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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 | import OAI.Probability.MatroidSecretary.Accounting.Contracts
namespace OAI
/-! Exact source interpretation of the cumulative count in `lem:all-orders`.
The implementation orders listed groups increasingly by integer weight level.
For an actual listed group, counting accepted labels by group index is exactly
counting them by the source's rounded-weight threshold. -/
namespace MatroidProphet.MainAlgorithm
open Finset
variable {n : ℕ}
lemma selected_level_mem_groups (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n)
{e : Fin n} (he : e ∈ selection M hE d w π) {j : ℤ}
(hw : w e = some j) : j ∈ groups M d w := by
have hel := layerGreedy_subset M _ _ _ _ he
obtain ⟨k, hwk, hek, _⟩ := (assign_spec M hE d w e _ _).mp
(assigned_some_of_eligible (assign M hE d w) w e hel)
have hjk : j = k := Option.some.inj (hw.symm.trans hwk)
exact hjk ▸ hek.2.2.1
/-- No selected label has an unlisted level, so the indexed and literal
weight-threshold cumulative accepted sets agree, not merely their sizes. -/
theorem indexed_selection_eq_threshold (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n)
{h : ℕ} {i : ℤ} (hi : (groups M d w)[h]? = some i) :
(selection M hE d w π).filter
(fun e => ∃ j, w e = some j ∧ h ≤ (groups M d w).idxOf j) =
(selection M hE d w π).filter
(fun e => ∃ j, w e = some j ∧ i ≤ j) := by
classical
obtain ⟨him, hidx⟩ := groups_index_of_get M d w hi
ext e
simp only [Finset.mem_filter]
constructor
· rintro ⟨he, j, hw, hj⟩
have hjm := selected_level_mem_groups M hE d w π he hw
have hlt := groups_idxOf_lt_iff M d w j i hjm him
exact ⟨he, j, hw, by omega⟩
· rintro ⟨he, j, hw, hj⟩
have hjm := selected_level_mem_groups M hE d w π he hw
have hlt := groups_idxOf_lt_iff M d w j i hjm him
exact ⟨he, j, hw, by omega⟩
/-- Literal source weight-threshold form of the deterministic clause. -/
theorem listedLambda_le_threshold_count (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n)
{h : ℕ} {i : ℤ} (hi : (groups M d w)[h]? = some i) :
listedLambda M hE d w h ≤
((selection M hE d w π).filter (fun e => ∃ j, w e = some j ∧ i ≤ j)).card := by
rw [← indexed_selection_eq_threshold M hE d w π hi]
exact (AccountingContracts.all_orders M hE d w h).1 π
end MatroidProphet.MainAlgorithm
end OAI
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