| import Mathlib |
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| namespace OAI |
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| namespace UniqueGames.Foundations.CorrelatedSampling |
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| noncomputable section |
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| variable {σ : Type*} [Fintype σ] |
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| def eventMass (w : σ → ℝ) (event : σ → Bool) : ℝ := |
| ∑ s, if event s then w s else 0 |
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| 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 |
|
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| def geometricMass (r : ℝ) : Nat → ℝ |
| | 0 => 0 |
| | n + 1 => 1 + r * geometricMass r n |
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| 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) |
|
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| 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 |
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|
| end |
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| end UniqueGames.Foundations.CorrelatedSampling |
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| end |
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| end |
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| end OAI |
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