openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /PreliminaryLevelPair.lean
| import OAI.Computability.PerfectCompleteness.Foundations.DualQuotientRowsLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.HierarchicalFixedAdviceFamilyLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.PreliminaryLevelPair | |
| open PreliminarySampler PreliminaryStrategy CandidateCoupling | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped BigOperators | |
| noncomputable section | |
| variable {v m n t : Nat} {branch rows repeats : Nat → Nat} | |
| def win (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (strategy : Strategy clauses branch n t rows repeats) | |
| (level : Fin n) (e : Shared clauses branch n t rows) : Bool := | |
| observe clauses (extend clauses branch n t rows repeats strategy) | |
| e.1 e.2.1 ⟨level, e.2.2 level⟩ | |
| theorem success_eq_mean [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) | |
| (hrows : ∀ k, 0 < rows (k + 1)) | |
| (strategy : Strategy clauses branch n t rows repeats) : | |
| (game clauses branch n t rows repeats hn hbranch hrows).success strategy = | |
| 𝔼 level : Fin n, | |
| (sharedLaw clauses rows repeats hbranch hrows).probability | |
| (win clauses strategy level) := by | |
| rw [success_eq_shared_candidate_mean clauses branch n t rows repeats | |
| hn hbranch hrows strategy] | |
| simp only [candidateFraction, expectation_div, SmallBias.expectation_sum, | |
| Fintype.expect_eq_sum_div_card, Fintype.card_fin] | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro level _ | |
| exact (probability_eq_expectation _ (win clauses strategy level)).symm | |
| theorem exists_pair [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) | |
| (hrows : ∀ k, 0 < rows (k + 1)) | |
| (strategy : Strategy clauses branch n t rows repeats) | |
| {ε : ℝ} (hε : 0 < ε) (hsize : 4 ≤ (n : ℝ) * ε) | |
| (hsuccess : ε ≤ | |
| (game clauses branch n t rows repeats hn hbranch hrows).success strategy) : | |
| ∃ i j : Fin n, i < j ∧ ε ^ 2 / 2 ≤ | |
| (sharedLaw clauses rows repeats hbranch hrows).probability | |
| (fun e => win clauses strategy i e && win clauses strategy j e) := by | |
| apply LevelPairCounting.exists_level_pair | |
| (sharedLaw clauses rows repeats hbranch hrows) (win clauses strategy) ε hε | |
| · rwa [success_eq_mean clauses hn hbranch hrows strategy] at hsuccess | |
| · exact hsize | |
| end | |
| end PerfectCompleteness.PreliminaryLevelPair | |
| end | |
| end OAI | |