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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | 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
end OAI
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