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namespace OAI
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namespace UniqueGames.Foundations.Repetition
open scoped BigOperators
noncomputable section
variable {ι α β : Type*} [Fintype ι] [DecidableEq α] [DecidableEq β]
def revealedProductWeight (weight : ι → α → β → ℝ) (mask : ι → Bool)
(fixedLeft : ι → α) (fixedRight : ι → β) (x : ι → α) (y : ι → β) : ℝ :=
∏ i, weight i (x i) (y i) *
(if mask i then (if y i = fixedRight i then 1 else 0)
else (if x i = fixedLeft i then 1 else 0))
def revealedLeftWeight (weight : ι → α → β → ℝ) (mask : ι → Bool)
(fixedLeft : ι → α) (fixedRight : ι → β) (x : ι → α) : ℝ :=
∏ i, if mask i then weight i (x i) (fixedRight i)
else (if x i = fixedLeft i then 1 else 0)
def revealedRightWeight (weight : ι → α → β → ℝ) (mask : ι → Bool)
(fixedLeft : ι → α) (fixedRight : ι → β) (y : ι → β) : ℝ :=
∏ i, if mask i then (if y i = fixedRight i then 1 else 0)
else weight i (fixedLeft i) (y i)
theorem revealedProductWeight_factorizes (weight : ι → α → β → ℝ) (mask : ι → Bool)
(fixedLeft : ι → α) (fixedRight : ι → β) (x : ι → α) (y : ι → β) :
revealedProductWeight weight mask fixedLeft fixedRight x y =
revealedLeftWeight weight mask fixedLeft fixedRight x *
revealedRightWeight weight mask fixedLeft fixedRight y := by
unfold revealedProductWeight revealedLeftWeight revealedRightWeight
rw [← Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro i _
by_cases hm : mask i = true
· by_cases hy : y i = fixedRight i <;> simp [hm, hy]
· by_cases hx : x i = fixedLeft i <;> simp [hm, hx]
theorem revealedProductWeight_local_restrictions
(weight : ι → α → β → ℝ) (mask : ι → Bool)
(fixedLeft : ι → α) (fixedRight : ι → β)
(leftTest : (ι → α) → ℝ) (rightTest : (ι → β) → ℝ)
(x : ι → α) (y : ι → β) :
revealedProductWeight weight mask fixedLeft fixedRight x y * leftTest x * rightTest y =
(revealedLeftWeight weight mask fixedLeft fixedRight x * leftTest x) *
(revealedRightWeight weight mask fixedLeft fixedRight y * rightTest y) := by
rw [revealedProductWeight_factorizes]
ring
variable {X Y : Type*} [Fintype X] [Fintype Y]
theorem productWeight_mass (left : X → ℝ) (right : Y → ℝ) :
(∑ xy : X × Y, left xy.1 * right xy.2) =
(∑ x, left x) * (∑ y, right y) := by
rw [Fintype.sum_prod_type]
simp only [← Finset.mul_sum, ← Finset.sum_mul]
def normalizedWeight (weight : X → ℝ) : X → ℝ :=
fun x => weight x / ∑ x', weight x'
theorem normalizedWeight_isProbability (weight : X → ℝ)
(hweight : ∀ x, 0 ≤ weight x) (hmass : 0 < ∑ x, weight x) :
(∀ x, 0 ≤ normalizedWeight weight x) ∧ (∑ x, normalizedWeight weight x) = 1 := by
constructor
· intro x
exact div_nonneg (hweight x) hmass.le
· simp only [normalizedWeight, div_eq_mul_inv, ← Finset.sum_mul, mul_inv_cancel₀ hmass.ne']
theorem normalized_productWeight (left : X → ℝ) (right : Y → ℝ)
(hleft : 0 < ∑ x, left x) (hright : 0 < ∑ y, right y) (xy : X × Y) :
normalizedWeight (fun z : X × Y => left z.1 * right z.2) xy =
normalizedWeight left xy.1 * normalizedWeight right xy.2 := by
unfold normalizedWeight
rw [productWeight_mass]
field_simp
theorem normalized_rectangle_restriction
(left : X → ℝ) (right : Y → ℝ) (leftTest : X → ℝ) (rightTest : Y → ℝ)
(hleft : 0 < ∑ x, left x * leftTest x)
(hright : 0 < ∑ y, right y * rightTest y) (xy : X × Y) :
normalizedWeight
(fun z : X × Y => left z.1 * right z.2 * leftTest z.1 * rightTest z.2) xy =
normalizedWeight (fun x => left x * leftTest x) xy.1 *
normalizedWeight (fun y => right y * rightTest y) xy.2 := by
have hfun : (fun z : X × Y => left z.1 * right z.2 * leftTest z.1 * rightTest z.2) =
(fun z : X × Y => (left z.1 * leftTest z.1) * (right z.2 * rightTest z.2)) := by
funext z
ring
rw [hfun]
exact normalized_productWeight _ _ hleft hright xy
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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