| import OAI.Probability.MatroidProphet.Pivots.Interleavings |
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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 : Type*} [Fintype α] |
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| lemma oldPrefix_subset_before |
| {α : Type u_1} [Fintype α] {r s : ℕ} (b : ℕ → α) (a : Fin s → α) |
| (time : Occurrence r s → ℕ) (horder : OldOrdered time) (i : Fin r) : |
| oldPrefix b i.val ⊆ occurrenceLabel b a '' {p | time p < time (Sum.inl i)} := by |
| rintro x ⟨k, hk, rfl⟩ |
| let j : Fin r := ⟨k, lt_trans hk i.isLt⟩ |
| exact ⟨Sum.inl j, horder j i hk, rfl⟩ |
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| lemma retained_old_initially_nonredundant {r s : ℕ} (M : Matroid α) (b : ℕ → α) |
| (a : Fin s → α) (time : Occurrence r s → ℕ) (horder : OldOrdered time) |
| (i : Fin r) (hi : Sum.inl i ∈ retained M b a time) : |
| b i.val ∉ M.closure (oldPrefix b i.val) := by |
| classical |
| intro hdep |
| exact (Finset.mem_filter.mp hi).2 |
| (M.closure_subset_closure (oldPrefix_subset_before b a time horder i) hdep) |
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| lemma card_retainedFamily_arbitrary_old {r s : ℕ} (M : Matroid α) (b : ℕ → α) |
| (a : Fin s → α) (he : ∀ k < r, b k ∈ M.E) : |
| (retainedFamily (r := r) M b a).card ≤ 4 ^ s := by |
| classical |
| let old : Finset (Occurrence r s) := |
| (Finset.univ.filter (fun i : Fin r => b i.val ∉ M.closure (oldPrefix b i.val))).image Sum.inl |
| let movable : Finset (Occurrence r s) := Finset.univ.image Sum.inr |
| let omissions : Finset (Occurrence r s) := old.filter (fun o => |
| match o with |
| | Sum.inl i => i.val ∈ vulnerable M b (Set.range a) r |
| | Sum.inr _ => False) |
| have hmov : movable.card ≤ s := by |
| simp [movable, Finset.card_image_of_injective, Sum.inr_injective] |
| have homit : omissions.card ≤ (vulnerable M b (Set.range a) r).card := by |
| apply Finset.card_le_card_of_injOn |
| (fun o : Occurrence r s => Sum.elim Fin.val (fun _ => 0) o) |
| · intro o ho |
| have ho' := (Finset.mem_filter.mp ho).2 |
| cases o with |
| | inl i => exact ho' |
| | inr j => exact False.elim ho' |
| · intro o ho p hp hop |
| have ho' := (Finset.mem_filter.mp ho).2 |
| have hp' := (Finset.mem_filter.mp hp).2 |
| cases o with |
| | inr i => exact False.elim ho' |
| | inl i => |
| cases p with |
| | inr j => exact False.elim hp' |
| | inl j => exact congrArg Sum.inl (Fin.ext hop) |
| have hrank : natRank M (Set.range a) ≤ (Set.range a).ncard := by |
| exact ENat.toNat_le_toNat (M.eRk_le_encard _) |
| (Set.toFinite (Set.range a)).encard_lt_top.ne |
| have hrange : (Set.range a).ncard ≤ s := by |
| simpa using (Set.ncard_image_le (f := a) (s := Set.univ)) |
| apply card_retained_family (retainedFamily M b a) old movable omissions s hmov |
| (homit.trans ((card_vulnerable_le_rank M b (Set.range a) r he).trans |
| (hrank.trans hrange))) |
| · intro J hJ o ho |
| obtain ⟨time, horder, rfl⟩ := (Finset.mem_filter.mp hJ).2 |
| have hoo := (Finset.mem_sdiff.mp ho).1 |
| obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hoo |
| have hind := (Finset.mem_filter.mp hi).2 |
| apply retains_nonomittable_old M b a time horder i |
| intro hom |
| have hom' := (Finset.mem_filter.mp hom).2 |
| have hv : i.val ∈ vulnerable M b (Set.range a) r := |
| Finset.mem_filter.mpr ⟨Finset.mem_range.mpr i.isLt, hind, hom'⟩ |
| exact (Finset.mem_sdiff.mp ho).2 (Finset.mem_filter.mpr ⟨hoo, hv⟩) |
| · intro J hJ o ho |
| obtain ⟨time, horder, rfl⟩ := (Finset.mem_filter.mp hJ).2 |
| cases o with |
| | inl i => |
| apply Finset.mem_union_left |
| exact Finset.mem_image.mpr ⟨i, Finset.mem_filter.mpr ⟨Finset.mem_univ _, |
| retained_old_initially_nonredundant M b a time horder i ho⟩, rfl⟩ |
| | inr j => exact Finset.mem_union_right _ (Finset.mem_image.mpr ⟨j, Finset.mem_univ _, rfl⟩) |
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| end Pivots |
| end MatroidProphet |
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| end OAI |
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