File size: 3,797 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Probability.MatroidProphet.Pivots.Interleavings
namespace OAI
namespace MatroidProphet
namespace Pivots
open Set Finset
variable {α : Type*} [Fintype α]
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⟩
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)
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⟩)
end Pivots
end MatroidProphet
end OAI
|