openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /FixedPreliminarySoundness.lean
| import OAI.Computability.PerfectCompleteness.Decoding.FixedStoppedUpperProbability | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedContradiction | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedPair | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedPreliminarySoundness | |
| noncomputable section | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ) | |
| theorem no_large_strategy (input : List Bool) | |
| (unsat : ¬ BinaryLanguage.language input) | |
| (strategy : FixedPreliminaryGame.Strategy parameters input) | |
| (hsuccess : InitialParameters.epsilon δ ≤ | |
| (FixedPreliminaryGame.game parameters input).success strategy) : False := by | |
| obtain ⟨i, j, hij, hreserve⟩ := FixedStoppedPair.exists_pair parameters input strategy hsuccess | |
| have hupper := FixedStoppedUpperProbability.meeting_ge_gamma | |
| parameters i j hij input strategy hreserve | |
| have hsmall := FixedStoppedContradiction.meeting_lt_gamma | |
| parameters i j hij input strategy unsat | |
| exact (not_lt_of_ge hupper) hsmall | |
| end | |
| end PerfectCompleteness.FixedPreliminarySoundness | |
| end | |
| end OAI | |