| import OAI.Probability.MatroidProphet.Pivots.Basic |
| import OAI.Probability.MatroidProphet.Pivots.GreedyScan |
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| namespace OAI |
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| namespace MatroidProphet |
| namespace Pivots |
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| open Set Finset |
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| variable {α β : Type*} [Fintype β] [DecidableEq β] |
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| lemma kept_label_injOn |
| {α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β] |
| (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (hground : ∀ o, label o ∈ M.E) (htime : Function.Injective time) : |
| Set.InjOn label (keptOccurrences M label time : Set β) := by |
| classical |
| have hd : ∀ x y, x ∈ keptOccurrences M label time → y ∈ keptOccurrences M label time → |
| time x < time y → label x ≠ label y := by |
| intro x y _ hy hxy hlabels |
| apply (Finset.mem_filter.mp hy).2 |
| rw [← hlabels] |
| exact M.mem_closure_of_mem ⟨x, hxy, rfl⟩ (by rintro _ ⟨z, _, rfl⟩; exact hground z) |
| intro x hx y hy hlabels |
| apply htime |
| rcases lt_trichotomy (time x) (time y) with hlt | heq | hgt |
| · exact False.elim (hd x y hx hy hlt hlabels) |
| · exact heq |
| · exact False.elim (hd y x hy hx hgt hlabels.symm) |
|
|
| noncomputable def occurrenceSupport (M : Matroid α) (label : β → α) |
| (J : Finset β) (e : α) : Finset β := by |
| classical |
| exact J.filter (fun o => label o ∈ support M (label '' (J : Set β)) e) |
|
|
| lemma mem_occurrenceSupport |
| {α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β] |
| {M : Matroid α} {label : β → α} {J : Finset β} {e : α} {o : β} : |
| o ∈ occurrenceSupport M label J e ↔ |
| o ∈ J ∧ label o ∈ support M (label '' (J : Set β)) e := by |
| classical |
| simp [occurrenceSupport] |
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| lemma mem_closure_iff_occurrenceSupport (M : Matroid α) (label : β → α) |
| (J : Finset β) (e : α) (hI : M.Indep (label '' (J : Set β))) |
| (he : e ∈ M.closure (label '' (J : Set β))) |
| (hinj : Set.InjOn label (J : Set β)) (P : β → Prop) : |
| e ∈ M.closure (label '' {o | o ∈ J ∧ P o}) ↔ |
| ∀ o ∈ occurrenceSupport M label J e, P o := by |
| classical |
| rw [mem_closure_iff_support_subset hI he (by |
| rintro _ ⟨o, ⟨ho, _⟩, rfl⟩; exact ⟨o, ho, rfl⟩)] |
| constructor |
| · intro h o ho |
| obtain ⟨hoJ, hol⟩ := mem_occurrenceSupport.mp ho |
| obtain ⟨p, ⟨hpJ, hpP⟩, hp⟩ := h hol |
| exact (hinj hpJ hoJ hp) ▸ hpP |
| · intro h x hx |
| obtain ⟨o, ho, rfl⟩ := support_subset he hx |
| exact ⟨o, ⟨ho, h o (mem_occurrenceSupport.mpr ⟨ho, hx⟩)⟩, rfl⟩ |
|
|
| def IsPivot (M : Matroid α) (label : β → α) (time : β → ℕ) (e : α) (o : β) : Prop := |
| e ∈ M.closure (label '' {p | time p ≤ time o}) ∧ |
| e ∉ M.closure (label '' {p | time p < time o}) |
|
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| lemma pivot_mem_support_and_max (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (hground : ∀ o, label o ∈ M.E) (htime : Function.Injective time) |
| {e : α} {o : β} (hpivot : IsPivot M label time e o) : |
| o ∈ occurrenceSupport M label (keptOccurrences M label time) e ∧ |
| ∀ p ∈ occurrenceSupport M label (keptOccurrences M label time) e, time p ≤ time o := by |
| classical |
| let J := keptOccurrences M label time |
| have hI := keptOccurrences_indep M label time hground htime |
| have hinj := kept_label_injOn M label time hground htime |
| have hup : e ∈ M.closure (label '' {p | p ∈ J ∧ time p ≤ time o}) := by |
| have h := hpivot.1 |
| have hcl := greedy_prefix_closure M label time hground (time o + 1) |
| simp only [Nat.lt_succ_iff] at hcl |
| rw [hcl] at h |
| exact h |
| have he : e ∈ M.closure (label '' (J : Set β)) := by |
| apply M.closure_subset_closure ?_ hup |
| rintro _ ⟨p, ⟨hp, _⟩, rfl⟩ |
| exact ⟨p, hp, rfl⟩ |
| have hmax := (mem_closure_iff_occurrenceSupport M label J e hI he hinj |
| (fun p => time p ≤ time o)).mp hup |
| have hnot : ¬ ∀ p ∈ occurrenceSupport M label J e, time p < time o := by |
| intro h |
| apply hpivot.2 |
| rw [greedy_prefix_closure M label time hground (time o)] |
| exact (mem_closure_iff_occurrenceSupport M label J e hI he hinj |
| (fun p => time p < time o)).mpr h |
| push Not at hnot |
| obtain ⟨p, hp, hge⟩ := hnot |
| have heq : p = o := htime (Nat.le_antisymm (hmax p hp) hge) |
| exact ⟨heq ▸ hp, hmax⟩ |
|
|
| end Pivots |
| end MatroidProphet |
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| end OAI |
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