| 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 | |