openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /FixedStoppedContradiction.lean
| import OAI.Computability.PerfectCompleteness.Decoding.DecoderFamilyLawLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedHighBound | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedStoppedContradiction | |
| 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) | |
| theorem meeting_lt_gamma | |
| (strategy : FixedPreliminaryGame.Strategy parameters input) | |
| (unsat : ¬BinaryLanguage.language input) : | |
| (FixedStoppedDecoderLaw.law parameters i j hij input strategy).probability | |
| (FixedStoppedDecoderLaw.meeting parameters i j hij input) < | |
| FixedRankContradiction.gamma parameters (j.val + 1) := by | |
| apply lt_of_not_ge | |
| intro hupper | |
| exact FixedDecoderFamilyContradiction.impossible parameters | |
| (FixedStoppedDecoderContext.geometry parameters i j hij input) input | |
| (FixedStoppedDecoderLaw.labeling parameters input strategy) | |
| (FixedStoppedDecoderLaw.useful parameters i j hij input strategy) | |
| parameters.plan.order parameters.plan.density | |
| (FixedStoppedDecoderContext.advice parameters i j hij input) | |
| (FixedStoppedDecoderContext.direction parameters i j hij input) | |
| (FixedStoppedDecoderLaw.threshold parameters j) | |
| (FixedStoppedDecoderContext.law parameters i j hij input) | |
| unsat (le_refl _) parameters.plan.density_pos | |
| (j.val + 1) (Nat.succ_le_of_lt j.isLt) hupper | |
| (FixedStoppedHighBound.probability_square_le parameters i j hij input strategy) | |
| end | |
| end PerfectCompleteness.FixedStoppedContradiction | |
| end | |
| end OAI | |