| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! Source-facing accounting lemmas. In particular, the thinning assertion is |
| proved on each fixed outside-test fiber, not only after averaging those bits. -/ |
|
|
| namespace MatroidProphet.AccountingContracts |
| open Set Finset Pivots |
|
|
| variable {α : Type*} [Fintype α] [DecidableEq α] |
|
|
| /-- Fixing all test memberships outside the focal group leaves its Bernoulli |
| rank lower bound intact. This is the finite conditional law in `lem:all-orders`. -/ |
| theorem thinning_final_fixed_outside |
| (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C : ℕ → Set α) (G : ℕ → Finset α) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) |
| (h final : ℕ) (O : Set α) (ε : Fin 2) |
| (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) |
| (A : Finset α) (hA : Disjoint A (G h)) : |
| t * (finalRankStatistic M hE κ D C |
| (fun j => (A : Set α) ∩ (G j : Set α)) h final |
| (G h) O Set.univ ε : ℝ) ≤ |
| bitsExpectation (fun _ : α => t) (G h) (fun B => |
| (finalRankStatistic M hE κ D C |
| (fun j => ((A ∪ B : Finset α) : Set α) ∩ (G j : Set α)) h final |
| (G h) O ((A ∪ B : Finset α) : Set α) ε : ℝ)) := by |
| classical |
| simp only [finalRankStatistic, Set.inter_univ, Nat.cast_sum, |
| bitsExpectation_sum, mul_sum] |
| apply Finset.sum_le_sum |
| intro b hb |
| let S := nominalLayerSet M hE κ D C h (G h) ε b \ O |
| have hS : S ⊆ (G h : Set α) := |
| fun e he => nominalLayerSet_subset M hE κ D C h (G h) ε b he.1 |
| have hin := independent_thinning_rank M hE (G h) S.toFinset |
| (fun e he => hS (by simpa only [Set.mem_toFinset] using he)) |
| (guardedPath M hE κ D C final (b.val.val - 2) ∪ |
| lowerCompetition M hE κ D C |
| (fun j => (A : Set α) ∩ (G j : Set α)) b.val.val h) t ht0 ht1 |
| simp only [Set.coe_toFinset, Finset.coe_inter] at hin |
| apply hin.trans_eq |
| apply bitsExpectation_congr |
| intro B hB |
| rw [lowerCompetition_ignore_focal M hE κ D C G hG b.val.val h A B hB] |
| have hs : S ∩ ((A ∪ B : Finset α) : Set α) = S ∩ (B : Set α) := by |
| ext e |
| simp only [Set.mem_inter_iff, Finset.mem_coe, Finset.mem_union] |
| constructor |
| · rintro ⟨he, ha | hb⟩ |
| · exact False.elim (Finset.disjoint_left.mp hA ha (hS he)) |
| · exact ⟨he, hb⟩ |
| · rintro ⟨he, hb⟩ |
| exact ⟨he, Or.inr hb⟩ |
| change (conditionalRank M (S ∩ (B : Set α)) _ : ℝ) = |
| (conditionalRank M (S ∩ ((A ∪ B : Finset α) : Set α)) _ : ℝ) |
| rw [hs] |
|
|
| /-- The manuscript's real-valued one-budget rank loss, before test thinning. |
| The bound is valid for the entire finite parity window, including empty layers. -/ |
| theorem nominal_rank_le_final_add_cost |
| (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (hκ : 0 < κ) (D C T : ℕ → Set α) (h later : ℕ) |
| (U O : Set α) (ε : Fin 2) : |
| (nominalRankStatistic M hE κ D C T h U O ε : ℝ) ≤ |
| (finalRankStatistic M hE κ D C T h (h+1+later) U O Set.univ ε : ℝ) + |
| (C h).ncard + ∑ j ∈ range later, |
| ((C (h+1+j)).ncard + (D (h+1+j)).ncard / (κ : ℝ)) := by |
| have hn := testRankStatistic_le_final_cost M hE κ hκ D C T h later U O Set.univ ε |
| have heq : testRankStatistic M hE κ D C T h U O Set.univ ε = |
| nominalRankStatistic M hE κ D C T h U O ε := by |
| simp only [testRankStatistic, nominalRankStatistic, Set.inter_univ] |
| rw [heq] at hn |
| have hr : (κ : ℝ) * (nominalRankStatistic M hE κ D C T h U O ε : ℝ) ≤ |
| (κ : ℝ) * (finalRankStatistic M hE κ D C T h (h+1+later) U O Set.univ ε : ℝ) + |
| (κ : ℝ) * (C h).ncard + ∑ j ∈ range later, |
| ((κ : ℝ) * (C (h+1+j)).ncard + (D (h+1+j)).ncard) := by |
| exact_mod_cast hn |
| have hk : (0 : ℝ) < κ := by exact_mod_cast hκ |
| apply (mul_le_mul_iff_right₀ hk).mp |
| have hs : (κ : ℝ) * (∑ j ∈ range later, |
| ((C (h+1+j)).ncard + (D (h+1+j)).ncard / (κ : ℝ))) = |
| ∑ j ∈ range later, ((κ : ℝ) * (C (h+1+j)).ncard + (D (h+1+j)).ncard) := by |
| rw [mul_sum] |
| apply sum_congr rfl |
| intro j hj |
| rw [mul_add] |
| congr 1 |
| field_simp |
| rw [mul_add, mul_add, hs] |
| exact hr |
| |
| open MainAlgorithm in |
| /-- Both source clauses of the order-uniform rank bound: deterministic domination |
| for every order, and test thinning on every fixed outside-group fiber. -/ |
| theorem all_orders {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) : |
| (∀ π : ArrivalOrder n, listedLambda M hE d w h ≤ |
| ((selection M hE d w π).filter (fun e => |
| ∃ i, w e = some i ∧ h ≤ (groups M d w).idxOf i)).card) ∧ |
| (∀ A : Finset (Fin n), Disjoint A (pathGroups M d w h) → |
| thinningRate * (finalRankStatistic M hE (2^100) |
| (groupMask M d w d.D) (groupMask M d w d.C) |
| (fun j => (A : Set (Fin n)) ∩ (pathGroups M d w j : Set (Fin n))) |
| h (groups M d w).length (pathGroups M d w h) |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) Set.univ |
| (boolParity d.odd) : ℝ) ≤ |
| bitsExpectation (fun _ : Fin n => thinningRate) (pathGroups M d w h) |
| (fun B => (listedLambda M hE {d with T := A ∪ B} w h : ℝ))) := by |
| classical |
| refine ⟨fun π => listedLambda_le_count M hE d w π h, ?_⟩ |
| intro A hA |
| simp only [listedLambda_update_T] |
| exact thinning_final_fixed_outside M hE (2^100) |
| (groupMask M d w d.D) (groupMask M d w d.C) (pathGroups M d w) |
| (pathGroups_disjoint M d w) h (groups M d w).length |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (boolParity d.odd) |
| thinningRate (by norm_num [thinningRate]) (by norm_num [thinningRate]) A hA |
| |
| open MainAlgorithm in |
| /-- The source telescoping rank-cost assertion for an actual listed group. |
| All later expansions use one cost over the whole auxiliary path. -/ |
| theorem rank_cost {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) |
| (hh : h < (groups M d w).length) : |
| (listedZ M hE d w h : ℝ) ≤ |
| (finalRankStatistic M hE (2^100) (groupMask M d w d.D) |
| (groupMask M d w d.C) (groupMask M d w d.T) h (groups M d w).length |
| (groupMask M d w univ h) |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) Set.univ |
| (boolParity d.odd) : ℝ) + (groupMask M d w d.C h).ncard + |
| ∑ j ∈ range ((groups M d w).length-(h+1)), |
| ((groupMask M d w d.C (h+1+j)).ncard + |
| (groupMask M d w d.D (h+1+j)).ncard / densityThreshold) := by |
| have hc := nominal_rank_le_final_add_cost M hE (2^100) (by positivity) |
| (groupMask M d w d.D) (groupMask M d w d.C) (groupMask M d w d.T) |
| h ((groups M d w).length-(h+1)) (groupMask M d w univ h) |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (boolParity d.odd) |
| have hlen : h+1+((groups M d w).length-(h+1)) = (groups M d w).length := by omega |
| rw [hlen] at hc |
| simpa only [listedZ, densityThreshold, Nat.cast_pow, Nat.cast_ofNat] using hc |
| |
| end MatroidProphet.AccountingContracts |
| |
| end OAI |
| |
