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c270adf | 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 | import Mathlib
namespace OAI
namespace IndependentSetsGames.Foundations.CorrelatedSampling
def firstAccepted {σ : Type*} (accept : σ → Bool) : List σ → Option σ
| [] => none
| proposal :: rest =>
if accept proposal then some proposal else firstAccepted accept rest
theorem first_union_common {σ : Type*} (left right : σ → Bool)
(proposals : List σ) (proposal : σ)
(hfirst : firstAccepted (fun s => left s || right s) proposals = some proposal)
(hleft : left proposal = true) (hright : right proposal = true) :
firstAccepted left proposals = some proposal ∧
firstAccepted right proposals = some proposal := by
induction proposals with
| nil => simp [firstAccepted] at hfirst
| cons head tail ih =>
by_cases hl : left head = true
· have heq : head = proposal := by
simpa [firstAccepted, hl] using hfirst
subst head
simp [firstAccepted, hleft, hright]
· by_cases hr : right head = true
· have heq : head = proposal := by
simpa [firstAccepted, hl, hr] using hfirst
subst head
exact False.elim (hl hleft)
· have htail : firstAccepted (fun s => left s || right s) tail = some proposal := by
simpa [firstAccepted, hl, hr] using hfirst
simpa [firstAccepted, hl, hr] using ih htail
def thresholdSample {α : Type*} (mass : α → Nat) (proposals : List (α × Nat)) :
Option α :=
(firstAccepted (fun s => decide (s.2 < mass s.1)) proposals).map Prod.fst
theorem thresholdSample_agreement {α : Type*} (left right : α → Nat)
(proposals : List (α × Nat)) (a : α) (height : Nat)
(hfirst : firstAccepted
(fun s => decide (s.2 < left s.1) || decide (s.2 < right s.1)) proposals =
some (a, height))
(hcommon : height < min (left a) (right a)) :
thresholdSample left proposals = some a ∧
thresholdSample right proposals = some a := by
have h := first_union_common
(fun s : α × Nat => decide (s.2 < left s.1))
(fun s : α × Nat => decide (s.2 < right s.1)) proposals (a, height) hfirst
(by simpa using (lt_min_iff.mp hcommon).1)
(by simpa using (lt_min_iff.mp hcommon).2)
simp [thresholdSample, h.1, h.2]
noncomputable section FiniteWeights
variable {α : Type*} [Fintype α]
def totalVariation (p q : α → ℝ) : ℝ := (∑ a, |p a - q a|) / 2
def overlapMass (p q : α → ℝ) : ℝ := ∑ a, min (p a) (q a)
def unionMass (p q : α → ℝ) : ℝ := ∑ a, max (p a) (q a)
theorem totalVariation_nonneg (p q : α → ℝ) : 0 ≤ totalVariation p q := by
unfold totalVariation
exact div_nonneg (Finset.sum_nonneg (fun _ _ => abs_nonneg _)) (by norm_num)
theorem overlapMass_nonneg (p q : α → ℝ) (hp : ∀ a, 0 ≤ p a)
(hq : ∀ a, 0 ≤ q a) : 0 ≤ overlapMass p q := by
exact Finset.sum_nonneg (fun a _ => le_min (hp a) (hq a))
private theorem min_abs_identity (x y : ℝ) : 2 * min x y + |x - y| = x + y := by
rcases le_total x y with h | h
· rw [min_eq_left h, abs_of_nonpos (sub_nonpos.mpr h)]
ring
· rw [min_eq_right h, abs_of_nonneg (sub_nonneg.mpr h)]
ring
private theorem max_abs_identity (x y : ℝ) : 2 * max x y - |x - y| = x + y := by
rcases le_total x y with h | h
· rw [max_eq_right h, abs_of_nonpos (sub_nonpos.mpr h)]
ring
· rw [max_eq_left h, abs_of_nonneg (sub_nonneg.mpr h)]
ring
theorem overlapMass_eq (p q : α → ℝ) (hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
overlapMass p q = 1 - totalVariation p q := by
have h := Finset.sum_congr (s₁ := Finset.univ) (s₂ := Finset.univ) rfl
(fun a _ => min_abs_identity (p a) (q a))
simp only [Finset.sum_add_distrib, ← Finset.mul_sum, hp, hq] at h
unfold overlapMass totalVariation
linarith
theorem unionMass_eq (p q : α → ℝ) (hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
unionMass p q = 1 + totalVariation p q := by
have h := Finset.sum_congr (s₁ := Finset.univ) (s₂ := Finset.univ) rfl
(fun a _ => max_abs_identity (p a) (q a))
simp only [Finset.sum_sub_distrib, Finset.sum_add_distrib, ← Finset.mul_sum, hp, hq] at h
unfold unionMass totalVariation
linarith
theorem totalVariation_le_one (p q : α → ℝ) (hp : ∀ a, 0 ≤ p a)
(hq : ∀ a, 0 ≤ q a) (hpsum : ∑ a, p a = 1) (hqsum : ∑ a, q a = 1) :
totalVariation p q ≤ 1 := by
have h := overlapMass_nonneg p q hp hq
rw [overlapMass_eq p q hpsum hqsum] at h
linarith
theorem overlap_union_ratio (p q : α → ℝ)
(hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
overlapMass p q / unionMass p q =
(1 - totalVariation p q) / (1 + totalVariation p q) := by
rw [overlapMass_eq p q hp hq, unionMass_eq p q hp hq]
theorem overlap_union_ratio_lower_bound (p q : α → ℝ)
(hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
1 - 2 * totalVariation p q ≤ overlapMass p q / unionMass p q := by
rw [overlap_union_ratio p q hp hq]
have htv := totalVariation_nonneg p q
have hden : 0 < 1 + totalVariation p q := by linarith
apply (le_div_iff₀ hden).2
nlinarith [sq_nonneg (totalVariation p q)]
theorem first_union_disagreement_upper_bound (p q : α → ℝ)
(hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
1 - overlapMass p q / unionMass p q ≤ 2 * totalVariation p q := by
linarith [overlap_union_ratio_lower_bound p q hp hq]
end FiniteWeights
end IndependentSetsGames.Foundations.CorrelatedSampling
end OAI
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