| import OAI.Probability.MatroidProphet.Main |
| import Mathlib.Data.Fintype.EquivFin |
|
|
| namespace OAI |
|
|
| /-! |
| # Timestamp representations of finite insertion orders |
|
|
| The marked-pivot proof uses injective natural-number timestamps, whereas the |
| manuscript speaks about positions in a finite interleaving. We prove that |
| timestamps can be normalized to a permutation of the finite position set |
| without changing any prefix, first-spanning pivot, or recorded mark vector. |
| No bound on gaps between timestamps is imposed. |
| -/ |
|
|
| namespace MatroidProphet.Pivots |
|
|
| open Set Finset |
|
|
| variable {α β : Type*} [Fintype β] |
|
|
| / |
| noncomputable def orderPosition (time : β → ℕ) (o : β) : ℕ := by |
| classical |
| exact (Finset.univ.filter (fun p => time p < time o)).card |
|
|
| lemma orderPosition_mono (time : β → ℕ) {o p : β} (h : time o ≤ time p) : |
| orderPosition time o ≤ orderPosition time p := by |
| classical |
| apply Finset.card_le_card |
| intro a ha |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, |
| lt_of_lt_of_le (Finset.mem_filter.mp ha).2 h⟩ |
|
|
| lemma orderPosition_strict (time : β → ℕ) {o p : β} (h : time o < time p) : |
| orderPosition time o < orderPosition time p := by |
| classical |
| apply Finset.card_lt_card |
| apply Finset.ssubset_iff_subset_ne.mpr |
| constructor |
| · intro a ha |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, |
| lt_trans (Finset.mem_filter.mp ha).2 h⟩ |
| · intro heq |
| have ho : o ∈ Finset.univ.filter (fun a => time a < time p) := by simp [h] |
| rw [← heq] at ho |
| simp at ho |
|
|
| lemma orderPosition_lt_iff (time : β → ℕ) (o p : β) : |
| orderPosition time o < orderPosition time p ↔ time o < time p := by |
| refine ⟨?_, orderPosition_strict time⟩ |
| intro h |
| by_contra hn |
| exact (not_lt_of_ge (orderPosition_mono time (Nat.le_of_not_gt hn))) h |
|
|
| lemma orderPosition_le_iff (time : β → ℕ) (o p : β) : |
| orderPosition time o ≤ orderPosition time p ↔ time o ≤ time p := by |
| simp only [← not_lt, orderPosition_lt_iff] |
|
|
| lemma orderPosition_lt_card (time : β → ℕ) (o : β) : |
| orderPosition time o < Fintype.card β := by |
| classical |
| change (Finset.univ.filter (fun p => time p < time o)).card < Fintype.card β |
| rw [← Finset.card_univ] |
| apply Finset.card_lt_card |
| apply Finset.ssubset_iff_subset_ne.mpr |
| refine ⟨Finset.filter_subset _ _, ?_⟩ |
| intro heq |
| have ho : o ∈ (Finset.univ : Finset β) := Finset.mem_univ _ |
| rw [← heq] at ho |
| simp at ho |
|
|
| / |
| noncomputable def finiteOrderPosition (time : β → ℕ) (o : β) : Fin (Fintype.card β) := |
| ⟨orderPosition time o, orderPosition_lt_card time o⟩ |
|
|
| lemma finiteOrderPosition_injective (time : β → ℕ) (htime : Function.Injective time) : |
| Function.Injective (finiteOrderPosition time) := by |
| intro o p heq |
| apply htime |
| have hval := congrArg Fin.val heq |
| change orderPosition time o = orderPosition time p at hval |
| apply Nat.le_antisymm |
| · exact (orderPosition_le_iff time o p).mp hval.le |
| · exact (orderPosition_le_iff time p o).mp hval.ge |
|
|
| lemma finiteOrderPosition_bijective (time : β → ℕ) (htime : Function.Injective time) : |
| Function.Bijective (finiteOrderPosition time) := by |
| rw [Fintype.bijective_iff_injective_and_card] |
| exact ⟨finiteOrderPosition_injective time htime, by simp⟩ |
|
|
| omit [Fintype β] in |
| / |
| lemma isPivot_iff_of_same_order (M : Matroid α) (label : β → α) |
| (time time' : β → ℕ) (horder : ∀ o p, time o < time p ↔ time' o < time' p) |
| (e : α) (o : β) : IsPivot M label time e o ↔ IsPivot M label time' e o := by |
| have hbefore : {p | time p < time o} = {p | time' p < time' o} := by |
| ext p |
| exact horder p o |
| have hthrough : {p | time p ≤ time o} = {p | time' p ≤ time' o} := by |
| ext p |
| simp only [mem_ofPred_eq, ← not_lt, horder o p] |
| simp only [IsPivot, hbefore, hthrough] |
|
|
| lemma isPivot_orderPosition_iff (M : Matroid α) (label : β → α) |
| (time : β → ℕ) (e : α) (o : β) : |
| IsPivot M label (orderPosition time) e o ↔ IsPivot M label time e o := |
| isPivot_iff_of_same_order M label (orderPosition time) time |
| (orderPosition_lt_iff time) e o |
|
|
| lemma recordsPivots_orderPosition_iff {r q 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) (t : Finset (Fin n)) : |
| RecordsPivots M label test (orderPosition time) oldMark movableMark t ↔ |
| RecordsPivots M label test time oldMark movableMark t := by |
| simp only [RecordsPivots, isPivot_orderPosition_iff] |
|
|
| lemma oldOrdered_orderPosition_iff {r q : ℕ} (time : Occurrence r q → ℕ) : |
| OldOrdered (orderPosition time) ↔ OldOrdered time := by |
| simp only [OldOrdered, orderPosition_lt_iff] |
|
|
| / |
| interleavings. Arbitrary natural-number gaps neither add nor remove patterns. -/ |
| lemma mem_markedPivotPatterns_iff_finite_order [Fintype α] {r q n : ℕ} |
| (M : Matroid α) (b : ℕ → α) (a : Fin q → α) (test : Fin n → α) |
| (oldMark : Fin r → Bool) (t : Finset (Fin n)) : |
| t ∈ markedPivotPatterns M b a test oldMark ↔ |
| ∃ position : Occurrence r q ≃ Fin (Fintype.card (Occurrence r q)), |
| OldOrdered (fun o => (position o).val) ∧ |
| ∃ movableMark : Fin q → Bool, |
| RecordsPivots M (occurrenceLabel b a) test (fun o => (position o).val) |
| oldMark movableMark t := by |
| classical |
| constructor |
| · intro ht |
| obtain ⟨time, horder, htime, movableMark, hrecords⟩ := (Finset.mem_filter.mp ht).2 |
| let position := Equiv.ofBijective (finiteOrderPosition time) |
| (finiteOrderPosition_bijective time htime) |
| refine ⟨position, ?_, movableMark, ?_⟩ |
| · change OldOrdered (orderPosition time) |
| exact (oldOrdered_orderPosition_iff time).mpr horder |
| · change RecordsPivots M (occurrenceLabel b a) test (orderPosition time) |
| oldMark movableMark t |
| exact (recordsPivots_orderPosition_iff M (occurrenceLabel b a) test time |
| oldMark movableMark t).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.Pivots |
|
|
| end OAI |
|
|
