openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionStoppedAcceptance.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.BudgetedCompressionTrial | |
| namespace OAI | |
| /-! # Bounded-sampler acceptance of the executed compression trial | |
| The accepted event requires a completed shared-budget execution, from | |
| array generation to the final residue. Watchdog failure returns false. | |
| -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.CompressionData | |
| variable {n : ℕ} (D : CompressionData n) | |
| def stoppedCompressionTrial (B : ℕ) (h : CompressionSetup.TrialHistory n) : Bool := | |
| match (BudgetedCompressionTrial.run D h.1 h.2 B (2 * trialWatchdog D.input)).1 with | |
| | none => false | |
| | some (v, _) => v | |
| theorem stoppedCompressionTrial_accepts (B : ℕ) (hn : 10000 ≤ n) | |
| (h : CompressionSetup.TrialHistory n) (hh : D.AcceptedWithCompactTests B h) : | |
| D.stoppedCompressionTrial B h = true := by | |
| obtain ⟨rest, he⟩ := BudgetedCompressionTrial.accepts_watchdog D h.1 h.2 B hn hh.1 hh.2 | |
| simp only [stoppedCompressionTrial, he] | |
| theorem stoppedCompressionTrial_sound (B : ℕ) (h : CompressionSetup.TrialHistory n) | |
| (hh : D.stoppedCompressionTrial B h = true) : Solvable D.input := by | |
| unfold stoppedCompressionTrial at hh | |
| cases he : (BudgetedCompressionTrial.run D h.1 h.2 B (2 * trialWatchdog D.input)).1 with | |
| | none => simp only [he, Bool.false_eq_true] at hh | |
| | some p => | |
| rcases p with ⟨v, rest⟩ | |
| simp only [he] at hh | |
| subst v | |
| exact BudgetedCompressionTrial.sound D h.1 h.2 B _ rest he | |
| theorem bounded_stoppedCompression_probability (B : ℕ) (hn : 168000000 ≤ n) | |
| (hsolve : Solvable D.input) | |
| (hmean : (153 / 20000 : ℝ) * n ≤ | |
| meanDeficiency D.weights Finset.univ (compressionHalfSize n)) | |
| (hlog : Real.logb 2 (inputScale D.input) ≤ (n : ℝ) / 1000000000) : | |
| (1 / 8 : ℝ) ≤ (D.sampler hn).boundedLaw.probability | |
| (fun o => ∃ h, o = some h ∧ D.stoppedCompressionTrial B h = true) := by | |
| apply (D.bounded_compact_acceptance_probability B hn hsolve hmean hlog).trans | |
| apply FiniteLaw.probability_mono | |
| rintro o ⟨h, ho, hh⟩ | |
| exact ⟨h, ho, D.stoppedCompressionTrial_accepts B (by omega) h hh⟩ | |
| end LowSpaceSubsetSum.CompressionData | |
| end | |
| end OAI | |