File size: 3,431 Bytes
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 | import OAI.Computability.UniqueGames.Decoding.ActualProjectedTransfer
import OAI.Computability.UniqueGames.Decoding.ActualSeedSamplingLemmas
import OAI.Computability.UniqueGames.PCP.SourceIncidence
namespace OAI
section
/-! The contradiction uses the same actual seed and local decoder on both
sides. Its lower bound comes from full-table variation followed by visible
conditioning; its upper bound comes from the exact incidence sampler. -/
namespace UniqueGamesTheorem.Decoder.ActualMatrixContradiction
open Integration.BinaryLinear Reduction ActualSource Foundations.Games
open scoped BigOperators Classical
noncomputable section
attribute [local instance] Classical.propDecidable
attribute [local instance] Fintype.ofFinite
variable {k s d r : ℕ}
private theorem univ_fintype_eq {X : Type*} (a b : Fintype X) :
@Finset.univ X a = @Finset.univ X b :=
congrArg (fun t : Fintype X => @Finset.univ X t) (Subsingleton.elim a b)
private def expectationOf {X : Type*} {i : Fintype X}
(μ : @FiniteDistribution X i) (f : X → ℝ) : ℝ :=
@FiniteDistribution.expectation X i μ f
private theorem card_to_natCard (X : Type*) (i : Fintype X) :
@Fintype.card X i = Nat.card X :=
(@Nat.card_eq_fintype_card X i).symm
theorem law_expectation_eq (S : Source) (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1)
(f : ActualSeedEvents.Seed S k s d r → ℝ) :
expectationOf (ActualSeedEvents.law S k s d r β hβ hβ') f =
expectationOf (AdviceLaw.fullSeedLaw (V := Alphabet s × Vector d)
(FiniteDistribution.uniform (Fin k → Fin S.occurrences))
(FiniteDistribution.uniform (Alphabet s →ₗ[F2] Vector r)) β hβ hβ') f := by
simp only [expectationOf, FiniteDistribution.expectation, ActualSeedEvents.law,
AdviceLaw.fullSeedLaw, FiniteDistribution.product, FiniteDistribution.uniform]
theorem contradiction (S : Source)
(labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
(α g₀ β : ℝ) (hα : 0 < α)
(hsmall : 1 / (2 : ℝ) ^ (s - r) < α / 8)
(hβ : 0 ≤ β) (hβ' : β ≤ 1)
(hgood : g₀ ≤ ActualGoodRows.adviceMass S k s d r labeling α)
(hvariation : ActualProjectedTransfer.slopeVariation k s d β ≤
ConstantSelection.variationBudget α g₀ s r)
(hrate : (1 - (β / (2 : ℝ) ^ (d + r)) / 3600) ^ k <
ConstantSelection.decodingMass α g₀ s r)
(hdistinct : S.DistinctNames)
(hopt : Clean.IncidenceGap.parityValue (ActualAdviceUpper.sourceIncidence S)
(FiniteDistribution.uniform (Fin S.occurrences)) ≤ (4 : ℝ) / 5) : False := by
have hlower := ActualProjectedTransfer.actual_success_lower S labeling α g₀ β
hα hsmall hβ hβ' hgood hvariation
have hnames : ∀ o i j, (ActualAdviceUpper.sourceIncidence S).name o i =
(ActualAdviceUpper.sourceIncidence S).name o j → i = j :=
Outer.SourceIncidence.names_distinct S hdistinct
have hupper := ActualSeedSampling.fullSeedLaw_upper S labeling
(SelectedPolicies.goodPredicate S labeling α hα hsmall) β hβ hβ' hnames hopt
have hbound : ConstantSelection.decodingMass α g₀ s r ≤
(1 - (β / (2 : ℝ) ^ (d + r)) / 3600) ^ k := by
apply hlower.trans
change expectationOf (ActualSeedEvents.law S k s d r β hβ hβ') _ ≤ _
rw [law_expectation_eq]
exact hupper
exact (not_lt_of_ge hbound) hrate
end
end UniqueGamesTheorem.Decoder.ActualMatrixContradiction
end
end OAI
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