openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /FixedStoppedPair.lean
| import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedError | |
| import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryOutputLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedStoppedPair | |
| open FixedParameters FixedRows | |
| noncomputable section | |
| variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ) | |
| theorem exists_pair (input : List Bool) (strategy : FixedPreliminaryGame.Strategy p input) | |
| (hsuccess : InitialParameters.epsilon δ ≤ | |
| (FixedPreliminaryGame.game p input).success strategy) : | |
| ∃ i j : Fin p.plan.depth, i < j ∧ | |
| InitialParameters.simultaneous δ + 2 * p.accuracy ≤ | |
| (StoppedSharedSampler.stoppedLaw | |
| (PCPSource.clauseFamily (BinaryLanguage.totalRename input)) | |
| (rows p.plan) (repeats p.plan) (cut := i.val) | |
| (Nat.succ_le_of_lt i.isLt) (fun k _ => branch_pos p k) | |
| (fun k => FixedPreliminaryGame.rows_pos p (k + 1))).probability | |
| (StoppedSharedEvents.pair | |
| (PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy | |
| (Nat.succ_le_of_lt i.isLt) (fun k _ => branch_pos p k) i j) := by | |
| obtain ⟨i, j, hij, hmass⟩ := PreliminaryPairReserve.exists_pair p input strategy hsuccess | |
| have hcut : i.val + 1 ≤ p.plan.depth := Nat.succ_le_of_lt i.isLt | |
| have hi : i.val + 1 ≤ i.val + 1 := le_refl _ | |
| have hj : i.val + 1 ≤ j.val + 1 := Nat.add_le_add_right (Nat.le_of_lt hij) 1 | |
| have htransfer := StoppedSharedComparison.pair_probability_lower | |
| (PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy | |
| hcut (fun k _ => branch_pos p k) | |
| (fun k => FixedPreliminaryGame.rows_pos p (k + 1)) i j hi hj hmass | |
| have herror := FixedStoppedError.stopped_error_lt p (le_refl p.plan.depth) hcut | |
| refine ⟨i, j, hij, ?_⟩ | |
| linarith | |
| end | |
| end PerfectCompleteness.FixedStoppedPair | |
| end | |
| end OAI | |