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5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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
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