| import OAI.Probability.EntangledGames.RepeatedModel |
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| namespace OAI |
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| universe u_I u_Ω |
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| noncomputable section |
| open scoped BigOperators |
| namespace ThresholdParallelRepetition.FiniteProbability |
| namespace Tests |
| open Law |
| variable {I : Type u_I} {Ω : Type u_Ω} [Fintype I] [DecidableEq I] [Nonempty I] [Fintype Ω] |
| (ρ : Law Ω) (win : Ω → I → Bool) |
| def event (c : Finset I) (ω : Ω) : ℝ := if ∀ i ∈ c, win ω i then 1 else 0 |
| def mass (c : Finset I) : ℝ := ρ.avg (event win c) |
| def rate (ω : Ω) : ℝ := (uniform (R := I)).avg (fun i => if win ω i then 1 else 0) |
| def moment (c : Finset I) (t : ℕ) : ℝ := ρ.avg (fun ω => event win c ω*(rate win ω)^t) |
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| omit [Nonempty I] [Fintype Ω] in |
| lemma event_nonneg (c : Finset I) (ω : Ω) : 0 ≤ event win c ω := by unfold event; split_ifs <;> norm_num |
| omit [Nonempty I] [Fintype Ω] in |
| lemma event_le_one (c : Finset I) (ω : Ω) : event win c ω ≤ 1 := by unfold event; split_ifs <;> norm_num |
| omit [Nonempty I] in |
| lemma mass_nonneg (c : Finset I) : 0 ≤ mass ρ win c := ρ.avg_nonneg (event_nonneg win c) |
| omit [Nonempty I] in |
| lemma mass_le_one (c : Finset I) : mass ρ win c ≤ 1 := ρ.avg_le_const (event_le_one win c) |
| omit [DecidableEq I] [Fintype Ω] in |
| lemma rate_nonneg (ω : Ω) : 0 ≤ rate win ω := by |
| apply avg_nonneg; intro i; split_ifs <;> norm_num |
| omit [DecidableEq I] [Fintype Ω] in |
| lemma rate_le_one (ω : Ω) : rate win ω ≤ 1 := by |
| apply avg_le_const; intro i; split_ifs <;> norm_num |
| omit [Nonempty I] [Fintype Ω] in |
| lemma event_empty (ω : Ω) : event win ∅ ω = 1 := by simp [event] |
| omit [Nonempty I] in |
| lemma mass_empty : mass ρ win ∅ = 1 := by |
| unfold mass |
| conv_lhs => arg 2; ext ω; rw [event_empty] |
| exact avg_const _ 1 |
| omit [Nonempty I] [Fintype Ω] in |
| lemma event_insert (c : Finset I) (i : I) (ω : Ω) : |
| event win (insert i c) ω = event win c ω*(if win ω i then 1 else 0) := by |
| simp only [event, Finset.forall_mem_insert] |
| split_ifs <;> simp_all |
| omit [Fintype Ω] in |
| lemma avg_event_insert (c : Finset I) (ω : Ω) : |
| (uniform (R := I)).avg (fun i => event win (insert i c) ω) = event win c ω*rate win ω := by |
| simp only [event_insert] |
| change (uniform (R := I)).avg (fun i => event win c ω • (if win ω i then (1:ℝ) else 0)) = _ |
| rw [avg_smul]; rfl |
| lemma moment_zero (c : Finset I) : moment ρ win c 0 = mass ρ win c := by simp only [moment, pow_zero, mul_one, mass] |
| lemma moment_step (c : Finset I) (t : ℕ) : |
| moment ρ win c (t+1) = (uniform (R := I)).avg (fun i => moment ρ win (insert i c) t) := by |
| unfold moment |
| rw [avg_comm] |
| apply congrArg ρ.avg |
| funext ω |
| have he : (fun i => event win (insert i c) ω * rate win ω^t) = |
| (fun i => (rate win ω^t) • event win (insert i c) ω) := by funext i; simp only [smul_eq_mul, mul_comm] |
| rw [he, avg_smul, avg_event_insert, pow_succ] |
| change _ = _*_ |
| ring |
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| lemma moment_upper {r e : ℝ} (hr : 0 ≤ r) (_he : 0 ≤ e) (T : ℕ) |
| (hstep : ∀ c : Finset I, c.card < T → |
| (uniform (R := I)).avg (fun i => mass ρ win (insert i c)) ≤ r*mass ρ win c+e) : |
| ∀ t : ℕ, ∀ c : Finset I, c.card+t ≤ T → |
| moment ρ win c t ≤ r^t*mass ρ win c + e*∑ s ∈ Finset.range t, r^s := by |
| intro t |
| induction t with |
| | zero => intro c hc; simp only [moment_zero, pow_zero, one_mul, Finset.range_zero, Finset.sum_empty, mul_zero, add_zero]; exact le_rfl |
| | succ t ih => |
| intro c hc |
| rw [moment_step] |
| have hbound := (uniform (R := I)).avg_mono (fun i => ih (insert i c) (by |
| have hi := Finset.card_insert_le i c |
| omega)) |
| have heq : (uniform (R := I)).avg (fun i => r^t*mass ρ win (insert i c)+e*∑ s ∈ Finset.range t, r^s) = |
| r^t*(uniform (R := I)).avg (fun i => mass ρ win (insert i c))+e*∑ s ∈ Finset.range t, r^s := by |
| rw [avg_add, avg_const] |
| change (uniform (R := I)).avg (fun i => (r^t) • mass ρ win (insert i c)) + _ = _ |
| rw [avg_smul]; rfl |
| rw [heq] at hbound |
| have hs := mul_le_mul_of_nonneg_left (hstep c (by omega)) (pow_nonneg hr t) |
| rw [Finset.sum_range_succ, pow_succ] |
| nlinarith |
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| lemma threshold_moment_lower {q : ℝ} (hq : 0 ≤ q) (t : ℕ) : |
| (ρ.avg (fun ω => if q ≤ rate win ω then (1:ℝ) else 0))*q^t ≤ moment ρ win ∅ t := by |
| have he : (ρ.avg (fun ω => if q ≤ rate win ω then (1:ℝ) else 0))*q^t = |
| ρ.avg (fun ω => if q ≤ rate win ω then q^t else 0) := by |
| rw [mul_comm] |
| change (q^t) • ρ.avg (fun ω => if q ≤ rate win ω then (1:ℝ) else 0) = _ |
| rw [← avg_smul] |
| congr 1; funext ω; split_ifs <;> simp |
| rw [he] |
| apply ρ.avg_mono |
| intro ω |
| simp only [event_empty, one_mul] |
| split_ifs with h |
| · exact pow_le_pow_left₀ hq h _ |
| · exact pow_nonneg (rate_nonneg win ω) _ |
| end Tests |
| end ThresholdParallelRepetition.FiniteProbability |
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| end |
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| end OAI |
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