| 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 |
|
|
