| import Mathlib |
| import OAI.Computability.MinUncut.PCP.SelectionSequence |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Outer |
| open MinUncut.Inner MinUncutGames.Foundations.Games |
| attribute [local instance] Classical.propDecidable |
| variable {Name S : Type*} [Fintype Name] [Fintype S] [Nonempty S] |
|
|
| lemma ordinary_repetition_formula (equations : S → Equation Name) (r : ℕ) |
| (strategy : Strategy (Fin r → Equation Name) (Fin r → Name) |
| (Fin r → Triple) (Fin r → F₂)) : |
| ((ordinaryGame equations).repetition r).success strategy = |
| 𝔼 w : Fin r → S × Fin 3, |
| if (∀ j, ordinaryAccepts (equations (w j).1) |
| ((equations (w j).1).names (w j).2) |
| (strategy.1 (fun i => equations (w i).1) j) |
| (strategy.2 (fun i => (equations (w i).1).names (w i).2) j)=true) |
| then (1 : ℝ) else 0 := by |
| unfold Game.success |
| change (((ordinaryGame equations).questions.iid r).transport |
| (Game.tupleQuestionEquiv r)).probability _ = _ |
| rw [FiniteDistribution.probability_transport] |
| change (((FiniteDistribution.uniform (S × Fin 3)).pushforward |
| (fun ip => (equations ip.1,(equations ip.1).names ip.2))).iid r).probability _ = _ |
| rw [FiniteDistribution.iid_pushforward,FiniteDistribution.iid_uniform, |
| FiniteDistribution.probability_pushforward] |
| change (∑ w : Fin r → S × Fin 3, |
| if decide (∀ j, ordinaryAccepts (equations (w j).1) |
| ((equations (w j).1).names (w j).2) |
| (strategy.1 (fun i => equations (w i).1) j) |
| (strategy.2 (fun i => (equations (w i).1).names (w i).2) j)=true) |
| then (1 : ℝ) / Fintype.card (Fin r → S × Fin 3) else 0) = _ |
| simp only [decide_eq_true_eq,Fintype.expect_eq_sum_div_card,Finset.sum_div,ite_div,zero_div] |
|
|
| theorem ordinary_repetition_on (equations : S → Equation Name) |
| (hsound : ∀ s : Name → F₂, equationFraction equations s ≤ 3/4) |
| {J : Type*} [Fintype J] |
| (a : (J → Equation Name) → J → Triple) (b : (J → Name) → J → F₂) : |
| (𝔼 w : J → S × Fin 3, |
| if (∀ j, ordinaryAccepts (equations (w j).1) |
| ((equations (w j).1).names (w j).2) |
| (a (fun i => equations (w i).1) j) |
| (b (fun i => (equations (w i).1).names (w i).2) j)=true) |
| then (1 : ℝ) else 0) ≤ repetitionRate ^ Fintype.card J := by |
| classical |
| let e : Fin (Fintype.card J) ≃ J := (Fintype.equivFin J).symm |
| let strategy : Strategy (Fin (Fintype.card J) → Equation Name) |
| (Fin (Fintype.card J) → Name) (Fin (Fintype.card J) → Triple) |
| (Fin (Fintype.card J) → F₂) := |
| (fun x i => a (fun j => x (e.symm j)) (e i), |
| fun y i => b (fun j => y (e.symm j)) (e i)) |
| have h := ordinary_repetition_success equations hsound (Fintype.card J) strategy |
| rw [ordinary_repetition_formula] at h |
| convert h using 1 |
| let ew : (J → S × Fin 3) ≃ (Fin (Fintype.card J) → S × Fin 3) := |
| Equiv.arrowCongr e.symm (Equiv.refl _) |
| apply Fintype.expect_equiv ew |
| intro w |
| simp only [ew,Equiv.arrowCongr_apply,Equiv.coe_refl,Equiv.symm_symm, |
| strategy,Function.comp_apply,id_eq,Equiv.apply_symm_apply] |
| congr 1 |
| exact propext e.forall_congr_right.symm |
|
|
| end MinUncut.Outer |
|
|
| end |
| end |
|
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| end OAI |
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|
