| import OAI.Probability.MatroidProphet.Pivots.OccurrenceSupport |
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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 exists_pivot |
| {α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β] |
| (M : Matroid α) (label : β → α) (time : β → ℕ) {e : α} |
| (hnonloop : e ∉ M.closure ∅) (hspan : e ∈ M.closure (Set.range label)) : |
| ∃ o, IsPivot M label time e o := by |
| classical |
| let P := fun k => e ∈ M.closure (label '' {o | time o ≤ k}) |
| have hex : ∃ k, P k := by |
| refine ⟨Finset.univ.sup time, M.closure_subset_closure ?_ hspan⟩ |
| rintro _ ⟨o, rfl⟩ |
| exact ⟨o, by |
| change time o ≤ Finset.univ.sup time |
| exact Finset.le_sup (Finset.mem_univ o), rfl⟩ |
| let k := Nat.find hex |
| have hk : P k := Nat.find_spec hex |
| have hnot : e ∉ M.closure (label '' {o | time o < k}) := by |
| intro hn |
| by_cases hk0 : k = 0 |
| · exact hnonloop (by simpa [hk0] using hn) |
| have hprev : P (k - 1) := by |
| apply M.closure_subset_closure ?_ hn |
| rintro _ ⟨o, ho, rfl⟩ |
| exact ⟨o, by |
| change time o ≤ k - 1 |
| change time o < k at ho |
| omega, rfl⟩ |
| have hmin : k ≤ k - 1 := Nat.find_min' hex hprev |
| omega |
| have hat : ∃ o, time o = k := by |
| by_contra h |
| apply hnot |
| apply M.closure_subset_closure ?_ hk |
| rintro _ ⟨o, ho, rfl⟩ |
| refine ⟨o, ?_, rfl⟩ |
| have hne : time o ≠ k := fun heq => h ⟨o, heq⟩ |
| change time o < k |
| change time o ≤ k at ho |
| omega |
| obtain ⟨o, ho⟩ := hat |
| exact ⟨o, by simpa [IsPivot, ho] using And.intro hk hnot⟩ |
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| lemma pivot_unique |
| {α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β] |
| (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (htime : Function.Injective time) {e : α} {o p : β} |
| (ho : IsPivot M label time e o) (hp : IsPivot M label time e p) : o = p := by |
| apply htime |
| apply le_antisymm |
| · by_contra h |
| apply ho.2 |
| apply M.closure_subset_closure ?_ hp.1 |
| rintro _ ⟨a, ha, rfl⟩ |
| exact ⟨a, lt_of_le_of_lt ha (Nat.lt_of_not_ge h), rfl⟩ |
| · by_contra h |
| apply hp.2 |
| apply M.closure_subset_closure ?_ ho.1 |
| rintro _ ⟨a, ha, rfl⟩ |
| exact ⟨a, lt_of_le_of_lt ha (Nat.lt_of_not_ge h), rfl⟩ |
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| lemma existsUnique_pivot (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (htime : Function.Injective time) {e : α} |
| (hnonloop : e ∉ M.closure ∅) (hspan : e ∈ M.closure (Set.range label)) : |
| ∃! o, IsPivot M label time e o := by |
| obtain ⟨o, ho⟩ := exists_pivot M label time hnonloop hspan |
| exact ⟨o, ho, fun p hp => pivot_unique M label time htime hp ho⟩ |
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| end Pivots |
| end MatroidProphet |
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| end OAI |
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