| import OAI.Probability.MatroidProphet.Main |
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| namespace OAI |
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| namespace MatroidProphet |
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| open Set |
| variable {α : Type*} [Fintype α] |
| attribute [local instance] Classical.propDecidable |
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| namespace DensityComparison |
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| noncomputable def unionExpansion (M : Matroid α) (_hE : M.E = univ) |
| (κ : ℕ) (D P : Set α) : Set α := |
| ⋃₀ {Q | IsDensityMax M κ D P Q} |
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| theorem unionExpansion_eq (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D P : Set α) : |
| unionExpansion M hE κ D P = densityExpansion M hE κ D P := by |
| apply Set.Subset.antisymm |
| · intro x hx |
| change x ∈ ⋃₀ {Q | IsDensityMax M κ D P Q} at hx |
| obtain ⟨Q, hQ, hxQ⟩ := Set.mem_sUnion.mp hx |
| exact (densityExpansion_spec M hE κ D P).2 Q hQ hxQ |
| · intro x hx |
| change x ∈ ⋃₀ {Q | IsDensityMax M κ D P Q} |
| exact Set.mem_sUnion.mpr |
| ⟨densityExpansion M hE κ D P, (densityExpansion_spec M hE κ D P).1, hx⟩ |
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| noncomputable def guardedBirth (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) : ℤ := |
| if hex : ∃ j : ℕ, e ∈ nominalPath M hE κ D C h (activation h + j) then |
| activation h + Nat.find hex |
| else activation h |
|
|
| theorem guardedBirth_eq (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) : |
| guardedBirth M hE κ D C h e = nominalBirth M hE κ D C h e := by |
| unfold guardedBirth |
| rw [dite_eq_left (exists_nominal_member M hE κ D C h e)] |
| rfl |
|
|
| end DensityComparison |
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| lemma densityExpansion_eq_chosen (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D P : Set α) : |
| densityExpansion M hE κ D P = (exists_largest_densityMax M hE κ D P).choose := by |
| rfl |
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| lemma nominalBirth_eq_find (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) : |
| nominalBirth M hE κ D C h e = |
| activation h + Nat.find (exists_nominal_member M hE κ D C h e) := by |
| rfl |
|
|
| end MatroidProphet |
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| end OAI |
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|
