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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 | import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedActualBounds
import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedMeetingEquality
import OAI.Computability.PerfectCompleteness.Sampling.FixedSourcePairVariation
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedUpperProbability
noncomputable section
attribute [local instance] FixedStoppedDecoderLaw.sampleFintype
UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype
variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ)
(i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool)
(strategy : FixedPreliminaryGame.Strategy parameters input)
theorem pairedVariation_le :
FixedStoppedInverseBound.pairedVariation parameters i j hij input strategy ≤
10 * parameters.accuracy :=
FixedSourcePairVariation.totalVariation_le_ten parameters
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(Nat.succ_le_of_lt j.isLt) hij (FixedStoppedPhysicalLaw.designated parameters i)
(FixedStoppedDecoderLaw.labeling parameters input strategy) (le_refl _)
(FixedStoppedInverseBound.outerLaw parameters j input)
theorem meeting_ge_gamma
(hreserve : InitialParameters.simultaneous δ + 2 * parameters.accuracy ≤
(StoppedSharedSampler.stoppedLaw
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability
(StoppedSharedEvents.pair (PCPSource.clauseFamily (BinaryLanguage.totalRename input))
strategy (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) i j)) :
FixedRankContradiction.gamma parameters (j.val + 1) ≤
(FixedStoppedDecoderLaw.law parameters i j hij input strategy).probability
(FixedStoppedDecoderLaw.meeting parameters i j hij input) := by
have hpaired := pairedVariation_le parameters i j hij input strategy
have hmass := FixedStoppedActualBounds.usefulMass_ge parameters i j hij input strategy hreserve
have hcoarse :=
(FixedStoppedActualBounds.coarseVariation_le_pairedVariation parameters i j hij input strategy).trans
hpaired
have hupper := FixedStoppedInverseBound.expectedMeeting_ge_gamma
parameters i j hij input strategy hmass hpaired hcoarse
exact hupper.trans_eq
(FixedStoppedMeetingEquality.meeting_probability_eq_expectedMeeting
parameters i j hij input strategy).symm
end
end PerfectCompleteness.FixedStoppedUpperProbability
end
end OAI
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