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760f90e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.Computability.UniqueGames.Decoding.CoefficientSampling
import OAI.Computability.UniqueGames.Decoding.Experiment
namespace OAI
section
namespace UniqueGamesTheorem.Clean.NativeExperiment
open Foundations.Games
open Soundness
open Soundness.ConditionalIncidences Soundness.ConditionalGameLaw
open Experiment
open scoped BigOperators
noncomputable section
private theorem expectation_transport {X Y : Type*} [Fintype X] [Fintype Y]
(μ : FiniteDistribution X) (e : X ≃ Y) (f : Y → ℝ) :
(μ.transport e).expectation f = μ.expectation (fun x => f (e x)) := by
unfold FiniteDistribution.expectation FiniteDistribution.transport
exact Fintype.sum_equiv e.symm _ _ (fun _ => by simp)
theorem iid_product_split {X Y : Type*} [Fintype X] [Fintype Y]
(μ : FiniteDistribution X) (ν : FiniteDistribution Y) (k : ℕ) :
((μ.product ν).iid k).transport (Game.tupleQuestionEquiv k) =
(μ.iid k).product (ν.iid k) := by
apply FiniteDistribution.eq_of_weight_eq
intro pair
change (∏ i, μ.weight (pair.1 i) * ν.weight (pair.2 i)) =
(∏ i, μ.weight (pair.1 i)) * ∏ i, ν.weight (pair.2 i)
exact Finset.prod_mul_distrib
theorem expectation_iid_product {X Y : Type*} [Fintype X] [Fintype Y]
(μ : FiniteDistribution X) (ν : FiniteDistribution Y) (k : ℕ)
(f : (Fin k → X × Y) → ℝ) :
((μ.product ν).iid k).expectation f =
(μ.iid k).expectation (fun x =>
(ν.iid k).expectation (fun y => f (fun i => (x i, y i)))) := by
have law := expectation_transport ((μ.product ν).iid k)
(Game.tupleQuestionEquiv k) (fun pair => f (fun i => (pair.1 i, pair.2 i)))
rw [iid_product_split, FiniteDistribution.expectation_product] at law
exact law.symm
variable {R O N : Type} [Fintype R] [DecidableEq R]
[Fintype O] [DecidableEq O] [Fintype N] [DecidableEq N]
def maskSet {k : ℕ} (mask : Fin k → Bool) : Finset (Fin k) :=
Finset.univ.filter fun i => mask i = true
abbrev MaskStrategy (k : ℕ) (R O N : Type) :=
(mask : Fin k → Bool) → RawCoefficients (maskSet mask) (Coefficient R) →
LocalStrategies k R O N
def liftStrategy {k : ℕ} (strategy : MaskStrategy k R O N)
(extra : Additional k R) (seed : SparseSeed k R) : LocalStrategies k R O N :=
strategy (fun i => (seed i).1) (coefficients extra seed)
/-- The actual law: independent singleton choices followed by one independent
uniform coefficient at the intercept and each surviving partner coordinate. -/
def success (k : ℕ) (μ : FiniteDistribution (O × Fin 3))
(g : IncidenceExtraction.Incidence O N)
(β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
(strategy : MaskStrategy k R O N) : ℝ :=
((bernoulli β hβ₀ hβ₁).iid k).expectation fun mask =>
(FiniteDistribution.uniform (RawCoefficients (maskSet mask) (Coefficient R))).expectation
fun gamma => UpperBound.fixedAdviceSuccess μ (maskSet mask) g gamma (strategy mask gamma)
omit [DecidableEq O] [Fintype N] [DecidableEq N] in
theorem success_eq_flat (k : ℕ) (μ : FiniteDistribution (O × Fin 3))
(g : IncidenceExtraction.Incidence O N)
(β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
(strategy : MaskStrategy k R O N) :
success k μ g β hβ₀ hβ₁ strategy =
Experiment.success k μ g (FiniteDistribution.uniform (Additional k R))
β hβ₀ hβ₁ (liftStrategy strategy) := by
unfold success Experiment.success singletonSlopeLaw
symm
calc
_ = (FiniteDistribution.uniform (Additional k R)).expectation fun extra =>
((bernoulli β hβ₀ hβ₁).iid k).expectation fun mask =>
(FiniteDistribution.uniform (Fin k → Coefficient R)).expectation fun single =>
UpperBound.fixedAdviceSuccess μ (maskSet mask) g
(CoefficientSampling.selectCoefficients (maskSet mask) extra single)
(strategy mask (CoefficientSampling.selectCoefficients (maskSet mask) extra single)) := by
apply FiniteDistribution.expectation_congr
intro extra
rw [expectation_iid_product, FiniteDistribution.iid_uniform]
apply FiniteDistribution.expectation_congr
intro mask
apply FiniteDistribution.expectation_congr
intro single
have hgamma : coefficients extra (fun i => (mask i, single i)) =
CoefficientSampling.selectCoefficients (maskSet mask) extra single := by
funext slot
cases slot with
| none => rfl
| some slot =>
cases slot with
| inl pair => rfl
| inr i => rfl
change UpperBound.fixedAdviceSuccess μ (maskSet mask) g
(coefficients extra (fun i => (mask i, single i)))
(strategy mask (coefficients extra (fun i => (mask i, single i)))) = _
rw [hgamma]
_ = ((bernoulli β hβ₀ hβ₁).iid k).expectation (fun mask =>
(FiniteDistribution.uniform (Additional k R)).expectation fun extra =>
(FiniteDistribution.uniform (Fin k → Coefficient R)).expectation fun single =>
UpperBound.fixedAdviceSuccess μ (maskSet mask) g
(CoefficientSampling.selectCoefficients (maskSet mask) extra single)
(strategy mask (CoefficientSampling.selectCoefficients (maskSet mask) extra single))) :=
FiniteDistribution.expectation_comm _ _ _
_ = _ := by
apply FiniteDistribution.expectation_congr
intro mask
exact CoefficientSampling.uniform_selectCoefficients (maskSet mask)
(fun gamma => UpperBound.fixedAdviceSuccess μ (maskSet mask) g gamma (strategy mask gamma))
end
end UniqueGamesTheorem.Clean.NativeExperiment
end
end OAI
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