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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | 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
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