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2f887db | 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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | 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
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