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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 | import OAI.Probability.MatroidProphet.Residual.LabeledFamily
import OAI.Probability.MatroidProphet.PathRankCost
namespace OAI
namespace MatroidProphet
open Set Finset Pivots
variable {α : Type*} [Fintype α] [LinearOrder α]
lemma safeLayer_le_witness_residual
{α : Type u_1} [Fintype α] [LinearOrder α] (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U I : Set α) (hI : I ⊆ D h)
(ε : Fin 2) (b : ParityWindow (activation h) ε) (W : Set α) :
(safeLayerSet M hE κ D C T h U I ε b).ncard ≤ κ * W.ncard +
(nominalLayerSet M hE κ D C h U ε b \
M.closure (nominalPath M hE κ D C h (b.val.val - 2) ∪
lowerCompetition M hE κ D C T b.val.val h ∪ W)).ncard := by
by_cases hv : activation h + 2 ≤ b.val.val ∧ b.val.val ≤ 0
· have hx : nominalPath M hE κ D C h (b.val.val - 2) =
densityExpansion M hE κ (D h) (guardedPath M hE κ D C h (b.val.val - 2)) := by
unfold nominalPath densityEnabled
rw [ite_eq_left (by omega)]
rw [hx]
apply safe_layer_le_witness_residual M hE κ (D h)
(guardedPath M hE κ D C h (b.val.val - 2)) (lowerCompetition M hE κ D C T b.val.val h)
W (nominalLayerSet M hE κ D C h U ε b) (safeLayerSet M hE κ D C T h U I ε b)
intro e he
exact ⟨⟨hI he.1.1, he.1.2⟩, he.2⟩
· have he : nominalLayerSet M hE κ D C h U ε b = ∅ := by
ext e
constructor
· exact fun he => (hv ⟨he.2.1, he.2.2.1⟩).elim
· simp
simp [safeLayerSet, he]
lemma safeStatistic_le_witness_residual (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U I : Set α) (hI : I ⊆ D h)
(ε : Fin 2) (W : ParityWindow (activation h) ε → Set α) :
safeLayerStatistic M hE κ D C T h U I ε ≤ κ * (∑ b, (W b).ncard) +
(nominalResidualSet M hE κ D C T h U ε W).ncard := by
unfold safeLayerStatistic
calc
_ ≤ ∑ b, (κ * (W b).ncard +
(nominalLayerSet M hE κ D C h U ε b \
M.closure ((nominalPath M hE κ D C h (b.val.val - 2) ∪
lowerCompetition M hE κ D C T b.val.val h) ∪ W b)).ncard) :=
sum_le_sum (fun b _ => safeLayer_le_witness_residual M hE κ D C T h U I hI ε b (W b))
_ = _ := by
rw [sum_add_distrib, ← mul_sum, ← residualUnion_ncard M _ _ W
(nominalLayerSet_disjoint M hE κ D C h U ε)]
rfl
lemma exists_captured_residual (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (hκ : 0 < κ) (D C T : ℕ → Set α) (h : ℕ) (U : Finset α) (O I : Set α)
(hDU : D h ⊆ (U : Set α)) (hI : I ⊆ D h)
(hnonloop : ∀ e ∈ U, e ∉ M.closure ∅) (ε : Fin 2)
(hzsmall : nominalRankStatistic M hE κ D C T h (U : Set α) O ε ≤ U.card / κ) :
∃ R ∈ labeledResidualFamily M U κ (guardedPath M hE κ D C h) (activation h) ε
(fun b => lowerCompetition M hE κ D C T b.val.val h),
(R : Set α) ⊆ O ∧ safeLayerStatistic M hE κ D C T h (U : Set α) I ε ≤
κ * nominalRankStatistic M hE κ D C T h (U : Set α) O ε + R.card := by
classical
obtain ⟨W, hW, hsum, hspan, Z, hZU, hZcard, β, hβ⟩ :=
nominal_rank_witnesses M hE κ D C T h U O ε
let R := (Set.toFinite (nominalResidualSet M hE κ D C T h (U : Set α) ε W)).toFinset
have hmask : (R : Set α) ⊆ O := by
intro e he
exact residualUnion_subset_mask M _ _ W O hspan ((Set.Finite.mem_toFinset _).mp he)
have hdet : safeLayerStatistic M hE κ D C T h (U : Set α) I ε ≤
κ * nominalRankStatistic M hE κ D C T h (U : Set α) O ε + R.card := by
have h := safeStatistic_le_witness_residual M hE κ D C T h (U : Set α) I hI ε W
rwa [hsum, Set.ncard_eq_toFinset_card] at h
refine ⟨R, ?_, hmask, hdet⟩
apply Finset.mem_image.mpr
refine ⟨integerResidual M (nominalPath M hE κ D C h) (groupLabel U) (activation h) ε
(fun b => lowerCompetition M hE κ D C T b.val.val h) (fun w : Z => w.val) β, ?_, ?_⟩
· apply nominal_residual_mem_family M hE κ hκ D C h U Z hDU hZU (hZcard.trans hzsmall) ε
(fun b => lowerCompetition M hE κ D C T b.val.val h)
(fun b => lowerCompetition_subset_guarded M hE κ D C T b.val.val h) β
· intro w
have hw : w.val ∈ W (β w) := by
rw [← hβ (β w)]
exact ⟨w, rfl, rfl⟩
exact (hW (β w) hw).1.2.2.2.1
· exact fun i => hnonloop _ (groupLabel_mem U i)
· exact nominalResidualSet_eq_labeled M hE κ D C T h U Z ε W β hβ
end MatroidProphet
end OAI
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