |
| κ * (d.C ∩ ((candidateLabels M w d.H).filter (fun e => ∃ k, w e = some k ∧ i < k))).card + |
| ((candidateLabels M w d.H).filter (fun e => ∃ k, w e = some k ∧ i < k)).card := by |
| classical |
| let J := range ((groups M d w).length-(h+1)) |
| let high := (candidateLabels M w d.H).filter (fun e => ∃ k, w e = some k ∧ i < k) |
| have hh := (List.getElem?_eq_some_iff.mp hi).1 |
| have hc : (∑ j ∈ J, (groupMask M d w d.C (h+1+j)).ncard) ≤ (d.C ∩ high).card := by |
| apply sum_groupMask_card_le M d w J (fun j => h+1+j) (by intro a _ b _ hab; change h+1+a = h+1+b at hab; omega) |
| intro j hj |
| exact groupMask_later_subset M d w d.C hi (by have := mem_range.mp hj; omega) (by omega) |
| have hd : (∑ j ∈ J, (groupMask M d w d.D (h+1+j)).ncard) ≤ high.card := by |
| apply sum_groupMask_card_le M d w J (fun j => h+1+j) (by intro a _ b _ hab; change h+1+a = h+1+b at hab; omega) |
| intro j hj e he |
| have hm := groupMask_later_subset M d w d.D hi |
| (by have := mem_range.mp hj; omega) (by omega) he |
| exact (Finset.mem_inter.mp hm).2 |
| have hf : (groupMask M d w d.C h).ncard = |
| (d.C ∩ ((candidateLabels M w d.H).filter (fun e => w e = some i))).card := by |
| rw [Set.ncard_eq_toFinset_card', groupMask_eq_focal M d w d.C hi] |
| rw [hf, sum_add_distrib, ← mul_sum] |
| have hm := Nat.mul_le_mul_left κ hc |
| change _ ≤ _ at hd |
| dsimp [J, high] at hm hd |
| omega |
|
|
| end MatroidProphet.MainAlgorithm |
|
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| end OAI |
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