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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Probability.MatroidProphet.RankAccounting
import Mathlib.Data.Finset.Max
namespace OAI
namespace MatroidProphet
open Set Finset
variable {α : Type*} [DecidableEq α]
noncomputable def layerPrior (E : Finset α) (layer : α → ℤ) (time : α → ℕ)
(b : ℤ) (k : ℕ) : Set α := {e | e ∈ E ∧ layer e = b ∧ time e < k}
noncomputable def layerGreedy (M : Matroid α) (E : Finset α)
(base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ) : Finset α := by
classical
exact E.filter fun e => e ∉ M.closure (base (layer e) ∪ layerPrior E layer time (layer e) (time e))
omit [DecidableEq α] in
lemma layerGreedy_subset (M : Matroid α) (E : Finset α)
(base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ) :
layerGreedy M E base layer time ⊆ E := by
classical
exact filter_subset _ _
omit [DecidableEq α] in
lemma layerPrior_mono (E : Finset α) (layer : α → ℤ) (time : α → ℕ)
(b : ℤ) {k k' : ℕ} (h : k ≤ k') :
layerPrior E layer time b k ⊆ layerPrior E layer time b k' := by
rintro e ⟨he, hb, ht⟩
exact ⟨he, hb, lt_of_lt_of_le ht h⟩
lemma layerGreedy_spans_prefix (M : Matroid α) (E : Finset α)
(base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ)
(hE : M.E = Set.univ) (e : α) (he : e ∈ E) :
e ∈ M.closure (base (layer e) ∪
{f | f ∈ layerGreedy M E base layer time ∧ layer f = layer e ∧ time f ≤ time e}) := by
have aux : ∀ k, ∀ e ∈ E, time e = k →
e ∈ M.closure (base (layer e) ∪
{f | f ∈ layerGreedy M E base layer time ∧ layer f = layer e ∧ time f ≤ k}) := by
intro k
induction k using Nat.strong_induction_on with
| h k ih =>
intro e he htime
by_cases hkeep : e ∈ layerGreedy M E base layer time
· exact M.mem_closure_of_mem' (Or.inr ⟨hkeep, rfl, htime.le⟩) (by simp [hE])
have hdep : e ∈ M.closure (base (layer e) ∪ layerPrior E layer time (layer e) (time e)) := by
simpa only [layerGreedy, mem_filter, he, true_and, not_not] using hkeep
apply M.closure_subset_closure_of_subset_closure ?_ hdep
rintro f (hf | ⟨hf, hl, ht⟩)
· exact M.mem_closure_of_mem' (Or.inl hf) (by simp [hE])
· have hfk : time f < k := by simpa [htime] using ht
have hrec := ih (time f) hfk f hf rfl
apply M.closure_subset_closure ?_ hrec
rintro g (hg | ⟨hg, hgl, hgt⟩)
· exact Or.inl (by simpa [hl] using hg)
· exact Or.inr ⟨hg, hgl.trans hl, hgt.trans hfk.le⟩
exact aux (time e) e he rfl
theorem layerGreedy_indep (M : Matroid α) (E : Finset α)
(base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ)
(hE : M.E = Set.univ) (htime : Function.Injective time)
(hlower : ∀ e ∈ E, ∀ f ∈ E, layer e < layer f → e ∈ base (layer f)) :
M.Indep (layerGreedy M E base layer time : Set α) := by
classical
suffices aux : ∀ A : Finset α, A ⊆ layerGreedy M E base layer time →
M.Indep (A : Set α) from aux _ Finset.Subset.rfl
intro A
induction A using Finset.strongInductionOn with
| _ A ih =>
intro hA
by_cases hne : A.Nonempty
· obtain ⟨p₀, hp₀, hmaxLayer⟩ := exists_max_image A layer hne
have htne : (A.filter (fun e => layer e = layer p₀)).Nonempty :=
⟨p₀, mem_filter.mpr ⟨hp₀, rfl⟩⟩
obtain ⟨p, hpt, hmaxTime⟩ := exists_max_image
(A.filter (fun e => layer e = layer p₀)) time htne
have hp : p ∈ A := (mem_filter.mp hpt).1
have hpl : layer p = layer p₀ := (mem_filter.mp hpt).2
have hpE := layerGreedy_subset M E base layer time (hA hp)
have hI := ih (A.erase p) (erase_ssubset hp)
(fun e he => hA (mem_of_mem_erase he))
have hnot : p ∉ M.closure (A.erase p : Set α) := by
intro hmem
have hpnot := (mem_filter.mp (hA hp)).2
apply hpnot
apply M.closure_subset_closure ?_ hmem
intro q hq
have hqA := mem_of_mem_erase hq
have hqE := layerGreedy_subset M E base layer time (hA hqA)
have hqle : layer q ≤ layer p := by simpa [hpl] using hmaxLayer q hqA
rcases lt_or_eq_of_le hqle with hqlt | hqeq
· exact Or.inl (hlower q hqE p hpE hqlt)
· refine Or.inr ⟨hqE, hqeq, ?_⟩
have hqt : q ∈ A.filter (fun e => layer e = layer p₀) :=
mem_filter.mpr ⟨hqA, hqeq.trans hpl⟩
exact lt_of_le_of_ne (hmaxTime q hqt) (fun h => (ne_of_mem_erase hq) (htime h))
have hins := hI.insert_indep_iff.mpr (Or.inl ⟨by simp [hE], hnot⟩)
simpa only [← Finset.coe_insert, insert_erase hp] using hins
· have hzero := Finset.not_nonempty_iff_eq_empty.mp hne
simp [hzero]
lemma layerGreedy_spans (M : Matroid α) (E : Finset α)
(base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ)
(hE : M.E = Set.univ) (b : ℤ) :
{e | e ∈ E ∧ layer e = b} ⊆
M.closure (base b ∪ {e | e ∈ layerGreedy M E base layer time ∧ layer e = b}) := by
rintro e ⟨he, hb⟩
have h := layerGreedy_spans_prefix M E base layer time hE e he
apply M.closure_subset_closure ?_ h
rintro f (hf | ⟨hf, hfb, _⟩)
· exact Or.inl (by simpa [hb] using hf)
· exact Or.inr ⟨hf, hfb.trans hb⟩
end MatroidProphet
end OAI
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