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namespace OAI
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namespace UniqueGames.Foundations.CorrelatedSampling
noncomputable section
variable {σ : Type*} [Fintype σ]
def eventMass (w : σ → ℝ) (event : σ → Bool) : ℝ :=
∑ s, if event s then w s else 0
theorem eventMass_nonneg (w : σ → ℝ) (event : σ → Bool) (hw : ∀ s, 0 ≤ w s) :
0 ≤ eventMass w event := by
apply Finset.sum_nonneg
intro s _
split <;> simp_all
theorem eventMass_complement (w : σ → ℝ) (event : σ → Bool)
(hw : ∑ s, w s = 1) :
eventMass w event + eventMass w (fun s => !(event s)) = 1 := by
unfold eventMass
rw [← Finset.sum_add_distrib]
calc
∑ s, ((if event s then w s else 0) + (if !event s then w s else 0)) = ∑ s, w s := by
apply Finset.sum_congr rfl
intro s _
cases event s <;> simp
_ = 1 := hw
def geometricMass (r : ℝ) : Nat → ℝ
| 0 => 0
| n + 1 => 1 + r * geometricMass r n
theorem geometricMass_nonneg (r : ℝ) (hr : 0 ≤ r) (n : Nat) :
0 ≤ geometricMass r n := by
induction n with
| zero => simp [geometricMass]
| succ n ih => exact add_nonneg (by norm_num) (mul_nonneg hr ih)
theorem geometricMass_identity (r : ℝ) (n : Nat) :
(1 - r) * geometricMass r n = 1 - r ^ n := by
induction n with
| zero => simp [geometricMass]
| succ n ih =>
simp only [geometricMass, pow_succ]
calc
(1 - r) * (1 + r * geometricMass r n) =
(1 - r) + r * ((1 - r) * geometricMass r n) := by ring
_ = 1 - r ^ n * r := by rw [ih]; ring
def firstHitWeight (w : σ → ℝ) (accept : σ → Bool) : Nat → Option σ → ℝ
| 0, none => 1
| 0, some _ => 0
| n + 1, none => eventMass w (fun s => !(accept s)) * firstHitWeight w accept n none
| n + 1, some s =>
(if accept s then w s else 0) +
eventMass w (fun s => !(accept s)) * firstHitWeight w accept n (some s)
theorem firstHitWeight_nonneg (w : σ → ℝ) (accept : σ → Bool)
(hw : ∀ s, 0 ≤ w s) (n : Nat) (output : Option σ) :
0 ≤ firstHitWeight w accept n output := by
induction n generalizing output with
| zero => cases output <;> simp [firstHitWeight]
| succ n ih =>
have hr := eventMass_nonneg w (fun s => !(accept s)) hw
cases output with
| none => exact mul_nonneg hr (ih none)
| some s =>
apply add_nonneg
· split <;> simp_all
· exact mul_nonneg hr (ih (some s))
theorem firstHitWeight_none (w : σ → ℝ) (accept : σ → Bool) (n : Nat) :
firstHitWeight w accept n none = eventMass w (fun s => !(accept s)) ^ n := by
induction n with
| zero => simp [firstHitWeight]
| succ n ih => simp [firstHitWeight, ih, pow_succ, mul_comm]
theorem firstHitWeight_some (w : σ → ℝ) (accept : σ → Bool) (n : Nat) (s : σ) :
firstHitWeight w accept n (some s) =
(if accept s then w s else 0) *
geometricMass (eventMass w (fun s => !(accept s))) n := by
induction n with
| zero => simp [firstHitWeight, geometricMass]
| succ n ih =>
rw [firstHitWeight, geometricMass, ih]
ring
theorem firstHitWeight_normalized (w : σ → ℝ) (accept : σ → Bool)
(hw : ∑ s, w s = 1) (n : Nat) : ∑ output, firstHitWeight w accept n output = 1 := by
rw [Fintype.sum_option, firstHitWeight_none]
simp_rw [firstHitWeight_some]
rw [← Finset.sum_mul]
change eventMass w (fun s => !(accept s)) ^ n +
eventMass w accept * geometricMass (eventMass w (fun s => !(accept s))) n = 1
have hc := eventMass_complement w accept hw
have hg := geometricMass_identity (eventMass w (fun s => !(accept s))) n
have ha : eventMass w accept = 1 - eventMass w (fun s => !(accept s)) := by linarith
rw [ha, hg]
ring
def goodFirstMass (w : σ → ℝ) (accept good : σ → Bool) (n : Nat) : ℝ :=
∑ s, if good s then firstHitWeight w accept n (some s) else 0
theorem goodFirstMass_eq (w : σ → ℝ) (accept good : σ → Bool) (n : Nat) :
goodFirstMass w accept good n =
eventMass w (fun s => accept s && good s) *
geometricMass (eventMass w (fun s => !(accept s))) n := by
unfold goodFirstMass eventMass
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro s _
rw [firstHitWeight_some]
cases ha : accept s <;> cases hg : good s <;> simp [ha, hg, eventMass]
theorem goodFirstMass_exact (w : σ → ℝ) (accept good : σ → Bool)
(hw : ∑ s, w s = 1) (ha : eventMass w accept ≠ 0) (n : Nat) :
goodFirstMass w accept good n =
eventMass w (fun s => accept s && good s) / eventMass w accept *
(1 - eventMass w (fun s => !(accept s)) ^ n) := by
rw [goodFirstMass_eq]
have hc := eventMass_complement w accept hw
have hg := geometricMass_identity (eventMass w (fun s => !(accept s))) n
have hgeom : eventMass w accept *
geometricMass (eventMass w (fun s => !(accept s))) n =
1 - eventMass w (fun s => !(accept s)) ^ n := by
have heq : eventMass w accept = 1 - eventMass w (fun s => !(accept s)) := by linarith
rw [heq, hg]
rw [← hgeom]
field_simp
theorem commonMass_le_acceptMass (w : σ → ℝ) (accept good : σ → Bool)
(hw : ∀ s, 0 ≤ w s) :
eventMass w (fun s => accept s && good s) ≤ eventMass w accept := by
apply Finset.sum_le_sum
intro s _
cases ha : accept s <;> cases hg : good s <;> simp [ha, hg, hw s]
theorem goodFirstMass_lower_bound (w : σ → ℝ) (accept good : σ → Bool)
(hw : ∀ s, 0 ≤ w s) (hw_sum : ∑ s, w s = 1)
(ha : 0 < eventMass w accept) (n : Nat) :
eventMass w (fun s => accept s && good s) / eventMass w accept -
eventMass w (fun s => !(accept s)) ^ n ≤ goodFirstMass w accept good n := by
rw [goodFirstMass_exact w accept good hw_sum (ne_of_gt ha) n]
have hratio : eventMass w (fun s => accept s && good s) / eventMass w accept ≤ 1 := by
apply (div_le_one ha).2
exact commonMass_le_acceptMass w accept good hw
have htail : 0 ≤ eventMass w (fun s => !(accept s)) ^ n :=
pow_nonneg (eventMass_nonneg w (fun s => !(accept s)) hw) n
nlinarith
end
end UniqueGames.Foundations.CorrelatedSampling
end
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end
end OAI
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