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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 | import OAI.Probability.MatroidProphet.Algorithm.Rule
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Finset
variable {n : ℕ}
@[simp] lemma mem_groups (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) (i : ℤ) :
i ∈ groups M d seen ↔ i ∈ listed M d seen := by
simp [groups]
lemma groups_nodup (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) : (groups M d seen).Nodup :=
Finset.sort_nodup _ _
lemma groups_sorted (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) : (groups M d seen).SortedLT :=
Finset.sortedLT_sort _
@[simp] lemma groups_length (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) : (groups M d seen).length = (listed M d seen).card := by
simp [groups]
lemma groups_get_lt (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) (i j : Fin (groups M d seen).length) (h : i < j) :
(groups M d seen).get i < (groups M d seen).get j :=
groups_sorted M d seen h
lemma groups_get?_idxOf (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) (i : ℤ) (hi : i ∈ groups M d seen) :
(groups M d seen)[(groups M d seen).idxOf i]? = some i :=
List.getElem?_idxOf hi
lemma groups_idxOf_lt_iff (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) (i j : ℤ)
(hi : i ∈ groups M d seen) (hj : j ∈ groups M d seen) :
(groups M d seen).idxOf i < (groups M d seen).idxOf j ↔ i < j := by
have hii := List.idxOf_lt_length_of_mem hi
have hjj := List.idxOf_lt_length_of_mem hj
have h := (groups_sorted M d seen).getElem_lt_getElem_iff (hi := hii) (hj := hjj)
simpa using h.symm
lemma groups_earlier_key (M : Matroid (Fin n)) (d : MainMasks n)
(seen : Fin n → Option ℤ) {j h : ℕ} {i : ℤ}
(hh : (groups M d seen)[h]? = some i) (hjh : j < h) :
∃ k, (groups M d seen)[j]? = some k ∧ k < i := by
obtain ⟨hh', hi⟩ := List.getElem?_eq_some_iff.mp hh
have hj : j < (groups M d seen).length := hjh.trans hh'
refine ⟨(groups M d seen)[j], List.getElem?_eq_getElem hj, ?_⟩
rw [← hi]
exact (groups_sorted M d seen).getElem_lt_getElem_of_lt hjh
end MatroidProphet.MainAlgorithm
end OAI
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