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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import Mathlib.Combinatorics.Matroid.Closure
import Mathlib.Data.Finset.Max
namespace OAI
namespace MatroidProphet
namespace Pivots
open Set Finset
variable {α β : Type*} [Fintype β] [DecidableEq β]
noncomputable def keptOccurrences (M : Matroid α) (label : β → α) (time : β → ℕ) :
Finset β := by
classical
exact Finset.univ.filter (fun o => label o ∉ M.closure (label '' {p | time p < time o}))
lemma mem_closure_kept_upto (M : Matroid α) (label : β → α) (time : β → ℕ)
(hground : ∀ o, label o ∈ M.E) (o : β) :
label o ∈ M.closure (label '' {p | p ∈ keptOccurrences M label time ∧ time p ≤ time o}) := by
classical
have aux : ∀ k, ∀ o, time o = k →
label o ∈ M.closure (label '' {p | p ∈ keptOccurrences M label time ∧ time p ≤ k}) := by
intro k
induction k using Nat.strong_induction_on with
| h k ih =>
intro o ho
by_cases hkeep : o ∈ keptOccurrences M label time
· exact M.mem_closure_of_mem ⟨o, ⟨hkeep, ho.le⟩, rfl⟩
(by rintro _ ⟨p, _, rfl⟩; exact hground p)
have hdep : label o ∈ M.closure (label '' {p | time p < time o}) := by
simpa [keptOccurrences] using hkeep
apply M.closure_subset_closure_of_subset_closure ?_ hdep
rintro _ ⟨p, hp, rfl⟩
have hpk : time p < k := by simpa [ho] using hp
have hih := ih (time p) hpk p rfl
apply M.closure_subset_closure ?_ hih
rintro _ ⟨q, ⟨hq, hqt⟩, rfl⟩
exact ⟨q, ⟨hq, hqt.trans hpk.le⟩, rfl⟩
exact aux (time o) o rfl
lemma greedy_prefix_closure (M : Matroid α) (label : β → α) (time : β → ℕ)
(hground : ∀ o, label o ∈ M.E) (k : ℕ) :
M.closure (label '' {o | time o < k}) =
M.closure (label '' {o | o ∈ keptOccurrences M label time ∧ time o < k}) := by
classical
apply Set.Subset.antisymm
· apply M.closure_subset_closure_of_subset_closure
rintro _ ⟨o, ho, rfl⟩
apply M.closure_subset_closure ?_ (mem_closure_kept_upto M label time hground o)
rintro _ ⟨p, ⟨hp, hpt⟩, rfl⟩
exact ⟨p, ⟨hp, lt_of_le_of_lt hpt ho⟩, rfl⟩
· apply M.closure_subset_closure
rintro _ ⟨o, ⟨_, ho⟩, rfl⟩
exact ⟨o, ho, rfl⟩
private lemma indep_image_of_notMem_prior
{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) (s : Finset β)
(hground : ∀ o ∈ s, label o ∈ M.E)
(hprior : ∀ o ∈ s, label o ∉ M.closure (label '' {p | p ∈ s ∧ time p < time o})) :
M.Indep (label '' (s : Set β)) := by
classical
induction s using Finset.strongInductionOn with
| _ s ih =>
by_cases hs : s = ∅
· simp [hs]
obtain ⟨p, hp, hmax⟩ := Finset.exists_max_image s time (Finset.nonempty_iff_ne_empty.mpr hs)
have hI := ih (s.erase p) (Finset.erase_ssubset hp)
(fun o ho => hground o (Finset.mem_of_mem_erase ho)) (by
intro o ho hmem
apply hprior o (Finset.mem_of_mem_erase ho)
apply M.closure_subset_closure ?_ hmem
rintro _ ⟨q, ⟨hq, hqt⟩, rfl⟩
exact ⟨q, ⟨Finset.mem_of_mem_erase hq, hqt⟩, rfl⟩)
have hnot : label p ∉ M.closure (label '' (s.erase p : Set β)) := by
intro hmem
apply hprior p hp
apply M.closure_subset_closure ?_ hmem
rintro _ ⟨q, hq, rfl⟩
refine ⟨q, ⟨Finset.mem_of_mem_erase hq, ?_⟩, rfl⟩
apply lt_of_le_of_ne (hmax q (Finset.mem_of_mem_erase hq))
intro htimeeq
exact (Finset.ne_of_mem_erase hq) (htime htimeeq)
have hins := hI.insert_indep_iff.mpr (Or.inl ⟨hground p hp, hnot⟩)
have hset : insert (label p) (label '' (s.erase p : Set β)) = label '' (s : Set β) := by
rw [← Set.image_insert_eq, ← Finset.coe_insert, Finset.insert_erase hp]
rwa [hset] at hins
lemma keptOccurrences_indep (M : Matroid α) (label : β → α) (time : β → ℕ)
(hground : ∀ o, label o ∈ M.E) (htime : Function.Injective time) :
M.Indep (label '' (keptOccurrences M label time : Set β)) := by
classical
apply indep_image_of_notMem_prior M label time htime
· intro o _
exact hground o
· intro o ho hmem
have hn := (Finset.mem_filter.mp ho).2
apply hn
apply M.closure_subset_closure ?_ hmem
rintro _ ⟨p, ⟨_, hpt⟩, rfl⟩
exact ⟨p, hpt, rfl⟩
end Pivots
end MatroidProphet
end OAI
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