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import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Basic.Real.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
namespace OAI
section
/-!
Finite prerequisites for Holenstein's consistent sampling argument (Lemma 5.2).
`firstAccepted` is an actual bounded rejection sampler on a shared finite list
of proposals. `first_union_common` proves its synchronization property.
The remaining theorems establish the exact overlap/union mass formulas and the
agreement lower bound used to analyze a first accepted shared proposal.
No infinite random process or locally computable embedding is asserted here.
-/
namespace UniqueGamesTheorem.Foundations.CorrelatedSampling
/-- Return the first proposal accepted by the given test, or report exhaustion. -/
def firstAccepted {σ : Type*} (accept : σ → Bool) : List σ → Option σ
| [] => none
| proposal :: rest =>
if accept proposal then some proposal else firstAccepted accept rest
/-- If the first proposal accepted by either sampler is accepted by both,
the two samplers return exactly that same proposal. -/
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
/-- Finite integer-threshold rejection sampling. A mass table can be sampled
using shared proposals `(label, height)` drawn from a finite rectangle. -/
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]
/-- Total variation for real-valued finite mass functions. Distribution
normalization is supplied explicitly in theorems that need it. -/
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
/-- The overlap of two probability masses is exactly one minus their TV. -/
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
/-- The union of two probability masses is exactly one plus their TV. -/
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
/-- The first-union-point agreement ratio appearing in Holenstein Lemma 5.2. -/
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]
/-- Exact algebra behind the linear agreement estimate. This is a proved
bound on the overlap/union ratio, not an assumed sampler law. -/
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)]
/-- Equivalent upper bound on the probability of the first union point
belonging to just one density. -/
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 UniqueGamesTheorem.Foundations.CorrelatedSampling
end
end OAI
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