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import OAI.Computability.VertexCover.Repetition.Reveal
namespace OAI
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namespace UniqueGames.Foundations.Repetition
open scoped BigOperators
noncomputable section
variable {S X Y : Type*} [Fintype S] [Fintype X] [Fintype Y]
[DecidableEq X] [DecidableEq Y]
def maskedJoint (p : S × (X × Y) → ℝ) : (S × (X ⊕ Y)) × (X × Y) → ℝ := fun z =>
(if z.1.2 = Sum.inl z.2.1 then p (z.1.1, z.2) / 2 else 0) +
(if z.1.2 = Sum.inr z.2.2 then p (z.1.1, z.2) / 2 else 0)
def maskedModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) :
(S × (X ⊕ Y)) × (X × Y) → ℝ := fun z =>
Information.firstMarginal (maskedJoint p) z.1 * (revealProfile μ z.1.2).weight z.2
def leftRevealModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) :
S × (X × Y) → ℝ := fun z =>
(∑ y, p (z.1, (z.2.1, y))) * (revealProfile μ (Sum.inl z.2.1)).weight z.2
def rightRevealModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) :
S × (X × Y) → ℝ := fun z =>
(∑ x, p (z.1, (x, z.2.2))) * (revealProfile μ (Sum.inr z.2.2)).weight z.2
theorem sum_maskedJoint (p : S × (X × Y) → ℝ) :
(∑ z, maskedJoint p z) = ∑ z, p z := by
classical
have hrow (s : S) (q : X × Y) :
(∑ r : X ⊕ Y, maskedJoint p ((s,r),q)) = p (s,q) := by
simp [maskedJoint]
calc
_ = ∑ s : S, ∑ r : X ⊕ Y, ∑ q : X × Y, maskedJoint p ((s,r),q) := by
simp only [Fintype.sum_prod_type]
_ = ∑ s : S, ∑ q : X × Y, p (s,q) := by
apply Finset.sum_congr rfl
intro s _
rw [Finset.sum_comm]
simp_rw [hrow]
_ = _ := (Fintype.sum_prod_type _).symm
theorem maskedJoint_isProbability_iff (p : S × (X × Y) → ℝ) :
Information.IsProbability (maskedJoint p) ↔ Information.IsProbability p := by
classical
constructor
· intro h
constructor
· intro z
have hp := h.1 ((z.1,Sum.inl z.2.1),z.2)
simp [maskedJoint] at hp
linarith
· rw [← sum_maskedJoint]
exact h.2
· intro h
constructor
· intro z
have hp : 0 ≤ p (z.1.1,z.2) / 2 := div_nonneg (h.1 _) (by norm_num)
dsimp [maskedJoint]
split_ifs <;> linarith
· rw [sum_maskedJoint]
exact h.2
omit [Fintype S] in
theorem maskedJoint_firstMarginal_inl (p : S × (X × Y) → ℝ) (s : S) (x : X) :
Information.firstMarginal (maskedJoint p) (s, Sum.inl x) =
(∑ y, p (s, (x, y))) / 2 := by
simp [Information.firstMarginal, maskedJoint, Fintype.sum_prod_type,
div_eq_mul_inv, Finset.sum_mul]
rw [Finset.sum_comm]
simp
omit [Fintype S] in
theorem maskedJoint_firstMarginal_inr (p : S × (X × Y) → ℝ) (s : S) (y : Y) :
Information.firstMarginal (maskedJoint p) (s, Sum.inr y) =
(∑ x, p (s, (x, y))) / 2 := by
simp [Information.firstMarginal, maskedJoint, Fintype.sum_prod_type,
div_eq_mul_inv, Finset.sum_mul]
private theorem abs_half_sub_half_mul (a b «c» : ℝ) :
|a / 2 - b / 2 * «c»| = |a - b * «c»| / 2 := by
rw [show a / 2 - b / 2 * «c» = (a - b * «c») / 2 by ring, abs_div]
norm_num
omit [Fintype S] in
theorem abs_masked_inl (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) (s : S) (u x : X) (y : Y) :
|maskedJoint p ((s, Sum.inl u), (x, y)) - maskedModel μ p ((s, Sum.inl u), (x, y))| =
if x = u then |p (s, (x, y)) - leftRevealModel μ p (s, (x, y))| / 2 else 0 := by
by_cases hx : x = u
· subst x
simp only [maskedModel, maskedJoint_firstMarginal_inl]
simpa [maskedJoint, leftRevealModel] using
abs_half_sub_half_mul (p (s, (u, y))) (∑ v, p (s, (u, v)))
((revealProfile μ (Sum.inl u)).weight (u, y))
· have hs := revealProfile_inl_support μ u (x, y) hx
simp [maskedJoint, maskedModel, hs, hx, Ne.symm hx]
omit [Fintype S] in
theorem abs_masked_inr (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) (s : S) (v : Y) (x : X) (y : Y) :
|maskedJoint p ((s, Sum.inr v), (x, y)) - maskedModel μ p ((s, Sum.inr v), (x, y))| =
if y = v then |p (s, (x, y)) - rightRevealModel μ p (s, (x, y))| / 2 else 0 := by
by_cases hy : y = v
· subst y
simp only [maskedModel, maskedJoint_firstMarginal_inr]
simpa [maskedJoint, rightRevealModel] using
abs_half_sub_half_mul (p (s, (x, v))) (∑ u, p (s, (u, v)))
((revealProfile μ (Sum.inr v)).weight (x, v))
· have hs := revealProfile_inr_support μ v (x, y) hy
simp [maskedJoint, maskedModel, hs, hy, Ne.symm hy]
theorem masked_totalVariation_identity (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) :
2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) =
Information.totalVariation p (leftRevealModel μ p) +
Information.totalVariation p (rightRevealModel μ p) := by
have hleft (s : S) :
(∑ u : X, ∑ q : X × Y,
|maskedJoint p ((s, Sum.inl u), q) - maskedModel μ p ((s, Sum.inl u), q)|) =
(∑ q : X × Y, |p (s, q) - leftRevealModel μ p (s, q)|) / 2 := by
simp_rw [Fintype.sum_prod_type, abs_masked_inl]
simp [Finset.sum_ite_irrel, div_eq_mul_inv, Finset.sum_mul]
have hright (s : S) :
(∑ v : Y, ∑ q : X × Y,
|maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|) =
(∑ q : X × Y, |p (s, q) - rightRevealModel μ p (s, q)|) / 2 := by
have hcollapse :
(∑ v : Y, ∑ q : X × Y,
|maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|) =
∑ v : Y, ∑ x : X, |p (s, (x, v)) - rightRevealModel μ p (s, (x, v))| / 2 := by
simp [Fintype.sum_prod_type, abs_masked_inr]
rw [hcollapse, Finset.sum_comm]
simp [Fintype.sum_prod_type, div_eq_mul_inv, Finset.sum_mul]
have hmass :
(∑ z : (S × (X ⊕ Y)) × (X × Y), |maskedJoint p z - maskedModel μ p z|) =
((∑ z : S × (X × Y), |p z - leftRevealModel μ p z|) +
(∑ z : S × (X × Y), |p z - rightRevealModel μ p z|)) / 2 := by
calc
_ = ∑ s : S,
((∑ u : X, ∑ q : X × Y, |maskedJoint p ((s, Sum.inl u), q) - maskedModel μ p ((s, Sum.inl u), q)|) +
(∑ v : Y, ∑ q : X × Y, |maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|)) := by
simp only [Fintype.sum_prod_type, Fintype.sum_sum_type]
_ = ∑ s : S,
((∑ q : X × Y, |p (s, q) - leftRevealModel μ p (s, q)|) / 2 +
(∑ q : X × Y, |p (s, q) - rightRevealModel μ p (s, q)|) / 2) := by
simp_rw [hleft, hright]
_ = _ := by
simp [Fintype.sum_prod_type, div_eq_mul_inv, Finset.sum_add_distrib, Finset.sum_mul, add_mul]
unfold Information.totalVariation
rw [hmass]
ring
theorem leftRevealModel_totalVariation_le (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) :
Information.totalVariation p (leftRevealModel μ p) ≤
2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) := by
rw [masked_totalVariation_identity]
linarith [Information.totalVariation_nonneg p (rightRevealModel μ p)]
theorem rightRevealModel_totalVariation_le (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) :
Information.totalVariation p (rightRevealModel μ p) ≤
2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) := by
rw [masked_totalVariation_identity]
linarith [Information.totalVariation_nonneg p (leftRevealModel μ p)]
end
end UniqueGames.Foundations.Repetition
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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