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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 | import OAI.Probability.MatroidProphet.Residual.Generators
import OAI.Probability.MatroidProphet.RankAccounting
namespace OAI
namespace MatroidProphet
open Set Finset
variable {α : Type*} [Fintype α]
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)
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)
end MatroidProphet
end OAI
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