| import OAI.Probability.MatroidSecretary.Pivots.MarkedPivotSpecification |
| import OAI.Probability.MatroidSecretary.Pivots.PivotOrderInvariance |
|
|
| namespace OAI |
|
|
| /-! |
| # Exact finite interleavings for the source-level marked-pivot family |
|
|
| This bridge applies directly to the finite-old specification, including an |
| empty label type. It does not extend an old list to an infinite sequence. |
| -/ |
|
|
| namespace MatroidProphet.MarkedPivots |
|
|
| variable {α : Type*} {r q n : ℕ} |
|
|
| lemma firstSpans_orderPosition_iff (M : Matroid α) |
| (labels : Occurrences r q → α) (time : Occurrences r q → ℕ) |
| (e : α) (o : Occurrences r q) : |
| FirstSpans M labels (Pivots.orderPosition time) e o ↔ |
| FirstSpans M labels time e o := |
| Pivots.isPivot_orderPosition_iff M labels time e o |
|
|
| lemma records_orderPosition_iff (M : Matroid α) |
| (old : Fin r → α) (movable : Fin q → α) (test : Fin n → α) |
| (time : Occurrences r q → ℕ) (oldMark : Fin r → Bool) |
| (movableMark : Fin q → Bool) (pattern : Finset (Fin n)) : |
| Records M old movable test (Pivots.orderPosition time) oldMark movableMark pattern ↔ |
| Records M old movable test time oldMark movableMark pattern := by |
| simp only [Records, firstSpans_orderPosition_iff] |
|
|
| lemma oldOrdered_orderPosition_iff (time : Occurrences r q → ℕ) : |
| OldOrdered (Pivots.orderPosition time) ↔ OldOrdered time := by |
| simp only [OldOrdered, Pivots.orderPosition_lt_iff] |
|
|
| / |
| occurrence set into its finite set of positions, keeping the old order. |
| No inhabitedness or distinctness assumption on labels is needed. -/ |
| theorem mem_patterns_iff_finite_order (M : Matroid α) |
| (old : Fin r → α) (movable : Fin q → α) (test : Fin n → α) |
| (oldMark : Fin r → Bool) (pattern : Finset (Fin n)) : |
| pattern ∈ patterns M old movable test oldMark ↔ |
| ∃ position : Occurrences r q ≃ Fin (Fintype.card (Occurrences r q)), |
| OldOrdered (fun o => (position o).val) ∧ |
| ∃ movableMark : Fin q → Bool, |
| Records M old movable test (fun o => (position o).val) |
| oldMark movableMark pattern := by |
| classical |
| constructor |
| · intro ht |
| obtain ⟨time, horder, htime, movableMark, hrecords⟩ := (Finset.mem_filter.mp ht).2 |
| let position := Equiv.ofBijective (Pivots.finiteOrderPosition time) |
| (Pivots.finiteOrderPosition_bijective time htime) |
| refine ⟨position, ?_, movableMark, ?_⟩ |
| · change OldOrdered (Pivots.orderPosition time) |
| exact (oldOrdered_orderPosition_iff time).mpr horder |
| · change Records M old movable test (Pivots.orderPosition time) |
| oldMark movableMark pattern |
| exact (records_orderPosition_iff M old movable test time |
| oldMark movableMark pattern).mpr hrecords |
| · rintro ⟨position, horder, movableMark, hrecords⟩ |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, (fun o => (position o).val), |
| horder, (fun o p h => position.injective (Fin.ext h)), movableMark, hrecords⟩ |
|
|
| end MatroidProphet.MarkedPivots |
|
|
| end OAI |
|
|
