| import OAI.Probability.MatroidProphet.Algorithm.Observed |
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| namespace OAI |
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| namespace MatroidProphet.MainAlgorithm |
| open Set Finset |
| variable {n : ℕ} |
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| noncomputable def selection (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) : Finset (Fin n) := |
| layerGreedy M (assignedEligible (assign M hE d w) w) |
| (fun b => finalPath M hE d w (b-2)) (assignedLayer (assign M hE d w) w) |
| (fun e => (π.symm e).val) |
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| lemma selection_not_observed (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) |
| (e : Fin n) (he : e ∈ selection M hE d w π) : e ∉ d.H ∪ d.D ∪ d.C := by |
| have hel := layerGreedy_subset M _ _ _ _ he |
| have ha := assigned_some_of_eligible (assign M hE d w) w e hel |
| obtain ⟨i, _, hei, _⟩ := (assign_spec M hE d w e _ _).mp ha |
| exact hei.1 |
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| theorem hiddenAccepted_eq_selection (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (r : Seed (mainSeedBits n)) (w : Weights n) (π : ArrivalOrder n) : |
| hiddenAcceptedThrough (hidden M hE) r w π n = |
| selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π := by |
| classical |
| have hA : accepted (core M hE) r (observed (hidden M hE) r w) w π = |
| selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π := by |
| unfold core |
| rw [layerOnlineRule_accepted] |
| obtain ⟨hf, _, ha⟩ := observed_preprocessing M hE r w |
| rw [hf, ha] |
| rfl |
| change accepted (core M hE) r (observed (hidden M hE) r w) w π \ (hidden M hE).mask r = _ |
| rw [hA] |
| ext e |
| simp only [Finset.mem_sdiff] |
| constructor |
| · exact fun he => he.1 |
| · intro he |
| exact ⟨he, selection_not_observed M hE (mainMasks r) _ π e he⟩ |
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| theorem hiddenReward_eq_selection (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (r : Seed (mainSeedBits n)) (w : Weights n) (π : ArrivalOrder n) : |
| hiddenReward (hidden M hE) r w π = |
| ∑ e ∈ selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π, w e := by |
| unfold hiddenReward |
| rw [hiddenAccepted_eq_selection] |
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| theorem rounded_selection_le_hiddenReward (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (r : Seed (mainSeedBits n)) (w : Weights n) (hw : ∀ e, 0 ≤ w e) (π : ArrivalOrder n) : |
| (∑ e ∈ selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π, |
| roundedWeight weightBase (w e)) ≤ hiddenReward (hidden M hE) r w π := by |
| rw [hiddenReward_eq_selection] |
| apply Finset.sum_le_sum |
| intro e he |
| exact roundedWeight_le (by norm_num [weightBase]) (hw e) |
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| end MatroidProphet.MainAlgorithm |
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| end OAI |
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