File size: 6,352 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 | import OAI.Probability.MatroidProphet.Main
import OAI.Probability.MatroidSecretary.Secretary.PrefixObservationModel
namespace OAI
open MeasureTheory Finset
namespace MatroidProphet
@[simp] lemma mem_orderPrefix {n : ℕ} (π : ArrivalOrder n) (K : ℕ) (e : Fin n) :
e ∈ orderPrefix π K ↔ (π.symm e).val < K := by
simp [orderPrefix]
@[simp] lemma prefixLabels_zero {n : ℕ} {k : Fin n} (h : History n k) :
prefixLabels 0 h = ∅ := by
simp [prefixLabels]
@[simp] lemma prefixWeights_zero {n : ℕ} {k : Fin n} (h : History n k) :
prefixWeights 0 h = 0 := by
funext e
simp [prefixWeights]
lemma prefixLabels_history {n : ℕ} (w : Weights n) (π : ArrivalOrder n)
(k : Fin n) (K : ℕ) (hK : K ≤ k.val + 1) :
prefixLabels K (history w π k) = orderPrefix π K := by
ext e
simp only [prefixLabels, mem_image, mem_filter, mem_univ, true_and, mem_orderPrefix]
constructor
· rintro ⟨j, hj, hje⟩
have hj' : prefixIndex k j = π.symm e := by
apply π.injective
simpa [history] using hje
have hv := congrArg Fin.val hj'
exact hv ▸ hj
· intro he
let j : Fin (k.val + 1) := ⟨(π.symm e).val, lt_of_lt_of_le he hK⟩
refine ⟨j, he, ?_⟩
have hj : prefixIndex k j = π.symm e := by
apply Fin.ext
rfl
simp [history, hj]
/-- Prefix-label extraction is measurable despite the real-valued history entries. -/
lemma measurable_prefixLabels {n : ℕ} (k : Fin n) (K : ℕ) :
Measurable (prefixLabels (k := k) K) := by
classical
have hl : Measurable (fun h : History n k => fun j => (h j).1) := by
apply Measurable.of_eval
intro j
exact measurable_fst.comp (measurable_pi_apply j)
exact (measurable_of_countable (fun l : Fin (k.val + 1) → Fin n =>
(Finset.univ.filter fun j => j.val < K).image l)).comp hl
/-- Changing the suffix entries of an available history does not change prefix data. -/
lemma prefixWeights_congr {n : ℕ} {k : Fin n} (K : ℕ) (h h' : History n k)
(hh : ∀ j, j.val < K → h j = h' j) :
prefixWeights K h = prefixWeights K h' := by
funext e
apply Finset.sum_congr rfl
intro j _
by_cases hj : j.val < K
· rw [hh j hj]
· simp [hj]
lemma prefixLabels_congr {n : ℕ} {k : Fin n} (K : ℕ) (h h' : History n k)
(hh : ∀ j, j.val < K → (h j).1 = (h' j).1) :
prefixLabels K h = prefixLabels K h' := by
apply Finset.image_congr
intro j hj
exact hh j (Finset.mem_filter.mp hj).2
lemma measurable_prefixWeights {n : ℕ} (k : Fin n) (K : ℕ) :
Measurable (prefixWeights (k := k) K) := by
classical
apply Measurable.of_eval
intro e
apply Finset.measurable_sum
intro j _
have hj : Measurable (fun h : History n k => h j) := measurable_pi_apply j
have he : MeasurableSet {h : History n k | (h j).1 = e} :=
(measurable_fst.comp hj) (measurableSet_singleton e)
by_cases hKj : j.val < K
· have hm : Measurable (fun h : History n k =>
if (h j).1 = e then (h j).2 else (0 : ℝ)) :=
Measurable.ite he (measurable_snd.comp hj) measurable_const
simpa only [hKj, true_and] using hm
· simp only [hKj, false_and, ite_false]
exact measurable_const
/-- Seed-dependent cutoff extraction remains measurable jointly with the history. -/
lemma measurable_prefixWeights_seed {n : ℕ} {Q : Type*} [MeasurableSpace Q]
(k : Fin n) (K : Q → Fin (n + 1)) (hK : Measurable K) :
Measurable (fun x : Q × History n k => prefixWeights (K x.1).val x.2) := by
have hm : Measurable (fun x : Fin (n + 1) × History n k =>
prefixWeights x.1.val x.2) :=
measurable_from_prod_countable_right fun j => measurable_prefixWeights k j.val
exact hm.comp ((hK.comp measurable_fst).prodMk measurable_snd)
lemma measurable_prefixLabels_seed {n : ℕ} {Q : Type*} [MeasurableSpace Q]
(k : Fin n) (K : Q → Fin (n + 1)) (hK : Measurable K) :
Measurable (fun x : Q × History n k => prefixLabels (K x.1).val x.2) := by
have hm : Measurable (fun x : Fin (n + 1) × History n k =>
prefixLabels x.1.val x.2) :=
measurable_from_prod_countable_right fun j => measurable_prefixLabels k j.val
exact hm.comp ((hK.comp measurable_fst).prodMk measurable_snd)
lemma prefixWeights_history {n : ℕ} (w : Weights n) (π : ArrivalOrder n)
(k : Fin n) (K : ℕ) (hK : K ≤ k.val + 1) (e : Fin n) :
prefixWeights K (history w π k) e =
if (π.symm e).val < K then w e else 0 := by
classical
by_cases he : (π.symm e).val < K
· rw [ite_eq_left he]
let j : Fin (k.val + 1) := ⟨(π.symm e).val, lt_of_lt_of_le he hK⟩
have hj : prefixIndex k j = π.symm e := by
apply Fin.ext
rfl
unfold prefixWeights
rw [Finset.sum_eq_single j]
· simp [history, hj, j, he]
· intro b _ hbj
have hne : π (prefixIndex k b) ≠ e := by
intro hb
apply hbj
apply Fin.ext
have hbi : prefixIndex k b = π.symm e := by
apply π.injective
simpa using hb
exact congrArg (fun z : Fin n => z.val) hbi
simp [history, hne]
· simp
· rw [ite_eq_right he]
apply Finset.sum_eq_zero
intro j _
by_cases hj : j.val < K
· have hne : π (prefixIndex k j) ≠ e := by
intro hje
apply he
have hji : prefixIndex k j = π.symm e := by
apply π.injective
simpa using hje
have hv := congrArg Fin.val hji
exact hv ▸ hj
simp [history, hne]
· simp [hj]
lemma prefixWeights_eq_observed {n bits : ℕ} (A : HiddenRule n bits)
(r : Seed bits) (w : Weights n) (π : ArrivalOrder n) (k : Fin n) (K : ℕ)
(hK : K ≤ k.val + 1) (hmask : A.mask r = orderPrefix π K) :
prefixWeights K (history w π k) = observed A r w := by
funext e
rw [prefixWeights_history w π k K hK e]
simp [observed, hmask]
lemma hiddenAcceptedThrough_prefix_empty {n bits : ℕ} (A : HiddenRule n bits)
(r : Seed bits) (w : Weights n) (π : ArrivalOrder n) (K t : ℕ)
(hmask : A.mask r = orderPrefix π K) (ht : t ≤ K) :
hiddenAcceptedThrough A r w π t = ∅ := by
apply Finset.eq_empty_iff_forall_notMem.mpr
intro e he
obtain ⟨he, hnot⟩ := Finset.mem_sdiff.mp he
apply hnot
rw [hmask, mem_orderPrefix]
exact lt_of_lt_of_le (acceptedThrough_arrived A.core r (observed A r w) w π t e he) ht
end MatroidProphet
end OAI
|