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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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import OAI.Probability.MatroidProphet.Algorithm.Feasible
import OAI.Probability.MatroidProphet.Maximum.Guarantee
import OAI.Probability.MatroidProphet.Priority
import OAI.Probability.MatroidProphet.Simulation
namespace OAI
namespace MatroidProphet
open MeasureTheory
noncomputable def branchOnlineRule {n bits : ℕ} (branch : Seed bits → Bool)
(A B : OnlineRule n bits) : OnlineRule n bits where
decide k r s h := if branch r then A.decide k r s h else B.decide k r s h
measurable_decide k := by
have hb : Measurable (fun x : Seed bits × (Weights n × History n k) => branch x.1) :=
(measurable_of_countable branch).comp measurable_fst
exact Measurable.ite (hb (measurableSet_singleton true))
(A.measurable_decide k) (B.measurable_decide k)
noncomputable def branchHiddenRule {n bits : ℕ} (branch : Seed bits → Bool)
(A B : HiddenRule n bits) : HiddenRule n bits where
mask r := if branch r then A.mask r else B.mask r
core := branchOnlineRule branch A.core B.core
lemma branchOnlineRule_acceptedThrough {n bits : ℕ} (branch : Seed bits → Bool)
(A B : OnlineRule n bits) (r : Seed bits) (s v : Weights n) (π : ArrivalOrder n) (t : ℕ) :
acceptedThrough (branchOnlineRule branch A B) r s v π t =
if branch r then acceptedThrough A r s v π t else acceptedThrough B r s v π t := by
classical
cases hb : branch r <;> simp [acceptedThrough, decisionAt, branchOnlineRule, hb] <;> rfl
lemma branchHiddenRule_observed {n bits : ℕ} (branch : Seed bits → Bool)
(A B : HiddenRule n bits) (r : Seed bits) (w : Weights n) :
observed (branchHiddenRule branch A B) r w =
if branch r then observed A r w else observed B r w := by
funext e
cases hb : branch r <;> simp only [observed, branchHiddenRule, hb, Bool.false_eq_true,
↓reduceIte]
lemma branchHiddenRule_acceptedThrough {n bits : ℕ} (branch : Seed bits → Bool)
(A B : HiddenRule n bits) (r : Seed bits) (w : Weights n) (π : ArrivalOrder n) (t : ℕ) :
hiddenAcceptedThrough (branchHiddenRule branch A B) r w π t =
if branch r then hiddenAcceptedThrough A r w π t else hiddenAcceptedThrough B r w π t := by
change acceptedThrough (branchOnlineRule branch A.core B.core) r
(observed (branchHiddenRule branch A B) r w) w π t \ (if branch r then A.mask r else B.mask r) = _
rw [branchHiddenRule_observed, branchOnlineRule_acceptedThrough]
cases branch r <;> simp only [Bool.false_eq_true, ↓reduceIte, hiddenAcceptedThrough]
lemma branchHiddenRule_reward {n bits : ℕ} (branch : Seed bits → Bool)
(A B : HiddenRule n bits) (r : Seed bits) (w : Weights n) (π : ArrivalOrder n) :
hiddenReward (branchHiddenRule branch A B) r w π =
if branch r then hiddenReward A r w π else hiddenReward B r w π := by
cases hb : branch r <;> simp [hiddenReward, branchHiddenRule_acceptedThrough, hb]
lemma branchHiddenRule_worst {n bits : ℕ} (branch : Seed bits → Bool)
(A B : HiddenRule n bits) (r : Seed bits) (w : Weights n) :
hiddenWorstReward (branchHiddenRule branch A B) w r =
if branch r then hiddenWorstReward A w r else hiddenWorstReward B w r := by
classical
cases hb : branch r <;> simp [hiddenWorstReward, branchHiddenRule_reward, hb]
instance withBotIntMeasurableSpace : MeasurableSpace (WithBot ℤ) := ⊤
instance withBotIntMeasurableSingletonClass : MeasurableSingletonClass (WithBot ℤ) :=
⟨fun _ => trivial⟩
noncomputable def roundedMaximumHidden {n : ℕ} (M : Matroid (Fin n))
{bits : ℕ} (mask : Seed bits → Finset (Fin n)) : HiddenRule n bits :=
Maximum.maximumHidden M (fun x => Priority.asLevel (roundedLevel weightBase x))
((measurable_of_countable Priority.asLevel).comp (measurable_roundedLevel weightBase))
(fun a => a ≠ ⊥) mask
noncomputable def completeHiddenRule {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ) :
HiddenRule n (mainSeedBits n) :=
branchHiddenRule mainBranch (MainAlgorithm.hidden M hE)
(roundedMaximumHidden M (fun r => (mainMasks r).H))
lemma branchHiddenRule_feasible {n bits : ℕ} (M : Matroid (Fin n))
(branch : Seed bits → Bool) (A B : HiddenRule n bits)
(hA : ∀ w, (∀ e, 0 ≤ w e) → ∀ r π t,
M.Indep (hiddenAcceptedThrough A r w π t : Set (Fin n)))
(hB : ∀ w, (∀ e, 0 ≤ w e) → ∀ r π t,
M.Indep (hiddenAcceptedThrough B r w π t : Set (Fin n)))
(w : Weights n) (hw : ∀ e, 0 ≤ w e) (r : Seed bits) (π : ArrivalOrder n) (t : ℕ) :
M.Indep (hiddenAcceptedThrough (branchHiddenRule branch A B) r w π t : Set (Fin n)) := by
rw [branchHiddenRule_acceptedThrough]
cases hb : branch r
· simp only [Bool.false_eq_true, ↓reduceIte]
exact hB w hw r π t
· simp only [↓reduceIte]
exact hA w hw r π t
lemma roundedMaximumHidden_feasible {n bits : ℕ} (M : Matroid (Fin n))
(mask : Seed bits → Finset (Fin n)) (w : Weights n) (hw : ∀ e, 0 ≤ w e)
(r : Seed bits) (π : ArrivalOrder n) (t : ℕ) :
M.Indep (hiddenAcceptedThrough (roundedMaximumHidden M mask) r w π t : Set (Fin n)) :=
Maximum.maximumHidden_feasible M _ _ _ _ w hw r π t
lemma completeHiddenRule_feasible {n : ℕ} (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 (completeHiddenRule M hE) r w π t : Set (Fin n)) :=
branchHiddenRule_feasible M mainBranch _ _ (MainAlgorithm.hidden_feasible M hE)
(roundedMaximumHidden_feasible M _) w hw r π t
end MatroidProphet
end OAI
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