openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /RecursiveSamplerLaws.lean
| import OAI.Computability.PerfectCompleteness.Algebra.UniformLinearImage | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FiniteProduct | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| {Ω : I → Type*} [∀ i, Fintype (Ω i)] | |
| theorem eval_pushforward (P : (i : I) → FiniteDistribution (Ω i)) (i : I) : | |
| (law P).pushforward (fun x => x i) = P i := by | |
| classical | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro a | |
| calc | |
| _ = (law P).expectation (fun x => if x i = a then 1 else 0) := by | |
| simp [FiniteDistribution.pushforward, FiniteDistribution.expectation, mul_ite] | |
| _ = (P i).expectation (fun b => if b = a then 1 else 0) := | |
| expectation_eval P i (fun b => if b = a then (1 : ℝ) else 0) | |
| _ = _ := by simp [FiniteDistribution.expectation, mul_ite] | |
| end | |
| end PerfectCompleteness.FiniteProduct | |
| end | |
| end OAI | |