openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /BucketResamplingLaw.lean
| import OAI.Computability.PerfectCompleteness.Sampling.BucketSampler | |
| import OAI.Computability.PerfectCompleteness.Sampling.UniformDifference | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.BucketResamplingLaw | |
| noncomputable section | |
| open scoped Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| open BucketSampler (F2 Direction Tape) | |
| variable {H : Type*} [AddCommGroup H] [Module F2 H] [Fintype H] | |
| private theorem pushforward_id {A : Type*} [Fintype A] (μ : FiniteDistribution A) : | |
| μ.pushforward (fun a => a) = μ := by | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro a | |
| simp [FiniteDistribution.pushforward] | |
| theorem assembled_difference_law (ℓ : Nat) (μ : FiniteDistribution (Tape ℓ H)) | |
| (a : Direction ℓ) : | |
| (μ.product (FiniteDistribution.uniform H)).pushforward | |
| (fun p => (BucketSampler.evaluate ℓ id p.1, p.2 - p.1 a)) = | |
| (μ.pushforward (BucketSampler.evaluate ℓ id)).product (FiniteDistribution.uniform H) := by | |
| rw [UniformDifference.observed_difference_law μ (fun t => t a) | |
| (BucketSampler.evaluate ℓ id)] | |
| have hp := FiniteDistribution.product_pushforward μ (FiniteDistribution.uniform H) | |
| (BucketSampler.evaluate ℓ (id : H → H)) (id : H → H) | |
| have hi : (FiniteDistribution.uniform H).pushforward (id : H → H) = | |
| FiniteDistribution.uniform H := pushforward_id (FiniteDistribution.uniform H) | |
| rw [hi] at hp | |
| simpa only [id_eq] using hp | |
| theorem resampled_array_pair_law (ℓ : Nat) (μ : FiniteDistribution (Tape ℓ H)) | |
| (a : Direction ℓ) : | |
| (μ.product (FiniteDistribution.uniform H)).pushforward | |
| (fun p => (BucketSampler.evaluate ℓ id p.1, | |
| BucketSampler.evaluate ℓ id (Function.update p.1 a p.2))) = | |
| ((μ.pushforward (BucketSampler.evaluate ℓ id)).product | |
| (FiniteDistribution.uniform H)).pushforward | |
| (fun p => (p.1, p.1 + BucketSampler.rankOne a.val p.2)) := by | |
| have h := congrArg | |
| (fun ν : FiniteDistribution ((Fin ℓ → H) × H) => ν.pushforward | |
| (fun p => (p.1, p.1 + BucketSampler.rankOne a.val p.2))) | |
| (assembled_difference_law ℓ μ a) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| simpa only [BucketSampler.evaluate_update, id_eq] using h | |
| theorem resampled_event_probability (ℓ : Nat) (μ : FiniteDistribution (Tape ℓ H)) | |
| (a : Direction ℓ) (event : (Fin ℓ → H) × (Fin ℓ → H) → Bool) : | |
| (μ.product (FiniteDistribution.uniform H)).probability | |
| (fun p => event (BucketSampler.evaluate ℓ id p.1, | |
| BucketSampler.evaluate ℓ id (Function.update p.1 a p.2))) = | |
| ((μ.pushforward (BucketSampler.evaluate ℓ id)).product | |
| (FiniteDistribution.uniform H)).probability | |
| (fun p => event (p.1, p.1 + BucketSampler.rankOne a.val p.2)) := by | |
| have h := congrArg (fun ν => FiniteDistribution.probability ν event) | |
| (resampled_array_pair_law ℓ μ a) | |
| simpa only [FiniteDistribution.probability_pushforward] using h | |
| end | |
| end PerfectCompleteness.BucketResamplingLaw | |
| end | |
| end OAI | |