| import OAI.Probability.MatroidProphet.Residual.Generators |
| import OAI.Probability.MatroidProphet.RankAccounting |
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|
| namespace OAI |
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|
| namespace MatroidProphet |
| open Set Finset |
| variable {α : Type*} [Fintype α] |
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| def commonPathGenerators (C J : ℕ → Set α) (h : ℕ) : ℕ → Set α |
| | 0 => C h |
| | t+1 => commonPathGenerators C J h t ∪ C (h+1+t) ∪ J (h+1+t) |
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|
| omit [Fintype α] in |
| lemma commonPathGenerators_cost (κ : ℕ) (C D J : ℕ → Set α) (h t : ℕ) |
| (hJ : ∀ j, κ * (J j).ncard ≤ (D j).ncard) : |
| κ * (commonPathGenerators C J h t).ncard ≤ κ * (C h).ncard + |
| ∑ j ∈ range t, (κ * (C (h+1+j)).ncard + (D (h+1+j)).ncard) := by |
| induction t with |
| | zero => simp [commonPathGenerators] |
| | succ t ih => |
| have hcard := ncard_union_le (commonPathGenerators C J h t ∪ C (h+1+t)) (J (h+1+t)) |
| have hcard' := ncard_union_le (commonPathGenerators C J h t) (C (h+1+t)) |
| have hc : (commonPathGenerators C J h (t+1)).ncard ≤ |
| (commonPathGenerators C J h t).ncard + (C (h+1+t)).ncard + (J (h+1+t)).ncard := by |
| exact hcard.trans (Nat.add_le_add_right hcard' _) |
| have hm := Nat.mul_le_mul_left κ hc |
| simp only [Nat.mul_add] at hm |
| rw [sum_range_succ] |
| have hj := hJ (h+1+t) |
| omega |
|
|
| lemma commonPathGenerators_spans (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C J : ℕ → Set α) (h t : ℕ) |
| (hJ : ∀ j k, nominalPath M hE κ D C j k ⊆ |
| M.closure (guardedPath M hE κ D C j k ∪ J j)) (k : ℤ) : |
| guardedPath M hE κ D C (h+1+t) k ⊆ |
| M.closure (nominalPath M hE κ D C h k ∪ commonPathGenerators C J h t) := by |
| induction t with |
| | zero => |
| simp only [Nat.add_zero, commonPathGenerators, guardedPath_succ] |
| exact M.closure_subset_closure (union_subset_union_right _ inter_subset_left) |
| | succ t ih => |
| have hidx : h+1+(t+1) = (h+1+t)+1 := by omega |
| rw [hidx, guardedPath_succ] |
| apply M.closure_subset_closure_of_subset_closure |
| apply Set.union_subset |
| · apply (hJ (h+1+t) k).trans |
| apply M.closure_subset_closure_of_subset_closure |
| apply Set.union_subset |
| · exact ih.trans (M.closure_subset_closure (by |
| intro e he |
| rcases he with he | he |
| · exact Or.inl he |
| · exact Or.inr (Or.inl (Or.inl he)))) |
| · intro e he |
| exact M.mem_closure_of_mem (Or.inr (Or.inr he)) (by simp [hE]) |
| · intro e he |
| exact M.mem_closure_of_mem (Or.inr (Or.inl (Or.inr he.1))) (by simp [hE]) |
|
|
| theorem exists_commonPathBudget (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (hκ : 0 < κ) (D C : ℕ → Set α) (h t : ℕ) : |
| ∃ Jtotal : Set α, |
| κ * Jtotal.ncard ≤ κ * (C h).ncard + |
| ∑ j ∈ range t, (κ * (C (h+1+j)).ncard + (D (h+1+j)).ncard) ∧ |
| ∀ k, guardedPath M hE κ D C (h+1+t) k ⊆ |
| M.closure (nominalPath M hE κ D C h k ∪ Jtotal) := by |
| choose J hJD hcard τ htime hgen using |
| (fun j => nominalPath_all_time_generators M hE κ hκ D C j) |
| refine ⟨commonPathGenerators C J h t, commonPathGenerators_cost κ C D J h t hcard, ?_⟩ |
| intro k |
| apply commonPathGenerators_spans M hE κ D C J h t _ k |
| intro j l |
| rw [hgen j l] |
| apply M.closure_subset_closure |
| exact union_subset_union_right _ (fun _ he => he.1) |
|
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| end MatroidProphet |
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| end OAI |
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