| import Mathlib.Combinatorics.Matroid.Closure |
| import Mathlib.Data.Finset.Max |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| namespace Pivots |
|
|
| open Set Finset |
|
|
| variable {α β : Type*} [Fintype β] [DecidableEq β] |
|
|
| noncomputable def keptOccurrences (M : Matroid α) (label : β → α) (time : β → ℕ) : |
| Finset β := by |
| classical |
| exact Finset.univ.filter (fun o => label o ∉ M.closure (label '' {p | time p < time o})) |
|
|
| lemma mem_closure_kept_upto (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (hground : ∀ o, label o ∈ M.E) (o : β) : |
| label o ∈ M.closure (label '' {p | p ∈ keptOccurrences M label time ∧ time p ≤ time o}) := by |
| classical |
| have aux : ∀ k, ∀ o, time o = k → |
| label o ∈ M.closure (label '' {p | p ∈ keptOccurrences M label time ∧ time p ≤ k}) := by |
| intro k |
| induction k using Nat.strong_induction_on with |
| | h k ih => |
| intro o ho |
| by_cases hkeep : o ∈ keptOccurrences M label time |
| · exact M.mem_closure_of_mem ⟨o, ⟨hkeep, ho.le⟩, rfl⟩ |
| (by rintro _ ⟨p, _, rfl⟩; exact hground p) |
| have hdep : label o ∈ M.closure (label '' {p | time p < time o}) := by |
| simpa [keptOccurrences] using hkeep |
| apply M.closure_subset_closure_of_subset_closure ?_ hdep |
| rintro _ ⟨p, hp, rfl⟩ |
| have hpk : time p < k := by simpa [ho] using hp |
| have hih := ih (time p) hpk p rfl |
| apply M.closure_subset_closure ?_ hih |
| rintro _ ⟨q, ⟨hq, hqt⟩, rfl⟩ |
| exact ⟨q, ⟨hq, hqt.trans hpk.le⟩, rfl⟩ |
| exact aux (time o) o rfl |
|
|
| lemma greedy_prefix_closure (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (hground : ∀ o, label o ∈ M.E) (k : ℕ) : |
| M.closure (label '' {o | time o < k}) = |
| M.closure (label '' {o | o ∈ keptOccurrences M label time ∧ time o < k}) := by |
| classical |
| apply Set.Subset.antisymm |
| · apply M.closure_subset_closure_of_subset_closure |
| rintro _ ⟨o, ho, rfl⟩ |
| apply M.closure_subset_closure ?_ (mem_closure_kept_upto M label time hground o) |
| rintro _ ⟨p, ⟨hp, hpt⟩, rfl⟩ |
| exact ⟨p, ⟨hp, lt_of_le_of_lt hpt ho⟩, rfl⟩ |
| · apply M.closure_subset_closure |
| rintro _ ⟨o, ⟨_, ho⟩, rfl⟩ |
| exact ⟨o, ho, rfl⟩ |
|
|
| private lemma indep_image_of_notMem_prior |
| {α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β] (M : Matroid α) (label : β → α) |
| (time : β → ℕ) (htime : Function.Injective time) (s : Finset β) |
| (hground : ∀ o ∈ s, label o ∈ M.E) |
| (hprior : ∀ o ∈ s, label o ∉ M.closure (label '' {p | p ∈ s ∧ time p < time o})) : |
| M.Indep (label '' (s : Set β)) := by |
| classical |
| induction s using Finset.strongInductionOn with |
| | _ s ih => |
| by_cases hs : s = ∅ |
| · simp [hs] |
| obtain ⟨p, hp, hmax⟩ := Finset.exists_max_image s time (Finset.nonempty_iff_ne_empty.mpr hs) |
| have hI := ih (s.erase p) (Finset.erase_ssubset hp) |
| (fun o ho => hground o (Finset.mem_of_mem_erase ho)) (by |
| intro o ho hmem |
| apply hprior o (Finset.mem_of_mem_erase ho) |
| apply M.closure_subset_closure ?_ hmem |
| rintro _ ⟨q, ⟨hq, hqt⟩, rfl⟩ |
| exact ⟨q, ⟨Finset.mem_of_mem_erase hq, hqt⟩, rfl⟩) |
| have hnot : label p ∉ M.closure (label '' (s.erase p : Set β)) := by |
| intro hmem |
| apply hprior p hp |
| apply M.closure_subset_closure ?_ hmem |
| rintro _ ⟨q, hq, rfl⟩ |
| refine ⟨q, ⟨Finset.mem_of_mem_erase hq, ?_⟩, rfl⟩ |
| apply lt_of_le_of_ne (hmax q (Finset.mem_of_mem_erase hq)) |
| intro htimeeq |
| exact (Finset.ne_of_mem_erase hq) (htime htimeeq) |
| have hins := hI.insert_indep_iff.mpr (Or.inl ⟨hground p hp, hnot⟩) |
| have hset : insert (label p) (label '' (s.erase p : Set β)) = label '' (s : Set β) := by |
| rw [← Set.image_insert_eq, ← Finset.coe_insert, Finset.insert_erase hp] |
| rwa [hset] at hins |
|
|
| lemma keptOccurrences_indep (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (hground : ∀ o, label o ∈ M.E) (htime : Function.Injective time) : |
| M.Indep (label '' (keptOccurrences M label time : Set β)) := by |
| classical |
| apply indep_image_of_notMem_prior M label time htime |
| · intro o _ |
| exact hground o |
| · intro o ho hmem |
| have hn := (Finset.mem_filter.mp ho).2 |
| apply hn |
| apply M.closure_subset_closure ?_ hmem |
| rintro _ ⟨p, ⟨_, hpt⟩, rfl⟩ |
| exact ⟨p, hpt, rfl⟩ |
|
|
| end Pivots |
| end MatroidProphet |
|
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| end OAI |
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|
