| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! |
| # The layer rule is the source's accepted-prefix greedy rule |
| |
| The implementation tests the span of all previous eligible labels. The source |
| algorithm tests the span of the previous accepted labels. Prefix span preservation |
| makes these tests exactly equal, not merely comparable in final reward. |
| -/ |
| |
| namespace MatroidProphet |
| open Set Finset |
| |
| lemma layerGreedy_closure_prefix {α : Type*} [DecidableEq α] |
| (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 ∪ |
| {e | e ∈ layerGreedy M E base layer time ∧ layer e = b ∧ time e < 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 only [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⟩ |
|
|
| / |
| theorem layerGreedy_mem_iff_accepted_prefix {α : Type*} [DecidableEq 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 ∧ e ∉ M.closure (base (layer e) ∪ |
| {f | f ∈ layerGreedy M E base layer time ∧ |
| layer f = layer e ∧ time f < time e}) := by |
| classical |
| rw [show e ∈ layerGreedy M E base layer time ↔ e ∈ E ∧ |
| e ∉ M.closure (base (layer e) ∪ layerPrior E layer time (layer e) (time e)) from |
| Finset.mem_filter] |
| rw [layerGreedy_closure_prefix M E base layer time hE] |
|
|
| / |
| theorem layerGreedy_mem_iff_rank_one {α : Type*} [Fintype α] [DecidableEq 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) ∪ |
| {f | f ∈ layerGreedy M E base layer time ∧ |
| layer f = layer e ∧ time f < time e}) = 1 := by |
| rw [layerGreedy_mem_iff_accepted_prefix M E base layer time hE] |
| constructor |
| · rintro ⟨he, hne⟩ |
| exact ⟨he, Pivots.conditionalRank_singleton_of_notMem_closure M _ |
| (by simp [hE]) hne⟩ |
| · rintro ⟨he, hr⟩ |
| refine ⟨he, ?_⟩ |
| intro hmem |
| have hz := Pivots.conditionalRank_singleton_of_mem_closure M _ hmem |
| omega |
|
|
| end MatroidProphet |
|
|
| end OAI |
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|
