| 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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|
