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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 | import OAI.Computability.PerfectCompleteness.Algebra.CanonicalMatrixTable
import OAI.Computability.PerfectCompleteness.Decoding.TwoResponseCollisionLemmas
namespace OAI
section
namespace PerfectCompleteness.CanonicalBucketCollision
noncomputable section
open scoped Classical
open ClauseSupport MixedSupport CanonicalKeys
open DirectionQuotient (F2)
open CanonicalMatrixTable TwoResponseCollision
open UniqueGamesTheorem.Foundations.Games
variable {n : Nat} {K Z : Type*} [AddCommGroup K] [Module F2 K]
(slots : Fin n → Slot) (H : Submodule F2 (Assignment slots → F2))
(other : Assignment slots → Z) (σ : KeyStrategy.Strategy n)
def bucketTable (useful : Matrix (K := K) slots H → Prop) (a : K) (X : Matrix (K := K) slots H)
(h : H) : Option (K × Z) :=
partialTable slots H other σ useful (EvaluationMatrix.shift H X a h)
def bucketAccepted (a : K) (X : Matrix (K := K) slots H) (h : H) : Bool :=
decide (Accepts slots H other σ a (EvaluationMatrix.shift H X a h))
theorem bucketTable_eq_some_iff (useful : Matrix (K := K) slots H → Prop)
(a : K) (X : Matrix (K := K) slots H) (h : H) (y : K × Z) :
bucketTable slots H other σ useful a X h = some y ↔
useful (EvaluationMatrix.shift H X a h) ∧
response slots H other σ (EvaluationMatrix.shift H X a h) = y := by
by_cases hu : useful (EvaluationMatrix.shift H X a h) <;>
simp [bucketTable, partialTable, hu]
theorem collision_probability_lower_bound [Finite K] [Finite Z] [Fintype H]
(μ : FiniteDistribution H) (useful : Matrix (K := K) slots H → Prop)
(a : K) (ha : a ≠ 0) (X : Matrix (K := K) slots H) :
μ.probability (goodDefined (bucketTable slots H other σ useful a X)
(bucketAccepted slots H other σ a X)) ^ 2 / 2 ≤
(μ.product μ).probability (collision (bucketTable slots H other σ useful a X)) := by
let : Fintype K := Fintype.ofFinite K
let : Fintype Z := Fintype.ofFinite Z
apply collision_probability_ge_of_response_set μ
(bucketTable slots H other σ useful a X) (bucketAccepted slots H other σ a X)
(fun y => ∃ h : H,
Accepts slots H other σ a (EvaluationMatrix.shift H X a h) ∧
response slots H other σ (EvaluationMatrix.shift H X a h) = y)
· exact card_accepted_responses_le_two slots H other σ a ha X
· intro h y hgood hresponse
refine ⟨h, ?_, ?_⟩
· simpa only [bucketAccepted, decide_eq_true_eq] using hgood
· exact ((bucketTable_eq_some_iff slots H other σ useful a X h y).mp hresponse).2
end
end PerfectCompleteness.CanonicalBucketCollision
end
end OAI
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