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