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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Probability.MatroidProphet.Main
import OAI.Probability.MatroidSecretary.Secretary.Model
import Mathlib.MeasureTheory.Integral.Bochner.Basic
namespace OAI
open MeasureTheory Finset
namespace MatroidProphet
variable {n : ℕ} {Q : Type*} [MeasurableSpace Q]
lemma secretaryDecisionAt_congr (A : SecretaryRule n Q) (q : Q)
(w w' : Weights n) (π π' : ArrivalOrder n) (k : Fin n)
(hh : history w π k = history w' π' k) :
secretaryDecisionAt A q w π k = secretaryDecisionAt A q w' π' k := by
simp only [secretaryDecisionAt, hh]
lemma secretaryAcceptedThrough_mono (A : SecretaryRule n Q) (q : Q)
(w : Weights n) (π : ArrivalOrder n) {t t' : ℕ} (ht : t ≤ t') :
secretaryAcceptedThrough A q w π t ⊆ secretaryAcceptedThrough A q w π t' := by
intro e he
simp only [secretaryAcceptedThrough, mem_filter, mem_univ, true_and] at he ⊢
exact ⟨lt_of_lt_of_le he.1 ht, he.2⟩
lemma secretaryAcceptedThrough_arrived (A : SecretaryRule n Q) (q : Q)
(w : Weights n) (π : ArrivalOrder n) (t : ℕ) (e : Fin n)
(he : e ∈ secretaryAcceptedThrough A q w π t) : (π.symm e).val < t :=
(Finset.mem_filter.mp he).2.1
lemma secretaryAcceptedThrough_not_prefix (A : SecretaryRule n Q) (q : Q)
(w : Weights n) (π : ArrivalOrder n) (t : ℕ) (e : Fin n)
(he : e ∈ secretaryAcceptedThrough A q w π t) :
(A.prefixLength q).val ≤ (π.symm e).val := by
have hd := (Finset.mem_filter.mp he).2.2
by_contra hn
have hlt : (π.symm e).val < (A.prefixLength q).val := Nat.lt_of_not_ge hn
simp [secretaryDecisionAt, hlt] at hd
lemma secretaryAcceptedThrough_prefix_empty (A : SecretaryRule n Q) (q : Q)
(w : Weights n) (π : ArrivalOrder n) (t : ℕ)
(ht : t ≤ (A.prefixLength q).val) :
secretaryAcceptedThrough A q w π t = ∅ := by
apply Finset.eq_empty_iff_forall_notMem.mpr
intro e he
exact (not_lt_of_ge (secretaryAcceptedThrough_not_prefix A q w π t e he))
(lt_of_lt_of_le (secretaryAcceptedThrough_arrived A q w π t e he) ht)
lemma secretaryReward_nonneg (A : SecretaryRule n Q) (q : Q)
(w : Weights n) (π : ArrivalOrder n) (hw : ∀ e, 0 ≤ w e) :
0 ≤ secretaryReward A q w π := by
exact Finset.sum_nonneg fun e _ => hw e
lemma secretaryReward_le_optimum (M : Matroid (Fin n)) (A : SecretaryRule n Q)
(hA : SecretaryFeasible M A) (q : Q) (w : Weights n)
(hw : ∀ e, 0 ≤ w e) (π : ArrivalOrder n) :
secretaryReward A q w π ≤ optimum M w :=
sum_le_optimum M w _ (hA q w hw π n)
lemma measurable_secretaryDecisionAt (A : SecretaryRule n Q)
(π : ArrivalOrder n) (k : Fin n) :
Measurable (fun x : Q × Weights n => secretaryDecisionAt A x.1 x.2 π k) := by
classical
have hp : MeasurableSet {x : Q × Weights n | k.val < (A.prefixLength x.1).val} :=
(A.measurable_prefixLength.comp measurable_fst)
(by trivial : MeasurableSet {K : Fin (n + 1) | k.val < K.val})
exact Measurable.ite hp measurable_const
((A.measurable_decide k).comp
(measurable_fst.prodMk ((measurable_history π k).comp measurable_snd)))
lemma secretaryReward_eq_sum (A : SecretaryRule n Q) (q : Q)
(w : Weights n) (π : ArrivalOrder n) :
secretaryReward A q w π =
∑ e : Fin n, if secretaryDecisionAt A q w π (π.symm e) = true then w e else 0 := by
classical
unfold secretaryReward secretaryAcceptedThrough
rw [Finset.sum_filter]
apply Finset.sum_congr rfl
intro e _
simp only [(π.symm e).isLt, true_and]
lemma measurable_secretaryReward_fixed_order (A : SecretaryRule n Q)
(π : ArrivalOrder n) :
Measurable (fun x : Q × Weights n => secretaryReward A x.1 x.2 π) := by
classical
simp_rw [secretaryReward_eq_sum]
apply Finset.measurable_sum
intro e _
have hd := measurable_secretaryDecisionAt A π (π.symm e)
have ht : MeasurableSet {x : Q × Weights n |
secretaryDecisionAt A x.1 x.2 π (π.symm e) = true} :=
hd (measurableSet_singleton true)
exact Measurable.ite ht ((measurable_pi_apply e).comp measurable_snd) measurable_const
/-- No independence from the entire seed is imposed on the measurable arrival order. -/
lemma measurable_secretaryReward (A : SecretaryRule n Q) :
Measurable (fun x : ArrivalOrder n × (Q × Weights n) =>
secretaryReward A x.2.1 x.2.2 x.1) :=
measurable_from_prod_countable_right (measurable_secretaryReward_fixed_order A)
lemma integrable_secretaryReward (M : Matroid (Fin n)) (A : SecretaryRule n Q)
(hA : SecretaryFeasible M A) (w : Weights n) (hw : ∀ e, 0 ≤ w e)
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(q : Ω → Q) (π : Ω → ArrivalOrder n) (hq : Measurable q) (hπ : Measurable π) :
Integrable (fun ω => secretaryReward A (q ω) w (π ω)) μ := by
have hm : Measurable (fun ω => secretaryReward A (q ω) w (π ω)) :=
(measurable_secretaryReward A).comp
(hπ.prodMk (hq.prodMk measurable_const))
apply (integrable_const (optimum M w)).mono' hm.aestronglyMeasurable
apply ae_of_all
intro ω
rw [Real.norm_eq_abs, abs_of_nonneg (secretaryReward_nonneg A (q ω) w (π ω) hw)]
exact secretaryReward_le_optimum M A hA (q ω) w hw (π ω)
end MatroidProphet
end OAI
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