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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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
/-! The manuscript writes the observed rank mask as `D ∪ C`, whereas the
algorithm also removes `H`. The two definitions coincide on every candidate
group, including empty and unlisted groups. These lemmas discharge that
semantic identification without adding a premise to a source theorem. -/
namespace MatroidProphet.SourceRankMask
open Set Finset MainAlgorithm
variable {α : Type*} [Fintype α]
omit [Fintype α] in
lemma sdiff_union_of_subset_disjoint (S U H O : Set α)
(hSU : S ⊆ U) (hUH : Disjoint U H) : S \ (H ∪ O) = S \ O := by
ext e
constructor
· rintro ⟨he, hn⟩
exact ⟨he, fun ho => hn (Or.inr ho)⟩
· rintro ⟨he, hn⟩
refine ⟨he, ?_⟩
rintro (hh | ho)
· exact Set.disjoint_left.mp hUH (hSU he) hh
· exact hn ho
lemma nominalLayerSet_sdiff_union (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (U H O : Set α) (ε : Fin 2)
(b : Pivots.ParityWindow (activation h) ε) (hUH : Disjoint U H) :
nominalLayerSet M hE κ D C h U ε b \ (H ∪ O) =
nominalLayerSet M hE κ D C h U ε b \ O :=
sdiff_union_of_subset_disjoint _ U H O
(nominalLayerSet_subset M hE κ D C h U ε b) hUH
lemma nominalRankStatistic_mask_union (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U H O : Set α) (ε : Fin 2)
(hUH : Disjoint U H) :
nominalRankStatistic M hE κ D C T h U (H ∪ O) ε =
nominalRankStatistic M hE κ D C T h U O ε := by
unfold nominalRankStatistic
apply Finset.sum_congr rfl
intro b _
rw [nominalLayerSet_sdiff_union M hE κ D C h U H O ε b hUH]
lemma testRankStatistic_mask_union (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U H O test : Set α) (ε : Fin 2)
(hUH : Disjoint U H) :
testRankStatistic M hE κ D C T h U (H ∪ O) test ε =
testRankStatistic M hE κ D C T h U O test ε := by
unfold testRankStatistic
apply Finset.sum_congr rfl
intro b _
rw [nominalLayerSet_sdiff_union M hE κ D C h U H O ε b hUH]
lemma finalRankStatistic_mask_union (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C T : ℕ → Set α) (h final : ℕ) (U H O test : Set α) (ε : Fin 2)
(hUH : Disjoint U H) :
finalRankStatistic M hE κ D C T h final U (H ∪ O) test ε =
finalRankStatistic M hE κ D C T h final U O test ε := by
unfold finalRankStatistic
apply Finset.sum_congr rfl
intro b _
rw [nominalLayerSet_sdiff_union M hE κ D C h U H O ε b hUH]
variable {n : ℕ}
lemma groupMask_disjoint_H (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (mask : Finset (Fin n)) (h : ℕ) :
Disjoint (groupMask M d w mask h) (d.H : Set (Fin n)) := by
apply Set.disjoint_left.mpr
intro e he hh
exact he.2.1.2.1 hh
/-- Exact source mask for the nominal statistic; no listing hypothesis. -/
lemma listedZ_sourceMask (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) :
listedZ M hE d w h =
nominalRankStatistic M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C)
(groupMask M d w d.T) h (groupMask M d w univ h)
((d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (boolParity d.odd) := by
unfold listedZ
simpa only [Finset.coe_union, Set.union_assoc] using
nominalRankStatistic_mask_union M hE (2^100) (groupMask M d w d.D)
(groupMask M d w d.C) (groupMask M d w d.T) h (groupMask M d w univ h)
(d.H : Set (Fin n)) ((d.D ∪ d.C : Finset (Fin n)) : Set (Fin n))
(boolParity d.odd) (groupMask_disjoint_H M d w univ h)
/-- Exact source mask for the final statistic used in the all-orders bound. -/
lemma listedLambda_sourceMask (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) :
listedLambda M hE d w h =
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.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (d.T : Set (Fin n))
(boolParity d.odd) := by
unfold listedLambda
simpa only [Finset.coe_union, Set.union_assoc] using
finalRankStatistic_mask_union 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 : Set (Fin n))
((d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (d.T : Set (Fin n))
(boolParity d.odd) (groupMask_disjoint_H M d w univ h)
/-- Exact nominal statistic in the transfer section, including its zero branch. -/
lemma listedRankStatistic_sourceMask (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(κ : ℕ) (d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ) (ε : Fin 2) :
listedRankStatistic M hE κ d w i ε =
if i ∈ groups M d w then
nominalRankStatistic M hE κ (groupMask M d w d.D) (groupMask M d w d.C)
(groupMask M d w d.T) ((groups M d w).idxOf i)
(trueGroup M d w i : Set (Fin n))
((d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) ε
else 0 := by
classical
have hUH : Disjoint (trueGroup M d w i : Set (Fin n)) (d.H : Set (Fin n)) := by
apply Set.disjoint_left.mpr
intro e he hh
exact ((mem_trueGroup M d w i e).mp he).1.2.1 hh
unfold listedRankStatistic
split_ifs
· simpa only [Finset.coe_union, Set.union_assoc] using
nominalRankStatistic_mask_union M hE κ (groupMask M d w d.D)
(groupMask M d w d.C) (groupMask M d w d.T) ((groups M d w).idxOf i)
(trueGroup M d w i : Set (Fin n)) (d.H : Set (Fin n))
((d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) ε hUH
· rfl
end MatroidProphet.SourceRankMask
end OAI
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