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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import OAI.Probability.MatroidProphet.SafeSamples
namespace OAI
namespace MatroidProphet
open Finset
variable {α : Type*} [Fintype α] [DecidableEq α]
attribute [local instance] Classical.propDecidable
noncomputable def safeEndpointParityCount (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (d : α) (C T : Finset α) (parity : Bool) : ℝ :=
∑ i ∈ range (pathHorizon h-1), if safeBirthEvent M hE κ D G h d (-(i:ℤ)) C T ∧
(-(i:ℤ)) % 2 = (if parity then 1 else 0) then 1 else 0
noncomputable def safeSampleCount (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) (parity : Bool) : ℝ :=
∑ d ∈ I, safeEndpointParityCount M hE κ D G h d C T parity
omit [DecidableEq α] in
lemma safeSampleCount_parity_split (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) :
safeSampleCount M hE κ D G h I C T false + safeSampleCount M hE κ D G h I C T true =
safeCountBeforeParity M hE κ D G h I C T := by
unfold safeSampleCount safeCountBeforeParity
rw [← sum_add_distrib]
apply sum_congr rfl
intro d hd
unfold safeEndpointParityCount safeEndpointCount
rw [← sum_add_distrib]
apply sum_congr rfl
intro i hi
have hm0 := Int.emod_nonneg (-(i:ℤ)) (by norm_num : (2:ℤ) ≠ 0)
have hm1 := Int.emod_lt_of_pos (-(i:ℤ)) (by norm_num : (0:ℤ) < 2)
have hm : (-(i:ℤ)) % 2 = 0 ∨ (-(i:ℤ)) % 2 = 1 := by omega
by_cases hs : safeBirthEvent M hE κ D G h d (-(i:ℤ)) C T
· rcases hm with hm | hm <;> simp [hs, hm]
· simp [hs]
noncomputable def fairParityExpectation (f : Bool → ℝ) : ℝ := (f false + f true) / 2
omit [DecidableEq α] in
lemma safeSampleCount_fair_parity (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) :
fairParityExpectation (safeSampleCount M hE κ D G h I C T) =
safeCountBeforeParity M hE κ D G h I C T / 2 := by
unfold fairParityExpectation
rw [safeSampleCount_parity_split]
theorem safeSampleCount_lower (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (hκ : 0 < κ) (D : ℕ → Set α) (G : ℕ → Finset α)
(hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1)
(h : ℕ) (I : Finset α) (hI : M.Indep (I : Set α)) :
(((2:ℝ)^12)⁻¹) / 4 * I.card - (D h).ncard / (κ:ℝ) ≤
bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (safeSampleCount M hE κ D G h I C T))) := by
have hpre := safeCountBeforeParity_lower M hE κ hκ D G hG q hq0 hq1 h I hI
have heq : bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (safeSampleCount M hE κ D G h I C T))) =
(bitsExpectation q univ (fun C => bitsExpectation q univ (safeCountBeforeParity M hE κ D G h I C))) / 2 := by
simp only [safeSampleCount_fair_parity, div_eq_mul_inv]
rw [show bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
safeCountBeforeParity M hE κ D G h I C T * (2:ℝ)⁻¹)) =
bitsExpectation q univ (fun C => bitsExpectation q univ (safeCountBeforeParity M hE κ D G h I C) * (2:ℝ)⁻¹) from by
apply bitsExpectation_congr; intro C hC; exact bitsExpectation_mul_right q univ _ _]
exact bitsExpectation_mul_right q univ _ _
rw [heq]
have hnonneg : 0 ≤ (D h).ncard / (κ:ℝ) := div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
linarith
omit [Fintype α] [DecidableEq α] in
lemma sum_unique_indicators {ι : Type*} (J : Finset ι) (p : ι → Prop) [DecidablePred p]
(huniq : ∀ i ∈ J, ∀ j ∈ J, p i → p j → i = j) :
(∑ i ∈ J, if p i then (1:ℝ) else 0) = if ∃ i ∈ J, p i then 1 else 0 := by
classical
by_cases hex : ∃ i ∈ J, p i
· obtain ⟨i, hi, hp⟩ := hex
rw [ite_eq_left ⟨i, hi, hp⟩]
calc
_ = if p i then (1:ℝ) else 0 := by
apply sum_eq_single i
· intro j hj hji
exact ite_eq_right (fun hjp => hji (huniq j hj i hi hjp hp))
· intro hnot
exact (hnot hi).elim
_ = 1 := ite_eq_left hp
· rw [ite_eq_right hex]
apply sum_eq_zero
intro i hi
exact ite_eq_right (fun hp => hex ⟨i, hi, hp⟩)
noncomputable def IsSafeSample (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (d : α) (C T : Finset α) (parity : Bool) : Prop :=
let b := nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d
activation h + 2 ≤ b ∧ b ≤ 0 ∧ b % 2 = (if parity then 1 else 0) ∧
d ∉ densityExpansion M hE κ (D h)
(guardedPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h (b-2) ∪
lowerCompetition M hE κ D (fun i => (C : Set α) ∩ (G i : Set α))
(fun i => (T : Set α) ∩ (G i : Set α)) b h)
omit [DecidableEq α] in
lemma isSafeSample_iff_endpoint (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (d : α) (C T : Finset α) (parity : Bool) :
IsSafeSample M hE κ D G h d C T parity ↔ ∃ i ∈ range (pathHorizon h-1),
safeBirthEvent M hE κ D G h d (-(i:ℤ)) C T ∧ (-(i:ℤ)) % 2 = (if parity then 1 else 0) := by
let b := nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d
have ht := activation_add_horizon h
have hH := pathHorizon_pos h
constructor
· rintro ⟨hlo, hhi, hp, hs⟩
have hi0 : 0 ≤ -b := by omega
have hieq : -(((-b).toNat:ℕ):ℤ) = b := by rw [Int.toNat_of_nonneg hi0]; omega
refine ⟨(-b).toNat, mem_range.mpr ?_, ?_, ?_⟩
· have hiNat := Int.toNat_of_nonneg hi0
change activation h + 2 ≤ b at hlo
omega
· rw [hieq]
exact ⟨rfl, hs⟩
· simpa only [hieq] using hp
· rintro ⟨i, hi, hs, hp⟩
have hbirth := hs.1
have hbound := mem_range.mp hi
change _ ∧ _ ∧ _ ∧ _
rw [hbirth]
exact ⟨by omega, by omega, hp, hs.2⟩
noncomputable def safeSampleSet (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) (parity : Bool) : Finset α :=
I.filter (fun d => IsSafeSample M hE κ D G h d C T parity)
omit [DecidableEq α] in
lemma safeSampleCount_eq_card (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) (parity : Bool) :
safeSampleCount M hE κ D G h I C T parity = (safeSampleSet M hE κ D G h I C T parity).card := by
unfold safeSampleCount safeSampleSet
have hpoint (d : α) : safeEndpointParityCount M hE κ D G h d C T parity =
if IsSafeSample M hE κ D G h d C T parity then 1 else 0 := by
unfold safeEndpointParityCount
rw [sum_unique_indicators]
· simp only [← isSafeSample_iff_endpoint]
· intro i hi j hj hsi hsj
have hbi := hsi.1.1
have hbj := hsj.1.1
omega
simp_rw [hpoint]
simp
end MatroidProphet
end OAI
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