| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! Exact agreement between the source's previous-accepted-label test and the |
| concrete implementation's previous-eligible-label test. No arrival-order |
| assumption beyond the explicit arrival times is used in span preservation. -/ |
|
|
| namespace MatroidProphet |
| open Set Finset |
|
|
| variable {α : Type*} [DecidableEq α] |
|
|
| /-- Rejected earlier eligible labels do not change the span of a layer prefix. -/ |
| theorem layerGreedy_closure_prior (M : Matroid α) (E : Finset α) |
| (base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ) |
| (hE : M.E = Set.univ) (b : ℤ) (k : ℕ) : |
| M.closure (base b ∪ layerPrior E layer time b k) = |
| M.closure (base b ∪ |
| layerPrior (layerGreedy M E base layer time) layer time b k) := by |
| apply Set.Subset.antisymm |
| · apply M.closure_subset_closure_of_subset_closure |
| rintro e (he | ⟨he, hb, ht⟩) |
| · exact M.mem_closure_of_mem' (Or.inl he) (by simp [hE]) |
| · have hs := layerGreedy_spans_prefix M E base layer time hE e he |
| apply M.closure_subset_closure ?_ hs |
| rintro f (hf | ⟨hf, hfb, hft⟩) |
| · exact Or.inl (by simpa [hb] using hf) |
| · exact Or.inr ⟨hf, hfb.trans hb, hft.trans_lt ht⟩ |
| · apply M.closure_subset_closure |
| rintro e (he | ⟨he, hb, ht⟩) |
| · exact Or.inl he |
| · exact Or.inr ⟨layerGreedy_subset M E base layer time he, hb, ht⟩ |
|
|
| /-- The online decision can be implemented using only earlier accepted labels |
| of the same layer, exactly as in step 3 of the manuscript algorithm. -/ |
| theorem layerGreedy_mem_iff_previous_accepted (M : Matroid α) (E : Finset α) |
| (base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ) |
| (hE : M.E = Set.univ) (e : α) : |
| e ∈ layerGreedy M E base layer time ↔ |
| e ∈ E ∧ e ∉ M.closure (base (layer e) ∪ |
| layerPrior (layerGreedy M E base layer time) layer time (layer e) (time e)) := by |
| classical |
| conv_lhs => unfold layerGreedy |
| rw [Finset.mem_filter] |
| rw [layerGreedy_closure_prior M E base layer time hE] |
|
|
| omit [DecidableEq α] in |
| /-- On a finite ground set the source's rank-one test is precisely nonmembership |
| in the closure, including rank-zero and loop cases. -/ |
| theorem conditionalRank_singleton_eq_one_iff [Fintype α] (M : Matroid α) |
| (hE : M.E = Set.univ) (P : Set α) (e : α) : |
| conditionalRank M {e} P = 1 ↔ e ∉ M.closure P := by |
| by_cases he : e ∈ M.closure P |
| · rw [Pivots.conditionalRank_singleton_of_mem_closure M P he] |
| simp [he] |
| · rw [Pivots.conditionalRank_singleton_of_notMem_closure M P (by simp [hE]) he] |
| simp [he] |
|
|
| /-- Exact rank form of the source's accept/reject condition. -/ |
| theorem layerGreedy_mem_iff_rank_previous_accepted [Fintype α] |
| (M : Matroid α) (E : Finset α) (base : ℤ → Set α) |
| (layer : α → ℤ) (time : α → ℕ) (hE : M.E = Set.univ) (e : α) : |
| e ∈ layerGreedy M E base layer time ↔ |
| e ∈ E ∧ conditionalRank M {e} (base (layer e) ∪ |
| layerPrior (layerGreedy M E base layer time) layer time (layer e) (time e)) = 1 := by |
| rw [layerGreedy_mem_iff_previous_accepted M E base layer time hE, |
| conditionalRank_singleton_eq_one_iff M hE] |
|
|
| /-- The concrete history-based online decision agrees with the rank test on |
| previously accepted labels in the current layer. -/ |
| theorem layerKeyDecision_eq_rank_previous_accepted {n : ℕ} {K : Type*} |
| (M : Matroid (Fin n)) (hE : M.E = Set.univ) (base : ℤ → Set (Fin n)) |
| (assign : Fin n → K → Option ℤ) (v : Fin n → K) |
| (π : ArrivalOrder n) (k : Fin n) : |
| layerKeyDecision M base assign k (keyHistory v π k) = true ↔ |
| π k ∈ assignedEligible assign v ∧ |
| conditionalRank M {π k} (base (assignedLayer assign v (π k)) ∪ |
| layerPrior |
| (layerGreedy M (assignedEligible assign v) base (assignedLayer assign v) |
| (fun e => (π.symm e).val)) |
| (assignedLayer assign v) (fun e => (π.symm e).val) |
| (assignedLayer assign v (π k)) k.val) = 1 := by |
| rw [layerKeyDecision_eq_greedy, |
| layerGreedy_mem_iff_rank_previous_accepted M _ _ _ _ hE] |
| simp only [Equiv.symm_apply_apply] |
|
|
| end MatroidProphet |
|
|
| end OAI |
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|
