openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /DependentDecoderSampling.lean
| import OAI.Computability.PerfectCompleteness.Decoding.TwoResponseCollisionLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.CompletionSoundness | |
| import OAI.Computability.PerfectCompleteness.Sampling.RecursiveSamplerLaws | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.DependentDecoderSampling | |
| noncomputable section | |
| open scoped BigOperators Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {Q₁ Q₂ E : Type*} [Fintype Q₁] [Fintype Q₂] [Fintype E] | |
| [DecidableEq Q₁] [DecidableEq Q₂] | |
| {A : Q₁ → Type*} {B : Q₂ → Type*} | |
| [∀ q, Fintype (A q)] [∀ q, Fintype (B q)] | |
| abbrev Seed (A : Q₁ → Type*) (B : Q₂ → Type*) := | |
| ((q : Q₁) → A q) × ((q : Q₂) → B q) | |
| def seedLaw (leftKernel : (q : Q₁) → FiniteDistribution (A q)) | |
| (rightKernel : (q : Q₂) → FiniteDistribution (B q)) : | |
| FiniteDistribution (Seed A B) := | |
| (FiniteProduct.law leftKernel).product (FiniteProduct.law rightKernel) | |
| theorem read_law (leftKernel : (q : Q₁) → FiniteDistribution (A q)) | |
| (rightKernel : (q : Q₂) → FiniteDistribution (B q)) (q₁ : Q₁) (q₂ : Q₂) : | |
| (seedLaw leftKernel rightKernel).pushforward (fun seed => (seed.1 q₁, seed.2 q₂)) = | |
| (leftKernel q₁).product (rightKernel q₂) := by | |
| have h := FiniteDistribution.product_pushforward | |
| (FiniteProduct.law leftKernel) (FiniteProduct.law rightKernel) | |
| (fun table : (q : Q₁) → A q => table q₁) | |
| (fun table : (q : Q₂) → B q => table q₂) | |
| rw [FiniteProduct.eval_pushforward, FiniteProduct.eval_pushforward] at h | |
| exact h | |
| def sampleLaw (μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂) | |
| (leftKernel : (q : Q₁) → FiniteDistribution (A q)) | |
| (rightKernel : (q : Q₂) → FiniteDistribution (B q)) : | |
| FiniteDistribution (Σ e : E, A (left e) × B (right e)) := | |
| CompletionSoundness.sigmaLaw μ (fun e => (leftKernel (left e)).product (rightKernel (right e))) | |
| theorem sample_probability_eq (μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂) | |
| (leftKernel : (q : Q₁) → FiniteDistribution (A q)) | |
| (rightKernel : (q : Q₂) → FiniteDistribution (B q)) | |
| (event : (e : E) → A (left e) → B (right e) → Bool) : | |
| (sampleLaw μ left right leftKernel rightKernel).probability | |
| (fun x => event x.1 x.2.1 x.2.2) = | |
| (seedLaw leftKernel rightKernel).expectation | |
| (fun seed => μ.probability (fun e => event e (seed.1 (left e)) (seed.2 (right e)))) := by | |
| rw [sampleLaw, CompletionSoundness.sigmaLaw_probability] | |
| calc | |
| _ = μ.expectation (fun e => (seedLaw leftKernel rightKernel).probability | |
| (fun seed => event e (seed.1 (left e)) (seed.2 (right e)))) := by | |
| apply FiniteDistribution.expectation_congr | |
| intro e | |
| have h := FiniteDistribution.probability_pushforward | |
| (seedLaw leftKernel rightKernel) (fun seed => (seed.1 (left e), seed.2 (right e))) | |
| (fun answer => event e answer.1 answer.2) | |
| rw [read_law] at h | |
| exact h | |
| _ = μ.expectation (fun e => (seedLaw leftKernel rightKernel).expectation | |
| (fun seed => if event e (seed.1 (left e)) (seed.2 (right e)) then 1 else 0)) := by | |
| simp only [FiniteDistribution.probability, FiniteDistribution.expectation, | |
| mul_ite, mul_one, mul_zero] | |
| _ = (seedLaw leftKernel rightKernel).expectation (fun seed => μ.expectation | |
| (fun e => if event e (seed.1 (left e)) (seed.2 (right e)) then 1 else 0)) := | |
| FiniteDistribution.expectation_comm μ (seedLaw leftKernel rightKernel) _ | |
| _ = _ := by | |
| simp only [FiniteDistribution.probability, FiniteDistribution.expectation, | |
| mul_ite, mul_one, mul_zero] | |
| theorem probability_le_of_supported_seeds | |
| (μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂) | |
| (leftKernel : (q : Q₁) → FiniteDistribution (A q)) | |
| (rightKernel : (q : Q₂) → FiniteDistribution (B q)) | |
| (event : (e : E) → A (left e) → B (right e) → Bool) (bound : ℝ) | |
| (hbound : ∀ seed, (seedLaw leftKernel rightKernel).weight seed ≠ 0 → | |
| μ.probability (fun e => event e (seed.1 (left e)) (seed.2 (right e))) ≤ bound) : | |
| (sampleLaw μ left right leftKernel rightKernel).probability | |
| (fun x => event x.1 x.2.1 x.2.2) ≤ bound := by | |
| rw [sample_probability_eq] | |
| exact DecoderTableAdmissibility.expectation_le_of_support _ _ bound hbound | |
| theorem supported_left (leftKernel : (q : Q₁) → FiniteDistribution (A q)) | |
| (rightKernel : (q : Q₂) → FiniteDistribution (B q)) | |
| (seed : Seed A B) (hs : (seedLaw leftKernel rightKernel).weight seed ≠ 0) (q : Q₁) : | |
| (leftKernel q).weight (seed.1 q) ≠ 0 := by | |
| intro hq | |
| apply hs | |
| have hz : (FiniteProduct.law leftKernel).weight seed.1 = 0 := by | |
| exact Finset.prod_eq_zero (Finset.mem_univ q) hq | |
| change (FiniteProduct.law leftKernel).weight seed.1 * | |
| (FiniteProduct.law rightKernel).weight seed.2 = 0 | |
| rw [hz, zero_mul] | |
| theorem supported_right (leftKernel : (q : Q₁) → FiniteDistribution (A q)) | |
| (rightKernel : (q : Q₂) → FiniteDistribution (B q)) | |
| (seed : Seed A B) (hs : (seedLaw leftKernel rightKernel).weight seed ≠ 0) (q : Q₂) : | |
| (rightKernel q).weight (seed.2 q) ≠ 0 := by | |
| intro hq | |
| apply hs | |
| have hz : (FiniteProduct.law rightKernel).weight seed.2 = 0 := by | |
| exact Finset.prod_eq_zero (Finset.mem_univ q) hq | |
| change (FiniteProduct.law leftKernel).weight seed.1 * | |
| (FiniteProduct.law rightKernel).weight seed.2 = 0 | |
| rw [hz, mul_zero] | |
| end | |
| end PerfectCompleteness.DependentDecoderSampling | |
| end | |
| end OAI | |