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a2ee19a | 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.Main
namespace OAI
/-! Exact agreement between the source's previous-accepted-label test and the
concrete implementation's previous-eligible-label test. No arrival-order
assumption beyond the explicit arrival times is used in span preservation. -/
namespace MatroidProphet
open Set Finset
variable {α : Type*} [DecidableEq α]
/-- Rejected earlier eligible labels do not change the span of a layer prefix. -/
theorem layerGreedy_closure_prior (M : Matroid α) (E : Finset α)
(base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ)
(hE : M.E = Set.univ) (b : ℤ) (k : ℕ) :
M.closure (base b ∪ layerPrior E layer time b k) =
M.closure (base b ∪
layerPrior (layerGreedy M E base layer time) layer time b k) := by
apply Set.Subset.antisymm
· apply M.closure_subset_closure_of_subset_closure
rintro e (he | ⟨he, hb, ht⟩)
· exact M.mem_closure_of_mem' (Or.inl he) (by simp [hE])
· have hs := layerGreedy_spans_prefix M E base layer time hE e he
apply M.closure_subset_closure ?_ hs
rintro f (hf | ⟨hf, hfb, hft⟩)
· exact Or.inl (by simpa [hb] using hf)
· exact Or.inr ⟨hf, hfb.trans hb, hft.trans_lt ht⟩
· apply M.closure_subset_closure
rintro e (he | ⟨he, hb, ht⟩)
· exact Or.inl he
· exact Or.inr ⟨layerGreedy_subset M E base layer time he, hb, ht⟩
/-- The online decision can be implemented using only earlier accepted labels
of the same layer, exactly as in step 3 of the manuscript algorithm. -/
theorem layerGreedy_mem_iff_previous_accepted (M : Matroid α) (E : Finset α)
(base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ)
(hE : M.E = Set.univ) (e : α) :
e ∈ layerGreedy M E base layer time ↔
e ∈ E ∧ e ∉ M.closure (base (layer e) ∪
layerPrior (layerGreedy M E base layer time) layer time (layer e) (time e)) := by
classical
conv_lhs => unfold layerGreedy
rw [Finset.mem_filter]
rw [layerGreedy_closure_prior M E base layer time hE]
omit [DecidableEq α] in
/-- On a finite ground set the source's rank-one test is precisely nonmembership
in the closure, including rank-zero and loop cases. -/
theorem conditionalRank_singleton_eq_one_iff [Fintype α] (M : Matroid α)
(hE : M.E = Set.univ) (P : Set α) (e : α) :
conditionalRank M {e} P = 1 ↔ e ∉ M.closure P := by
by_cases he : e ∈ M.closure P
· rw [Pivots.conditionalRank_singleton_of_mem_closure M P he]
simp [he]
· rw [Pivots.conditionalRank_singleton_of_notMem_closure M P (by simp [hE]) he]
simp [he]
/-- Exact rank form of the source's accept/reject condition. -/
theorem layerGreedy_mem_iff_rank_previous_accepted [Fintype α]
(M : Matroid α) (E : Finset α) (base : ℤ → Set α)
(layer : α → ℤ) (time : α → ℕ) (hE : M.E = Set.univ) (e : α) :
e ∈ layerGreedy M E base layer time ↔
e ∈ E ∧ conditionalRank M {e} (base (layer e) ∪
layerPrior (layerGreedy M E base layer time) layer time (layer e) (time e)) = 1 := by
rw [layerGreedy_mem_iff_previous_accepted M E base layer time hE,
conditionalRank_singleton_eq_one_iff M hE]
/-- The concrete history-based online decision agrees with the rank test on
previously accepted labels in the current layer. -/
theorem layerKeyDecision_eq_rank_previous_accepted {n : ℕ} {K : Type*}
(M : Matroid (Fin n)) (hE : M.E = Set.univ) (base : ℤ → Set (Fin n))
(assign : Fin n → K → Option ℤ) (v : Fin n → K)
(π : ArrivalOrder n) (k : Fin n) :
layerKeyDecision M base assign k (keyHistory v π k) = true ↔
π k ∈ assignedEligible assign v ∧
conditionalRank M {π k} (base (assignedLayer assign v (π k)) ∪
layerPrior
(layerGreedy M (assignedEligible assign v) base (assignedLayer assign v)
(fun e => (π.symm e).val))
(assignedLayer assign v) (fun e => (π.symm e).val)
(assignedLayer assign v (π k)) k.val) = 1 := by
rw [layerKeyDecision_eq_greedy,
layerGreedy_mem_iff_rank_previous_accepted M _ _ _ _ hE]
simp only [Equiv.symm_apply_apply]
end MatroidProphet
end OAI
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