| import OAI.Probability.MatroidProphet.Main | |
| namespace OAI | |
| namespace MatroidProphet | |
| namespace TransferContracts | |
| open Finset Set Pivots MainAlgorithm | |
| section Pivots | |
| variable {α : Type*} {r q n : ℕ} | |
| /-- An injectively timed sequence determines one recorded pivot-mark vector. -/ | |
| lemma recordsPivots_unique (M : Matroid α) (label : Occurrence r q → α) | |
| (test : Fin n → α) (time : Occurrence r q → ℕ) | |
| (htime : Function.Injective time) (oldMark : Fin r → Bool) | |
| (movableMark : Fin q → Bool) {t u : Finset (Fin n)} | |
| (ht : RecordsPivots M label test time oldMark movableMark t) | |
| (hu : RecordsPivots M label test time oldMark movableMark u) : t = u := by | |
| ext f | |
| obtain ⟨o, ho, hto⟩ := ht f | |
| obtain ⟨p, hp, hup⟩ := hu f | |
| have hop := pivot_unique M label time htime ho hp | |
| subst p | |
| exact hto.symm.trans hup | |
| variable [Fintype α] | |
| /-- Complete marked-pivot contract: arbitrary repeated/redundant old labels, | |
| all placements/orders/marks of at most `s` movable occurrences, and the exact | |
| binomial bound. Source old-spanning/nonloop assumptions ensure every permitted | |
| interleaving actually contributes exactly one vector. No positivity assumption | |
| on `s`, `q`, `r`, or `n` is imposed. -/ | |
| theorem marked_pivot_contract (M : Matroid α) (b : ℕ → α) (a : Fin q → α) | |
| (test : Fin n → α) (oldMark : Fin r → Bool) (s : ℕ) (hqs : q ≤ s) | |
| (hground : ∀ o : Occurrence r q, occurrenceLabel b a o ∈ M.E) | |
| (holdspan : M.closure (oldPrefix b r) = M.E) | |
| (htest : ∀ f, test f ∈ M.E ∧ test f ∉ M.closure ∅) : | |
| (markedPivotPatterns M b a test oldMark).card ≤ | |
| 4 ^ s * ∑ k ∈ Finset.Iic s, n.choose k ∧ | |
| ∀ (time : Occurrence r q → ℕ), OldOrdered time → Function.Injective time → | |
| ∀ movableMark : Fin q → Bool, | |
| ∃! t, t ∈ markedPivotPatterns M b a test oldMark ∧ | |
| RecordsPivots M (occurrenceLabel b a) test time oldMark movableMark t := by | |
| refine ⟨card_markedPivotPatterns_le M b a test oldMark s hqs hground, ?_⟩ | |
| intro time horder htime movableMark | |
| obtain ⟨t, htmem, ht⟩ := recorded_pattern_exists_of_old_spanning M b a test oldMark | |
| holdspan htest time horder htime movableMark | |
| refine ⟨t, ⟨htmem, ht⟩, ?_⟩ | |
| intro u hu | |
| exact recordsPivots_unique M (occurrenceLabel b a) test time htime oldMark movableMark hu.2 ht | |
| end Pivots | |
| section Transfer | |
| variable {n : ℕ} | |
| /-- The independent comparison group is constructed by priority greedy and | |
| fair-mask deletion, exactly as in the manuscript (`Y_i = (I* \ H) ∩ U_i`). -/ | |
| lemma comparison_group_indep_subset (M : Matroid (Fin n)) (hE : M.E = Set.univ) | |
| (d : MainMasks n) (a : Fin n → Option ℤ) (i : ℤ) : | |
| let Y := (positiveGreedy M a \ d.H).filter (fun e => a e = some i) | |
| M.Indep (Y : Set (Fin n)) ∧ (Y : Set (Fin n)) ⊆ (trueGroup M d a i : Set (Fin n)) := by | |
| classical | |
| dsimp only | |
| constructor | |
| · exact (positiveGreedy_survivors_indep M hE a d.H).subset (Finset.filter_subset _ _) | |
| · intro e he | |
| obtain ⟨heY, hei⟩ := Finset.mem_filter.mp he | |
| apply (mem_trueGroup M d a i e).mpr | |
| refine ⟨?_, hei⟩ | |
| exact (Finset.mem_filter.mp (positiveGreedy_survivors_subset M a d.H heY)).2 | |
| /-- Full sample-to-rank inequality for the actual rounded weight vector. | |
| `withMasks` averages all density, guard and test bits while fixing `H`; | |
| `fairParityExpectation` averages parity. `listedRankStatistic` is explicitly | |
| zero on the unlisted branch. Thus there is no conditioning on listing and no | |
| independence assumption about the selected residual. -/ | |
| theorem sample_to_rank_contract (M : Matroid (Fin n)) (hE : M.E = Set.univ) | |
| (w : Weights n) (d : MainMasks n) (i : ℤ) | |
| (hn : densityThreshold ≤ | |
| ((trueGroup M d (fun e => roundedLevel weightBase (w e)) i).card : ℝ)) : | |
| let a := fun e => roundedLevel weightBase (w e) | |
| let Y := (positiveGreedy M a \ d.H).filter (fun e => a e = some i) | |
| (retainedFraction * (Y.card : ℝ) - ((2 : ℝ) ^ 23)⁻¹ * | |
| (trueGroup M d a i).card) / densityThreshold ≤ | |
| tripleMaskExpectation (fun _ => (1 : ℝ) / 4) | |
| (fun _ => thinningRate) (fun _ => thinningRate) Finset.univ | |
| (fun D C T => fairParityExpectation (fun p => | |
| (listedRankStatistic M hE (2 ^ 100) (withMasks d D C T) a i | |
| (boolParity p) : ℝ))) := by | |
| classical | |
| let a := fun e => roundedLevel weightBase (w e) | |
| let Y := (positiveGreedy M a \ d.H).filter (fun e => a e = some i) | |
| have hY := comparison_group_indep_subset M hE d a i | |
| have h := listedRankStatistic_lower M hE d a i (Y : Set (Fin n)) hY.2 hY.1 hn | |
| simpa only [Set.ncard_coe_finset] using h | |
| end Transfer | |
| end TransferContracts | |
| end MatroidProphet | |
| end OAI | |