| import OAI.Probability.MatroidProphet.Pivots.Guarantee |
| import OAI.Probability.MatroidProphet.Pivots.Existence |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| namespace Pivots |
|
|
| open Set Finset |
|
|
| variable {α : Type*} [Fintype α] {r q n : ℕ} |
|
|
| lemma recordsPivots_exists |
| {α : Type u_1} [Fintype α] {r : ℕ} {q : ℕ} {n : ℕ} (M : Matroid α) (label : Occurrence r q → α) |
| (test : Fin n → α) (time : Occurrence r q → ℕ) |
| (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) |
| (hnonloop : ∀ f, test f ∉ M.closure ∅) |
| (hspan : ∀ f, test f ∈ M.closure (Set.range label)) : |
| ∃ t, RecordsPivots M label test time oldMark movableMark t := by |
| classical |
| choose pivot hpivot using fun f => exists_pivot M label time (hnonloop f) (hspan f) |
| refine ⟨Finset.univ.filter (fun f => occurrenceMark oldMark movableMark (pivot f) = true), ?_⟩ |
| intro f |
| exact ⟨pivot f, hpivot f, by simp⟩ |
|
|
| lemma recorded_pattern_exists_of_old_spanning (M : Matroid α) (b : ℕ → α) (a : Fin q → α) |
| (test : Fin n → α) (oldMark : Fin r → Bool) |
| (holdspan : M.closure (oldPrefix b r) = M.E) |
| (htest : ∀ f, test f ∈ M.E ∧ test f ∉ M.closure ∅) |
| (time : Occurrence r q → ℕ) (horder : OldOrdered time) (htime : Function.Injective time) |
| (movableMark : Fin q → Bool) : |
| ∃ t ∈ markedPivotPatterns M b a test oldMark, |
| RecordsPivots M (occurrenceLabel b a) test time oldMark movableMark t := by |
| classical |
| have hspan : ∀ f, test f ∈ M.closure (Set.range (occurrenceLabel (r := r) b a)) := by |
| intro f |
| have htest' : test f ∈ M.closure (oldPrefix b r) := holdspan.symm ▸ (htest f).1 |
| apply M.closure_subset_closure ?_ htest' |
| rintro _ ⟨k, hk, rfl⟩ |
| exact ⟨Sum.inl ⟨k, hk⟩, rfl⟩ |
| obtain ⟨t, ht⟩ := recordsPivots_exists M (occurrenceLabel b a) test time oldMark movableMark |
| (fun f => (htest f).2) hspan |
| exact ⟨t, Finset.mem_filter.mpr ⟨Finset.mem_univ _, time, horder, htime, movableMark, ht⟩, ht⟩ |
|
|
| end Pivots |
| end MatroidProphet |
|
|
| end OAI |
|
|
