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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 82 83 84 85 86 87 88 89 90 91 92 93 94 | import OAI.Probability.MatroidProphet.CommonBudget
import OAI.Probability.MatroidProphet.Reverse.Paths
import OAI.Probability.MatroidProphet.LayerRank
namespace OAI
namespace MatroidProphet
open Set Finset
variable {α : Type*} [Fintype α]
lemma lowerCompetition_subset_guarded (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C T : ℕ → Set α) (b : ℤ) (h : ℕ) :
lowerCompetition M hE κ D C T b h ⊆ guardedPath M hE κ D C h b := by
induction h with
| zero => exact Set.empty_subset _
| succ h ih =>
apply Set.union_subset
· exact ih.trans ((guarded_subset_nominal M hE κ D C h b).trans
(nominal_subset_guarded M hE κ D C h b))
· exact fun _ he => nominal_subset_guarded M hE κ D C h b he.2.1
noncomputable def unobservedBirthRegion (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C : ℕ → Set α) (G O : Set α) (h : ℕ) (b : ℤ) : Set α :=
if activation h + 2 ≤ b then
(G ∩ (nominalPath M hE κ D C h b \ nominalPath M hE κ D C h (b-1))) \ O
else ∅
lemma unobservedBirthRegion_subset (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C : ℕ → Set α) (G O : Set α) (h : ℕ) (b : ℤ) :
unobservedBirthRegion M hE κ D C G O h b ⊆ nominalPath M hE κ D C h b := by
unfold unobservedBirthRegion
split_ifs
· exact fun _ he => he.1.2.1
· exact Set.empty_subset _
theorem actual_path_rank_cost (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (hκ : 0 < κ) (D C T : ℕ → Set α) (h later layers : ℕ)
(start : ℤ) (S : ℤ → Set α)
(hS : ∀ b, S b ⊆ nominalPath M hE κ D C h b) :
κ * (∑ j ∈ range layers, conditionalRank M (S (start+2*j+2))
(nominalPath M hE κ D C h (start+2*j) ∪
lowerCompetition M hE κ D C T (start+2*j+2) h)) ≤
κ * (∑ j ∈ range layers, conditionalRank M (S (start+2*j+2))
(guardedPath M hE κ D C (h+1+later) (start+2*j) ∪
lowerCompetition M hE κ D C T (start+2*j+2) h)) +
κ * (C h).ncard +
∑ j ∈ range later, (κ * (C (h+1+j)).ncard + (D (h+1+j)).ncard) := by
obtain ⟨J, hJcost, hJ⟩ := exists_commonPathBudget M hE κ hκ D C h later
have hXfinal (b : ℤ) : nominalPath M hE κ D C h b ⊆
guardedPath M hE κ D C (h+1+later) b := by
apply (nominal_subset_guarded M hE κ D C h b).trans
have hm : Monotone (fun j => guardedPath M hE κ D C j b) :=
monotone_nat_of_le_succ (fun j => (guarded_subset_nominal M hE κ D C j b).trans
(nominal_subset_guarded M hE κ D C j b))
exact hm (by omega)
have hr := rankCost_chain M J layers
(fun j => nominalPath M hE κ D C h (start+2*j))
(fun j => nominalPath M hE κ D C h (start+2*j) ∪
lowerCompetition M hE κ D C T (start+2*j+2) h)
(fun j => guardedPath M hE κ D C (h+1+later) (start+2*j) ∪
lowerCompetition M hE κ D C T (start+2*j+2) h)
(fun j => S (start+2*j+2))
(fun _ _ => Set.subset_union_left)
(fun j _ => Set.union_subset_union_left _ (hXfinal _))
(by
intro j hj
apply Set.union_subset
· exact (hJ _).trans (M.closure_subset_closure (by
intro e he
rcases he with he | he
· exact Or.inl (Or.inl he)
· exact Or.inr he))
· intro e he
exact M.mem_closure_of_mem (Or.inl (Or.inr he)) (by simp [hE]))
(by
intro j hj
have heq : start+2*(↑(j+1):ℤ) = start+2*j+2 := by push_cast; omega
rw [heq]
apply Set.union_subset
· apply Set.union_subset
· exact nominalPath_mono M hE κ D C h (by omega)
· exact (lowerCompetition_subset_guarded M hE κ D C T _ h).trans
(guarded_subset_nominal M hE κ D C h _)
· exact hS _)
have hmul := Nat.mul_le_mul_left κ hr
rw [Nat.mul_add] at hmul
have hcard := natRank_le_ncard M J
have hbound := (Nat.mul_le_mul_left κ hcard).trans hJcost
omega
end MatroidProphet
end OAI
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