| import OAI.Probability.MatroidProphet.Residual.LabeledFamily |
| import OAI.Probability.MatroidProphet.PathRankCost |
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| namespace OAI |
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| namespace MatroidProphet |
| open Set Finset Pivots |
| variable {α : Type*} [Fintype α] [LinearOrder α] |
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| 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] |
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| 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 |
|
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| 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 |
|
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| end MatroidProphet |
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| end OAI |
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