| import OAI.Probability.MatroidProphet.Pivots.Blocks | |
| namespace OAI | |
| namespace MatroidProphet.Pivots | |
| open Set Finset | |
| variable {α A T : Type*} [Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T] {n : ℕ} | |
| def blockLabel (F : T → Set α) (a : A → α) : BlockOld F ⊕ A → α := | |
| Sum.elim (blockOldLabel (F := F)) a | |
| def blockPrefix (F : T → Set α) (a : A → α) (τ : A → T) (k : T) : Set α := | |
| blockLabel F a '' {o | blockStage F τ o ≤ k} | |
| def blockBefore (F : T → Set α) (a : A → α) (τ : A → T) (k : T) : Set α := | |
| blockLabel F a '' {o | blockStage F τ o < k} | |
| noncomputable def blockBirthPattern (M : Matroid α) (F : T → Set α) | |
| (a : A → α) (τ : A → T) (test : Fin n → α) (mark : T → Bool) : Finset (Fin n) := by | |
| classical | |
| exact Finset.univ.filter fun f => ∃ k, mark k = true ∧ | |
| test f ∈ M.closure (blockPrefix F a τ k) ∧ test f ∉ M.closure (blockBefore F a τ k) | |
| lemma pivot_block_birth (M : Matroid α) (F : T → Set α) (a : A → α) | |
| (τ : A → T) {e : α} {o : BlockOld F ⊕ A} | |
| (hp : IsPivot M (blockLabel F a) (blockTime F τ) e o) : | |
| e ∈ M.closure (blockPrefix F a τ (blockStage F τ o)) ∧ | |
| e ∉ M.closure (blockBefore F a τ (blockStage F τ o)) := by | |
| constructor | |
| · apply M.closure_subset_closure (Set.image_mono ?_) hp.1 | |
| exact fun p h => blockTime_le_implies_stage_le F τ h | |
| · intro h | |
| apply hp.2 | |
| apply M.closure_subset_closure (Set.image_mono ?_) h | |
| exact fun p h => stage_lt_implies_blockTime_lt F τ h | |
| lemma block_birth_unique | |
| {α : Type u_1} {A : Type u_2} {T : Type u_3} | |
| [Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T] | |
| (M : Matroid α) (F : T → Set α) (a : A → α) | |
| (τ : A → T) {e : α} {k l : T} | |
| (hk : e ∈ M.closure (blockPrefix F a τ k) ∧ e ∉ M.closure (blockBefore F a τ k)) | |
| (hl : e ∈ M.closure (blockPrefix F a τ l) ∧ e ∉ M.closure (blockBefore F a τ l)) : | |
| k = l := by | |
| rcases lt_trichotomy k l with h | rfl | h | |
| · exact (hl.2 (M.closure_subset_closure (Set.image_mono (fun _ hx => lt_of_le_of_lt hx h)) hk.1)).elim | |
| · rfl | |
| · exact (hk.2 (M.closure_subset_closure (Set.image_mono (fun _ hx => lt_of_le_of_lt hx h)) hl.1)).elim | |
| lemma blockBirthPattern_mem_freeMarkedPatterns (M : Matroid α) (F : T → Set α) | |
| (a : A → α) (τ : A → T) (test : Fin n → α) (mark : T → Bool) | |
| (hnonloop : ∀ f, test f ∉ M.closure ∅) | |
| (hspan : ∀ f, test f ∈ M.closure (Set.range (blockLabel F a))) : | |
| blockBirthPattern M F a τ test mark ∈ | |
| freeMarkedPatterns M (blockOldLabel (F := F)) a test (fun o : BlockOld F => mark (blockOldStage (F := F) o)) := by | |
| classical | |
| apply Finset.mem_filter.mpr | |
| refine ⟨Finset.mem_univ _, blockTime F τ, blockTime_old_strictMono F τ, | |
| blockTime_injective F τ, mark ∘ τ, ?_⟩ | |
| intro f | |
| obtain ⟨o, hp⟩ := exists_pivot M (blockLabel F a) (blockTime F τ) (hnonloop f) (hspan f) | |
| refine ⟨o, hp, ?_⟩ | |
| have hb := pivot_block_birth M F a τ hp | |
| change Sum.elim (fun o : BlockOld F => mark (blockOldStage (F := F) o)) (mark ∘ τ) o = true ↔ _ | |
| have hm : Sum.elim (fun o : BlockOld F => mark (blockOldStage (F := F) o)) (mark ∘ τ) o = | |
| mark (blockStage F τ o) := by cases o <;> rfl | |
| rw [hm] | |
| simp only [blockBirthPattern, Finset.mem_filter, Finset.mem_univ, true_and] | |
| constructor | |
| · intro h | |
| exact ⟨blockStage F τ o, h, hb⟩ | |
| · rintro ⟨k, hmark, hk⟩ | |
| have heq := block_birth_unique M F a τ hb hk | |
| simpa only [heq] using hmark | |
| lemma blockPrefix_eq | |
| {α : Type u_1} {A : Type u_2} {T : Type u_3} | |
| [Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T] | |
| (F : T → Set α) (hF : Monotone F) | |
| (a : A → α) (τ : A → T) (k : T) : | |
| blockPrefix F a τ k = F k ∪ a '' {g | τ g ≤ k} := by | |
| ext e | |
| constructor | |
| · rintro ⟨o, ho, rfl⟩ | |
| cases o with | |
| | inl o => exact Or.inl (hF ho o.property) | |
| | inr g => exact Or.inr ⟨g, ho, rfl⟩ | |
| · rintro (he | ⟨g, hg, rfl⟩) | |
| · exact ⟨Sum.inl ⟨toLex (k, e), he⟩, by change k ≤ k; exact le_rfl, rfl⟩ | |
| · exact ⟨Sum.inr g, hg, rfl⟩ | |
| lemma blockBefore_eq_iUnion | |
| {α : Type u_1} {A : Type u_2} {T : Type u_3} | |
| [Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T] | |
| (F : T → Set α) (a : A → α) (τ : A → T) | |
| (k : T) : | |
| blockBefore F a τ k = (⋃ j : T, ⋃ (_ : j < k), F j) ∪ a '' {g | τ g < k} := by | |
| ext e | |
| constructor | |
| · rintro ⟨o, ho, rfl⟩ | |
| cases o with | |
| | inl o => exact Or.inl (mem_iUnion.mpr ⟨blockOldStage (F := F) o, mem_iUnion.mpr ⟨ho, o.property⟩⟩) | |
| | inr g => exact Or.inr ⟨g, ho, rfl⟩ | |
| · rintro (he | ⟨g, hg, rfl⟩) | |
| · obtain ⟨j, hj, he⟩ := mem_iUnion.mp he |>.imp fun _ h => mem_iUnion.mp h | |
| exact ⟨Sum.inl ⟨toLex (j, e), he⟩, hj, rfl⟩ | |
| · exact ⟨Sum.inr g, hg, rfl⟩ | |
| end MatroidProphet.Pivots | |
| end OAI | |