File size: 2,452 Bytes
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 63 64 | 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
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