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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.Algorithm.Rule
namespace OAI
namespace MatroidProphet
open Set Finset
lemma guardedPath_group_mono {α : Type*} [Fintype α]
(M : Matroid α) (hE : M.E = Set.univ) (κ : ℕ) (D C : ℕ → Set α) (k : ℤ) :
Monotone (fun h => guardedPath M hE κ D C h k) := by
apply monotone_nat_of_le_succ
intro h
exact (guarded_subset_nominal M hE κ D C h k).trans
(nominal_subset_guarded M hE κ D C h k)
lemma assigned_some_of_eligible {n : ℕ} {K : Type*}
(assign : Fin n → K → Option ℤ) (v : Fin n → K) (e : Fin n)
(he : e ∈ assignedEligible assign v) :
assign e (v e) = some (assignedLayer assign v e) := by
classical
cases h : assign e (v e) <;> simp_all [assignedEligible, assignedLayer]
namespace MainAlgorithm
variable {n : ℕ}
lemma assign_spec (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (seen : Fin n → Option ℤ) (e : Fin n) (w : Option ℤ) (b : ℤ) :
assign M hE d seen e w = some b ↔
∃ i, w = some i ∧ eligible M hE d seen e i ∧ birth M hE d seen e i = b := by
classical
cases w with
| none => simp [assign]
| some i =>
by_cases h : eligible M hE d seen e i <;> simp [assign, h]
lemma eligible_mem_final (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (seen : Fin n → Option ℤ) (e : Fin n) (i : ℤ)
(he : eligible M hE d seen e i) :
e ∈ finalPath M hE d seen (birth M hE d seen e i) := by
have hi : i ∈ groups M d seen := he.2.2.1
have hidx : (groups M d seen).idxOf i < (groups M d seen).length :=
List.idxOf_lt_length_of_mem hi
have hm := nominalBirth_mem M hE (2^100) (groupMask M d seen d.D)
(groupMask M d seen d.C) ((groups M d seen).idxOf i) e
have hs := nominal_subset_guarded M hE (2^100) (groupMask M d seen d.D)
(groupMask M d seen d.C) ((groups M d seen).idxOf i) (birth M hE d seen e i) hm
exact guardedPath_group_mono M hE (2^100) (groupMask M d seen d.D)
(groupMask M d seen d.C) _ (Nat.succ_le_of_lt hidx) hs
lemma assigned_lower_in_base (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (seen v : Fin n → Option ℤ)
(e : Fin n) (he : e ∈ assignedEligible (assign M hE d seen) v)
(f : Fin n) (hf : f ∈ assignedEligible (assign M hE d seen) v)
(hlt : assignedLayer (assign M hE d seen) v e < assignedLayer (assign M hE d seen) v f) :
e ∈ finalPath M hE d seen (assignedLayer (assign M hE d seen) v f - 2) := by
obtain ⟨i, _, hei, hbi⟩ := (assign_spec M hE d seen e _ _).mp
(assigned_some_of_eligible (assign M hE d seen) v e he)
obtain ⟨j, _, hfj, hbj⟩ := (assign_spec M hE d seen f _ _).mp
(assigned_some_of_eligible (assign M hE d seen) v f hf)
have heiParity := hei.2.2.2.2.2.1
have hfjParity := hfj.2.2.2.2.2.1
rw [hbi] at heiParity
rw [hbj] at hfjParity
have hgap : assignedLayer (assign M hE d seen) v e ≤
assignedLayer (assign M hE d seen) v f - 2 := by omega
have hm := eligible_mem_final M hE d seen e i hei
rw [hbi] at hm
exact guardedPath_mono M hE (2^100) (groupMask M d seen d.D)
(groupMask M d seen d.C) (groups M d seen).length hgap hm
theorem core_feasible (M : Matroid (Fin n)) (hE : M.E = Set.univ) :
Feasible M (core M hE) := by
apply layerOnlineRule_feasible M hE (roundedLevel weightBase) (measurable_roundedLevel weightBase)
intro r seen v e he f hf hlt
exact assigned_lower_in_base M hE (mainMasks r) seen v e he f hf hlt
theorem hidden_feasible (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(w : Weights n) (hw : ∀ e, 0 ≤ w e) (r : Seed (mainSeedBits n))
(π : ArrivalOrder n) (t : ℕ) :
M.Indep (hiddenAcceptedThrough (hidden M hE) r w π t : Set (Fin n)) := by
have hobs : ∀ e, 0 ≤ observed (hidden M hE) r w e := by
intro e
unfold observed
split_ifs
· exact hw e
· exact le_rfl
exact (core_feasible M hE r (observed (hidden M hE) r w) w π t hobs hw).subset
(by intro e he; exact (Finset.mem_sdiff.mp he).1)
end MainAlgorithm
end MatroidProphet
end OAI
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