openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /CanonicalBucketCollision.lean
| 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 | |