| import OAI.Probability.MatroidProphet.Residual.Statistics |
| import Mathlib.Data.Set.Card.Arithmetic |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Set Finset Pivots |
| variable {α B : Type*} [Fintype α] [Fintype B] |
|
|
| lemma exists_rankWitness (M : Matroid α) (hE : M.E = Set.univ) (A S : Set α) : |
| ∃ Z : Set α, Z ⊆ S ∧ Z.ncard = conditionalRank M S A ∧ S ⊆ M.closure (A ∪ Z) := by |
| obtain ⟨Z, hZ, hgen, hcard⟩ := exists_extension_generators M hE (M.closure A) S |
| (M.closure (M.closure A ∪ S)) (Matroid.isFlat_closure (M := M) _) rfl |
| refine ⟨Z, hZ.trans sdiff_subset, ?_, ?_⟩ |
| · simp only [natRank_closure] at hcard |
| have hrank : natRank M (M.closure A ∪ S) = natRank M (A ∪ S) := by |
| simp only [natRank, M.eRk_union_closure_left_eq] |
| rw [hrank] at hcard |
| have hr := conditionalRank_add_base M S A |
| rw [union_comm S A] at hr |
| omega |
| · rw [M.closure_union_closure_left_eq, M.closure_union_closure_left_eq] at hgen |
| rw [hgen] |
| exact subset_union_right.trans (M.subset_closure _ (by simp [hE])) |
|
|
| lemma collect_disjoint_witnesses (W : B → Set α) (U : Finset α) |
| (hW : ∀ b, W b ⊆ (U : Set α)) |
| (hdisj : Pairwise (fun b c => Disjoint (W b) (W c))) : |
| ∃ Z : Finset α, Z ⊆ U ∧ Z.card ≤ ∑ b, (W b).ncard ∧ |
| ∃ β : Z → B, ∀ b, (fun w : Z => w.val) '' {w | β w = b} = W b := by |
| classical |
| let Z := (⋃ b, W b).toFinset |
| have hmem (w : Z) : ∃ b, w.val ∈ W b := by |
| exact mem_iUnion.mp (Set.mem_toFinset.mp w.property) |
| let β : Z → B := fun w => (hmem w).choose |
| have hβ (w : Z) : w.val ∈ W (β w) := (hmem w).choose_spec |
| refine ⟨Z, ?_, ?_, β, ?_⟩ |
| · intro e he |
| obtain ⟨b, hb⟩ := mem_iUnion.mp (Set.mem_toFinset.mp he) |
| exact hW b hb |
| · rw [show Z.card = (⋃ b, W b).ncard from (Set.ncard_eq_toFinset_card' _).symm] |
| exact Set.ncard_iUnion_le_of_fintype W |
| · intro b |
| ext e |
| constructor |
| · rintro ⟨w, hw, rfl⟩ |
| rw [← hw] |
| exact hβ w |
| · intro he |
| let w : Z := ⟨e, Set.mem_toFinset.mpr (mem_iUnion.mpr ⟨b, he⟩)⟩ |
| have heq : β w = b := by |
| by_contra hne |
| exact Set.disjoint_left.mp (hdisj hne) (hβ w) he |
| exact ⟨w, heq, rfl⟩ |
|
|
| lemma residualUnion_subset_mask |
| {α : Type u_1} {B : Type u_2} [Fintype α] [Fintype B] |
| (M : Matroid α) (P A W : B → Set α) (O : Set α) |
| (hspan : ∀ b, P b \ O ⊆ M.closure (A b ∪ W b)) : |
| (⋃ b, P b \ M.closure (A b ∪ W b)) ⊆ O := by |
| intro e he |
| obtain ⟨b, hb⟩ := mem_iUnion.mp he |
| by_contra hn |
| exact hb.2 (hspan b ⟨hb.1, hn⟩) |
|
|
| lemma residualUnion_ncard (M : Matroid α) (P A W : B → Set α) |
| (hdisj : Pairwise (fun b c => Disjoint (P b) (P c))) : |
| (⋃ b, P b \ M.closure (A b ∪ W b)).ncard = |
| ∑ b, (P b \ M.closure (A b ∪ W b)).ncard := by |
| have hd : Pairwise (fun b c => Disjoint (P b \ M.closure (A b ∪ W b)) |
| (P c \ M.closure (A c ∪ W c))) := by |
| intro b c hbc |
| exact (hdisj hbc).mono sdiff_subset sdiff_subset |
| simpa only [finsum_eq_sum_of_fintype] using Set.ncard_iUnion_of_finite (fun b => Set.toFinite _) hd |
|
|
| lemma nominal_rank_witnesses (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (h : ℕ) (U : Finset α) (O : Set α) (ε : Fin 2) : |
| ∃ W : ParityWindow (activation h) ε → Set α, |
| (∀ b, W b ⊆ nominalLayerSet M hE κ D C h (U : Set α) ε b \ O) ∧ |
| (∑ b, (W b).ncard) = nominalRankStatistic M hE κ D C T h (U : Set α) O ε ∧ |
| (∀ b, nominalLayerSet M hE κ D C h (U : Set α) ε b \ O ⊆ |
| 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)) ∧ |
| ∃ Z : Finset α, Z ⊆ U ∧ Z.card ≤ nominalRankStatistic M hE κ D C T h (U : Set α) O ε ∧ |
| ∃ β : Z → ParityWindow (activation h) ε, |
| ∀ b, (fun w : Z => w.val) '' {w | β w = b} = W b := by |
| classical |
| choose W hW hc hs using fun b : ParityWindow (activation h) ε => exists_rankWitness M hE |
| (nominalPath M hE κ D C h (b.val.val - 2) ∪ lowerCompetition M hE κ D C T b.val.val h) |
| (nominalLayerSet M hE κ D C h (U : Set α) ε b \ O) |
| have heq : (∑ b, (W b).ncard) = nominalRankStatistic M hE κ D C T h (U : Set α) O ε := by |
| unfold nominalRankStatistic |
| exact sum_congr rfl (fun b _ => hc b) |
| have hd : Pairwise (fun b c => Disjoint (W b) (W c)) := by |
| intro b c hbc |
| exact (nominalLayerSet_disjoint M hE κ D C h (U : Set α) ε hbc).mono |
| ((hW b).trans sdiff_subset) ((hW c).trans sdiff_subset) |
| obtain ⟨Z, hZU, hZcard, β, hβ⟩ := collect_disjoint_witnesses W U |
| (fun b => ((hW b).trans sdiff_subset).trans (nominalLayerSet_subset M hE κ D C h _ ε b)) hd |
| refine ⟨W, hW, heq, hs, Z, hZU, ?_, β, hβ⟩ |
| rwa [heq] at hZcard |
|
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| end MatroidProphet |
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| end OAI |
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