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ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | 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
end OAI
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